Tetrahedra and Their Nets

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TETRAHEDRA AND THEIR NETS

MATHEMATICAL AND PEDAGOGICAL IMPLICATIONS

Derege Haileselassie Mussa

Submitted in partial fulfillment of the

Requirements for the degree of

Doctor of Philosophy

Under the Executive Committee

Of the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2013

Derege Haileselassie Mussa

TETRAHEDRA AND THEIR NETS

MATHEMATICAL AND PEDAGOGICAL IMPLICATIONS

Derege Haileselassie Mussa

If one has three sticks (lengths), when can you make a triangle with the sticks? As long as any two of the lengths sum to a value strictly larger than the third length one can make a triangle.

Perhaps surprisingly, if one is given 6 sticks (lengths) there is no simple way of telling if one can

build a tetrahedron with the sticks. In fact, even though one can make a triangle with any triple of

three lengths selected from the six, one still may not be able to build a tetrahedron. At the other extreme, if one can make a tetrahedron with the six lengths, there may be as many 30 different

(incongruent) tetrahedra with the six lengths.

Although tetrahedra have been studied in many cultures (Greece, India, China, etc.) Over

thousands of years, there are surprisingly many simple questions about them that still have not

been answered. This thesis answers some new questions about tetrahedra, as well raising many more new questions for researchers, teachers, and students. It also shows in an appendix how tetrahedra can be used to illustrate ideas about arithmetic, algebra, number theory, geometry, and

combinatorics that appear in the Common Cores State Standards for Mathematics (CCSS -M).

In particular it addresses representing three-dimensional polyhedra in the plane. Specific topics addressed are a new classification system for tetrahedra based on partitions of an integer n, existence of tetrahedra with different edge lengths, unfolding tetrahedra by cutting edges of tetrahedra, and other combinatorial aspects of tetrahedra. TABLE OF CONTENTS

List of Figures…………………………………………………………………………...……….iv

List of Tables ………………………………………………………………………………….....xi

Acknowledgements…………………………………………………………………………...…xii

Dedication……………………………………………………………………………………….xiv

CHAPTER I …………………………………………………………………………..…………1

Introduction………………………………………………………………………………………..1

Purpose of the study………………………………………………………………………………2

Procedure of the study…………………………………………………………………………….3

CHAPTER II ……………………………………………………………………………..……...5

New mathematical insights into tetrahedra…………………….………………………………….5

Polygons …………………………………………………………………………………………..5

Polyhedra………………………………………………………………………………………….7

Unfoldings of polyhedra ……………………………………………………………………11

Tetrahedron .….…………………………………………………………………………………..13

Nets of tetrahedron ……………………………………………………..………………..……...18

i CHAPTER III …………………………………………………………………………………..32

Tetrahedron………………………………………………………………………………………32

Standard form of labeling a tetrahedron…………………………………………………………39

Partition type of tetrahedron……………………………………………………………………..40

Partition of Tetrahedron……………………………………………………………………….…62

Degenerate tetrahedron………………………………………………………………….………126

CHAPTER IV…………………………………………………………….. ………… …….....139

Nets of tetrahedra……………………………………………………………………….…….…139

CHAPTER V ……………………………………………………………………………….….164

Summary, conclusion and Recommendation………………………………………………...….164

REFERENCES……………………………………………………...... 169

APPENDICES ………………………..……………………………………………………...... 172

New ideas to augment traditional topics and pedagogy

Appendix A ………………………………………………………………...……………..…….172

Geometry course

Appendix B……………………………………………………………..…………………..…..176

Unit I: Back ground information

Points, Lines, and Polygons

Appendix C……………………………………………………………..…………………..…..212

Unit II: Classification of degenerate tetrahedra in the Euclidean plane

ii Appendix D ……………………………………………………………………………….…….217

Unit III: Nets of tetrahedron

Appendix E ……………………………………………………………………………….…….218

Unit IV: Classification of tetrahedron in 3D

iii LIST OF FIGURES

Figure 2.1: Different Polygons…………………………………….…………………………….6

Figure 2.2: Convex 4-gon………………………………………………………………………..8

Figure 2.3: Non convex 7-gon…………………………………….. …………………………....9

Figure 2.4a: spanning cut tree and their nets ……………………………………………………12

Figure 2.4b: spanning cut tree and their nets……………………………………………………13

Figure 2.5a: Labeling of Tetrahedron …………………………………………………………..14

Figure 2.5b: Labeling of Tetrahedron …………………………………………………………..15

Figure 2.6: Star net…………………………………………………….. ……. . ……………. …17

Figure 2.7: Edges of a spanning tree……………………………………………………………..19

Figure 2.8: Star net and Path net ………………………………………………………………...20

Figure 2.9: Overlapping tetrahedron…………………………………………………………….21

Figure 2.10: Grünbaum overlapping tetrahedron………………………………………………. 22

Figure 2.11 :{ 2, 1, 1, 1, 1} Overlapping tetrahedron……………………………………………23

Figure 2.12: Incongruent tetrahedra ……………………………………. ………………………24

Figure 2.13: {3, 3}Tetrahedra ………………….……………………………………………...... 25

Figure 2.14 :{2, 2, 1, 1}Tetrahedra ……………………………………………………………..26

iv Figure 2.15: Dual tetrahedra ……………………………………………………………………29

Figure 2.16: Degenerate tetrahedra …………………………………………………………….30

Figure 2.17: Degenerate tetrahedron ……………………………………………………………31

Figure 3.1: Potential tetrahedron………………………………………………………………..33

Figure 3.2: Tetrahedron …………………………………………………………………………34

Figure 3.3: Tetrahedron in terms of faces ………………………………………………….…..35

Figure 3.4: Tetrahedron in terms of vertices ……………………………………………………36

Figure 3.5: Tetrahedron with negative determinant …………………………………………….37

Figure 3.6: Tetrahedron with non facial ………………………………………………………...37

Figure 3.7a: Dual tetrahedron…………………………………………………………..……….38

Figure 3.7b: Dual tetrhedron……………………………………………………………………..39

Figure 3.8: labeling of tetrahedra………………………………………………………………..40

Figure 3.9: Construction of congruent tetrahedra …………………………………………...... 50

Figure 3.10: Congruent tetrahedra ……………………………………………………………....54

Figure 3.11: Dual tetrhedra ……………………………………………………………………...60

Figure 3.12: {2, 1, 1, 1, 1} tetrhedron with equal product matching edge………………………61

v Figure 3.13a :{ 1, 1, 1, 1, 1, 1} Tetrahedron with equal product matching edge……………...... 61

Figure 3.13b :{ 1, 1, 1, 1, 1, 1} Tetrahedron with equal product matching edge… ……………62

Figure 3.14: 11 Partition classes of tetrhedra ………………………….. ….…………………..68

Figure 3.15: 25 Types of tetrahedra by partition type …………………………………………….82

Figure 3.16a: {3, 2, 1} Tetrahedron regard to the numbers of congruent triangles……………..83

Figure 3.16b: {3, 2, 1} Tetrahedron regard to the numbers of congruent triangles……….. …..84

Figure 3.17: Congruent face partition with edge partition behavior……………………………88

Figure 3.18a: Congruent face partition with edge partition behavior…………………………..90

Figure 3.18b: Congruent face partition with edge partition behavior………. ………………....91

Figure 3.18c: Congruent face partition with edge partition behavior……….. ………………...92

Figure 3.18d: Congruent face partition with edge partition behavior………. ………………....93

Figure 3.19: Partitions of a tetrahedron with integer lengths …………………………………106

Figure 3.20: Incongruent Scalene tetrahedron with integer length ……………………………114

Figure 3.21: {2, 1, 1, 1, 1} Incongruent tetrahedra with integer length………………………..118

Figure 3.22 a : monotone up and monotone down series………………………………………122

Figure 3.22 b: Monotone up and Monotone down series……………….. …………………….125

Figure 3.23a: Degenerate tetrahedra…………………………………………………………....136

Figure 3.23b: Degenerate tetrahedra……………………………………………………...... 137

Figure 3.23c: Degenerate tetrahedra …………………………………………………………...137

Figure 3.24: labeled tetrahedron………………………………………………………………..138

Figure 4.1: Spanning cut tree and their nets………………………………………………...... 139

Figure 4.2: Spanning cut tree and their nets………………………………………...………….140

Figure 4.3: Spanning cut tree ……………………………………………………...……...... 140

Figure 4.4: {5, 1} Tetrahedron ……………………………………………………………. …..141

Figure 4.5a: Star net for {5, 1}………………………………………………………………....142

Figure 4.5b: Path net for {5, 1}…………………………………………………………..…….143

Figure 4.6a :{2, 1, 1, 1, 1} Type1 tetrahedron………………………….………………...... 144

Figure 4.6b :{2, 1, 1, 1, 1} Type 2 tetrahedron…………………………………………….…144

Figure 4.6a :{ 2, 1, 1, 1, 1} Type 1 tetrahedron……………………………………………....145

Figure 4.7a: Star net for {2, 1, 1, 1, 1} Type 1 tetrahedron….……………………………..…146

Figure 4.7b: Path net for {2, 1, 1, 1, 1} Type 1 tetrahedron……………………………….....150

Figure 4.6b :{2, 1, 1, 1, 1} Type 2 tetrahedron……………………………………………….151

Figure 4.8a: Star net for {2, 1, 1, 1, 1} Type 2 tetrahedron…………………………………..153

Figure 4.8b: Path net for {2, 1, 1, 1, 1} Type 2 tetrahedron………………………………….157

Figure 4.9a: Path net for {2, 1, 1, 1, 1} Type 2 tetrahedron………………………………….158

Figure 4.9b: Path net for {2, 1, 1, 1, 1} Type 2 tetrahedron………………………………….158

Figure 4.10a: Path net for {2, 1, 1, 1, 1} Type 2 tetrahedron…………………………………159

Figure 4.10b: Path net for {2, 1, 1, 1, 1} Type 1 tetrahedron………………………………...159

vii

Figure 4.11: Dual tetrahedron with the same path for {2, 1, 1, 1, 1} type……… …………….160

Figure 4.12: Dual tetrahedra with a matching edge for {2, 1, 1, 1, 1} and their nets…………..161

Figure 4.13a: {2, 1, 1, 1, 1} Tetrahedron……….………………………….…………………….162

Figure 4.13b :{ 2, 1, 1, 1, 1} Overlapping tetrahedron ….….…………………………………163

Figure 1.1: A point...... ………………………………………………………………………....176

Figure1. 2: A line..……………………………………………………………………………...176

Figure 1.3: Line segments …………………………………………….………………………..177

Figure 1.4: A plane …………………………………………………….………….……………177

Figure 1.5: Three collinear points ………………………………….…………………………..177

Figure1. 6: Coplanar points..…………………………………………………………………...178

Figure 1.7: Rays ………………………………………………………………………………..178

Figure 1.8: Two congruent segments ……………………………………………………….....179

Figure 1.9: Midpoint of line segment………………………………………………………….179

Figure 1.10: Angle …………………………………………………………………………….180

Figure1.11: Right angle………………………………………………………………………..180

Figure 1.12: Acute angle………………………………………………………………………181

Figure 1.13: obtuse angle……………………………………………………………………...181

Figure 1.14: Reflex angle……………………………………………………………………...182

Figure 1.15: Adjacent…………………………………………………………………………..182

viii

Figure 1.16: Vertical angles…………………………………………………………………..183

Figure 1.17: Complementary angles………………………………………………………….183

Figure 1.18: Supplementary angles…………………………………………………………184

Figure 1.19: Point, line, and Plane……………………………………………………………184

Figure 1.20: Line segment……………………………………………………………………185

Figure 1.21a: Polygons…………………………………………………………………….….186

Figure 1.21b: Polygons………………………………………………………………………..187

Figure 1.22: Diagonal of Polygons…………………………………………………………....187

Figure 1.23: Non Convex polygon……………………………………………………………188

Figure 1.24: Equilateral Polygon…………………………………………………………….188

Figure 1.25: Equiangular polygon…………………………………………………………….189

Figure 1.26: Regular polygon…………………………………………………………………189

Figure 1.27: Triangle………………………………………………………………………….190

Figure 1.28: Kite……………………………………………………………………………….191

Figure 1.29: Isosceles trapezoid……………………………………………………………….192

Figure 1.30: Rhombus…………………………………………………………………………192

Figure 1.31: Parallelogram…………………………………………………………………… 193

Figure 1.32: Rectangle…………………………………………………………………………194

Figure 1.33: Square……………………………………………………………………………194

Figure 1.34: Triangle ………………………………………………………………………….195

Figure 1.35: Equilateral, Isosceles, and Scalene triangles……………………………………..196

Figure 1.36: Triangle…………………………………………………………………………..196 ix

Figure 1.37a: Congruent triangles………………………………………………………………197

Figure 1.37b: Congruent triangles……………………………………………………………...198

Figure 1.38a: Congruent triangles………………………………………………………………198

Figure 1.38b: Congruent triangles……………………………………………………………...199

Figure 1.39: Right angle triangle ………………………………………………………………199

Figure 1.40: Heronian triangles…………………………………………………… …………..200

Figure 1.41a: Quadrilaterals……………………………………………………………………202

Figure 1.41b: Quadrilaterals………………………………………………….………………..204

Figure 1.41c: Quadrilaterals…………………………………………………..………………...207

Figure 1.42: Hierarchies of quadrilaterals……………………………………………………...207

Figure 1.43a: Cyclic quadrilaterals……………………………………………………………. 209

Figure 1.43b: Cyclic quadrilaterals…………………………………………………………….210

Figure 1.44: Traiangle…………………………………………………………………………..212

Figure 1.45: Degenerate tetrahedra…………………………………………………………….215

LIST OF TABLES

Table 3.1: Congruence Property by face and vertex…. . ..……………………………..…….85

Table 3.2: Congruent face partition with edge partition behavior…………………………...87

Table 3.3: Congruent vertex partition with edge partition behavior…………………………89

Table 5.1: Congruence Property by face and vertex.. . .………………………………………180

Table 5.2: Congruent vertex partition with edge partition behavior ………………………….181

Table 5.3: Congruent face partition with edge partition behavior……………………………..182

ACKNOWLEDGEMENTS

It is a pleasure to thank the many people who made this thesis possible.

My time, three years and seven months, at Teachers college, Columbia University very enjoyable

and my work there became a part of my life. I am indebted to many Students and colleagues for

providing a stimulating and exciting environment in which to learn and grow.

At Teachers College I had the opportunity to study under and work with prestigious and highly qualified mathematicians, mathematics educators and leaders in other educational fields.

They have made a great impact in my achievements and consolidation of career goals and dreams. I thank them all.

I am particularly honored to have studied with Prof. Joseph Malkevitch who has facilitated and

inspired me to deepen my knowledge in the branches of geometry. I have been amazingly

fortunate to have him as my advisor. I want to extend my thanks to Prof. Bruce Vogeli for

helping me during the writing of the dissertation proposal process, for the valuable advice, suggestions, and comments he made on my work.

I feel especially very blessed and profoundly happy with all the advice and the tremendous work

Prof. Malkevitch has dedicated to my research work as my dissertation Sponsor. I want to thank

him and all the members of the Dissertation Committee: Prof. Vogeli, Prof. J. Smith,

Prof. O. Roger Anderson and Prof Patrick Gallagher for their encouragement, insightful comments and guidance which were helpful in refining the draft version of my dissertation into its final form. I am deeply indebted to my friends whose encouragement made this work possible towards its final stages. They have consistently helped me keep perspective on what is important in life. They always gave me support throughout my studies.

Most importantly none of this would have been possible without the support of my parents.

xii

I would like to thank my parents for all their love. They guided me in all my pursuits.

They raised me, supported me, taught me, and loved me. I wish my father could have lived to see me finish this work .I know it would have meant so much to him. This thesis is dedicated to him.

X111

DEDICATION

This dissertation is dedicated to my father Ato Haileselassie Mussa

XlV 1

CHAPTER I

INTRODUCTION

This thesis investigates new mathematical properties of tetrahedra motivated by showing novel

approaches to traditional questions addressed by mathematics educators interested in geometry and how geometry is connected to other parts of Mathematics.

Definitions of geometric objects such as polygon and polyhedron have evolved since antiquity in

response to new mathematical problems. Ordinary usage of such terms contrasts with usage in

the mathematics research literature and the definitions found in texts at different levels of school

mathematics. This may cause confusion for students. Variations in definitions from one school

level to another are especially troublesome.

This study investigates these issues in the context of the transition between 2-dimensional

geometrical objects and 3-dimensional objects. In particular it addresses representing three-

dimensional polyhedra in the plane. Historically, such representations have included isometric

drawings, projective drawings (period of the Renaissance in the work of artists such as da Vinci),

nets (Durer), and the use of graph theory ideas. One focus of this study builds on the idea of

using partitions of an integer to provide organized thinking about geometrical objects such as

quadrilaterals and tetrahedra. Specifically studying the behavior of the existence of tetrahedra in

a partition environment is an excellent framework for implementation of small cooperative

learning projects for the goal of improving student skills with visualization and the role of

definitions. 2

A polyhedron is a 3-dimensional object that contains flat faces and straight edges. Examples

include the Platonic solids and tetrahedra in general, combinatorial cubes, and other convex polyhedra. The tetrahedron is the simplest three dimensional solid, having four vertices, four faces (all triangles) and six edges. Partitions of 6, the number of edges of a tetrahedron offers a way to classify tetrahedra, and to address questions of research interest to mathematicians while

offering K-12 students a “laboratory” to make models, and get insight into representation of geometrical objects, visualizing these objects, and telling when the objects are the same or different. New theorems about tetrahedra are developed with these educational goals as motivation.

Purpose of the study

The purpose of this study is to give a new mathematical approach to the study of tetrahedra and

their nets for use by students, teachers and researchers.

Goals include:

* showing ways to transition from geometrical problems arising in 2-dimensions (2D) to ones in

3-dimensions (3D)

* showing connections between different mathematics topics (algebra, geometry, combinatorics,

etc.) that should be treated in a more integrated way

* enhancing sense making for the role of definitions in geometry and to design activities containing exploration problems and classroom activities that enable students to study geometry 3

by investigating, exploring, and establishing geometric conjectures, which eventually will

develop into proofs.

* showing that there are new mathematical questions about polygons and tetrahedra accessible at the college and pre-college level.

The following research issues will be considered in the study:

1. How to classify tetrahedron into different types based on the notion of a partition.

2. Existence questions for tetrahedron with integer lengths.

3. Relating the existence of nets of tetrahedra to the type of spanning tree that is used to generate the net, and under what circumstances attempts to generate a net for a tetrahedra fail due to overlap of the faces when the tetrahedron is flattened into the plane. (Some tetrahedra have an edge unfolding which overlaps, a rather unexpected recent discovery in elementary geometry.)

4. Which partition types of tetrahedra admit spanning tree with cuts that lead to overlaps?

5. What are current views about pedagogy involving three-dimensional polyhedra, in particular

tetrahedra?

Procedure of the study

In order to address these research questions the following procedure was used

1. Develop new results, examples, and theorems related to triangles, quadrilaterals and tetrahedra which can provide to mathematicians, mathematics educators and students a model for seeing how mathematics grows and develops. 4

2. Generate research questions for high school students and mathematicians.

Different tasks will be considered and a set of problems and motivational activities will be

designed to enhance student understanding of the geometry they learn. Activities for exploration will be accompanied by comments intended to help teachers use this approach. This material

appears in the appendices.

3. Exploration of the mathematics of polygons and polyhedra.

Polyhedra and drawings of them in the plane were studied. The use of partitions enables

classification of tetrahedra and addresses mathematical research questions that offer ways to

obtain new insights about geometrical objects.

4. Develop student activities involving triangles, quadrilaterals, and tetrahedra which grow out of the new mathematical work. 5

CHAPTER II

NEW MATHEMATICAL INSIGHTS INTO TETRAHEDRON

Some Definitions

Some basic definitions and concepts needed in what follows in more detail are introduced rather

than be extremely formal these terms are introduced in an intuitive way that would benefit their

use in school.

Polygons

A polygon is a collection of points A1 , A2 , A3 , …., An called vertices joined by line segments

A1 A2 , A2 A3 , …., An A1 joining the points as in Figure 1 where for convenience a notation avoiding subscripts is used. The points are known as the vertices of the polygon and the segments are known as the edges of the polygon. The segments are said to make up the boundary of the polygon. Thus, when the polygon has 3 points it is called a triangle, when it has 4 points it

is called a quadrilateral, and when it has n points it is called an n-gon. Usually, the points are

such that no three of them are allowed to be on a line (no 3 collinear). So a triangle with an extra

vertex on one of its sides will not be thought of as a triangle but as a 4-gon, though in some

situations it can be considered as a "degenerate triangle." Usually, when talks about a polygon P

means only the "rods" that make up the boundary of the polygon. If wants to refer to the interior

region that the rods bound, strictly speaking should talk about the interior of the polygon, but

will be more informal here and what is meant should be clear from the context. A polygon whose

vertices lie in a single plane is called planar. Note, that the approach given restricts our attention 6

to planar polygons though more general polygons are of great interest to researchers in geometry.

(a) (b) ( c)

Figure 2.1: Different Polygons

A simple polygon P is one whose edges only meet at vertices. A simple polygon P divides the

plane into two regions called the interior and exterior of P. In Figure 2.1a and 2.1b shows simple polygons but Figure 2.1b is non-simple, it intersects itself. Sometimes is useful to view polygons from a graph theory point of view. From this point of view an n-gon is a cycle with n vertices.

The interior of the polygon is "bounded" since it can be completely enclosed in a suitably large

circle and the exterior region is "unbounded." The exterior region cannot be enclosed in a circle

with a finite radius. Note that for a simple polygon the points at which two edges (line segments) meet are vertices of the polygon. Two edges that meet at a vertex are known as adjacent edges.

Two of the edges in Figure 2.1b meet but this point is not a vertex of the polygon, which is not simple. Polygons which are not simple cannot be assigned an area in a natural way.

A polygon that contains a pair of adjacent edges or, for that matter non-adjacent edge along a straight line is called a degenerate polygon. Typically this is ruled out by the assumption that no

three points are collinear; however, for many interesting questions about polygons allowing 7

adjacent or non-adjacent sides of the polygon to lie along the same line is of interest - for

example, in discussions of "art gallery theorems."

Polyhedra

A polyhedron (plural, polyhedra) is the three-dimensional version of polygon and of the more general objects that can be defined in arbitrary dimensions. Polyhedra appear early in the history of geometry, including in Euclid’s Elements. The word “polyhedron” (in geometry) derives from the Greek poly (many) plus the Indo-European hedron (seat).

The regular convex polyhedra or Platonic solids (polyhedra whose faces are regular convex

polygons and which have the same number of edges at each vertex) have been known for at least

2500 years, and were studied in detail in Euclid's Elements.

A set X is convex if when M and N are points of X then the whole line segment joining M and N

is completely contained in X. From this it follows that in a convex polygon each of its interior

angles has measure at most 180 degrees. (When the polygon is “degenerate “there can be straight

angles in the boundary of the polygon.) Note that when thinks of a polygon only as the rods that

makes it up, then polygons strictly speaking cannot be convex. Strictly speaking one has to talk

about a polygon and its interior in asking if convexity holds. Figure 2.1c shows a convex

polygon and Figure 2.1a is non-convex. Figure 2.1 b is not simple and, thus, of necessity is non-

convex. 8

Convex 4-gon convex 4-gon and its diagonal

(a) (b)

Figure 2.2: Convex 4-gon

I will be discussing non-convex simple polygons in addition to convex polygons. Figure 2.2a shows a convex polygon (4-gon) ABCD and Figure 2.2b the same polygon with its two diagonals. AC and BD are the diagonals. A 4-gon, together with its two diagonals is known as a

complete quadrilateral.

A non-convex polygon is a polygon that has at least one interior angle that measures more than

180 degrees (but less than 360 degrees). These angles are known as reflex angles. 9

Figure 2.3: Non convex 7-gon

Figure 2.3 shows a non-convex 7-gon with three reflex angles.

The convex regular solids are the tetrahedron (4 faces which are equilateral triangles), cube (6 faces which are squares), octahedron (8 faces which are equilateral triangles), icosahedron (20 faces which are equilateral triangles), and the dodecahedron (12 faces which are regular pentagons).The faces of each of these solids are congruent to each other, and they have the same number of faces meeting at every vertex which means that, considered as graphs, they have regular graphs. No general definition of polyhedron appeared until much later than Euclid.

Euclid proves theorems without telling the reader that he is dealing only with convex polyhedra.

Even Euler discussed his famous formula (Euler's polyhedral formula) V – E+ F = 2

(V = vertices, E = edges, F = faces) without specifying the type of polyhedra he had in mind.

During the 19th century, the center of attention concerning polyhedra expanded beyond the convex ones. Ernst Steinitz formulated the criteria that are necessary and sufficient for the 10

existence of a convex 3-dimensional polyhedron in the period from 1910 -1920. His famous result, Steinitz's Theorem, is that a (vertex-edge) graph is the graph of some convex

3-dimensional polyhedron (also known as a 3-polytope) if and only if the graph is planar (can be drawn in the plane with edges meeting at vertices) and 3-connected (for every pair of vertices u and v in the graph there are at least three paths between the vertices u and v which have only u and v in common). In the 19th century, certain non-convex polyhedra including various ones that intersected themselves were discussed and described by different authors; however no one gave a meaningful definition of non-convex polyhedron. Even in the discussion of non-convex regular polyhedra, 19th century work does not meet modern standards of rigor.

Here we will adopt a version of an informal definition due to Joseph O’Rourke (Smith College)

who has written about polyhedra for audiences at many levels. A polyhedron is the surface of a

3-dimensional object composed of flat, convex polygon faces such that:

Every side of a polygon belongs to just one other polygon.

The faces that share a vertex form a "chain" of polygons in which every pair of consecutive polygons share a side.

Summarizing the discussion above in slightly different terms, an object (polygon or polyhedron)

is convex if for every pair of points that belongs to the shape (its boundary or its interior), the object contains the whole straight line segment connecting the two points. Convex polygons have

no reflex vertices. A reflex angle is an angle whose measure is greater than 180 and less than 360

(Figure 2.3). Every face of a convex polyhedron is simply connected (no holes). The tetrahedron

has 4 vertices, 4 faces and 6 edges. The interior angle at vertex v in a face incident to v is called

the face angle. The “curvature” at the vertex v is 2π - (the sum of the face angles at v). 11

Descartes’ Theorem: the sum of the curvature for a polyhedron of genus zero (topologically like

a sphere) is 4π.

Sometimes the curvature at a vertex is called the angle defect at the vertex. The genus of a graph

is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with

n handles (an oriented surface of genus n).A planar graph has genus zero because it can be drawn

on a sphere without self-crossings. Intuitively, the genus of a surface counts the number of holes

or handles it has. Thus, a sphere and the plane have genus zero since they have no holes or handles. A donut, or torus, has genus 1 because it can be thought of as having one hole, or being a sphere with one handle.

One of the most important theorems for polyhedra is the Euler formula relating the number of vertices, edges and faces of a convex polyhedron.

For any convex polyhedron P with no holes, |V (P) - |E (P)| +|F (P)| = 2 where V (P), E (P), and F (P) represent the sets of vertices, edges and faces of the polyhedron. Here |X| denotes the number of elements in set X.

Unfolding of a Polyhedron

One unfolds a polyhedron P by cutting along some of its edges and flattening out its faces into the plane without overlap. The result is called a simple unfolding or a net and it is created by cutting edges of the polyhedron P that form a spanning tree of the vertex-edge graph of P. Note that nets are treated as metrical objects when drawn in the plane, and that nets include fold lines that show up as diagonals of the polygon that makes up the net. 12

A spanning tree is a sub graph of a graph G that is a tree (a connected graph which has no

circuits) and includes all the vertices of G. The set of cut edges must be connected because the cut edges unfold to the boundary of a polygon and the boundary is connected. Thus, the cut edges form a spanning tree. If one cuts edges containing a circuit the result would be to separate what is obtained into two pieces. The boundary of a polygon of the unfolding (Figure 4) consists

of twice the number of cut edges. Each cut edge appears exactly twice on the boundary.

. Figure 2.4a: spanning cut tree and their nets

For the cut edges of CABD and BCAD (in Figure 2.4a and 2.4b) the boundary of a polygon of the unfolding consists of twice the number of cut edges, and is hexagon, which need not be

convex. 13

Figure 2.4b: spanning cut tree and their nets

Conjecture (Geoffrey Shephard’s Conjecture): Every convex polyhedron can be cut along some of its edges (a spanning tree) and unfolded into the plane without overlap.

Many mathematicians have attempted to prove the conjecture. However, a counterexample has

been found for every algorithm proposed that some special type of spanning tree leads to a simple unfolding for any convex polyhedron. On the other hand experimental results suggest that

a random spanning tree of a random polytope causes overlap with probability approaching one as

the number of vertices approaches infinity (C. Schevon).

Tetrahedra

A tetrahedron is a three-dimensional solid having four vertices, four triangular faces and six

edges that don’t lie in a single plane. Tetrahedron is of Greek origin (tetra- four and hedra -seat) and refers to its four plane faces.

The tetrahedron is the only convex polyhedron that has four faces. It is one kind of pyramid,

which is a polyhedron with a flat polygons base (n- gon) and triangular faces connecting the base 14

to a common point so a tetrahedron is a triangular pyramid. There are two types of angles connected with a tetrahedron: six dihedral angles formed by all pairs of faces and four trihedra angles formed by all possible triples of faces. The sum of the dihedral angles varies between

2πand 3π while the sum of the trihedral angles varies between 0 and 2π.

Standard Forms of Labeling a Plane Diagram of a Tetrahedron

There are two standard forms for drawing a labeled tetrahedron in the plane. Both involve

isomorphic copies of K4 (the complete graph with 4 vertices) with typically different symmetry

groups as drawings in the plane.

Figure 2.5a: Labeling of Tetrahedron

Figure 2.5a has 8 symmetries and it enables one to see the edge of the perfect matching’s of a

tetrahedron more clearly. A perfect matching of a graph with an even number of vertices is a

collection of edges that are disjoint but include all of the vertices in the graph. 15

(Note the following edges are equal: b=c=d and a=e=f.)

Figure 2.5b: Labeling of Tetrahedron

The other drawing (Figure 2.5b) can have as many as 6 symmetries but in Figure 2.5b as shown there are two symmetries.

According to the labeling of vertices and edges shown in the Figures 2.5a and Figure 2.5b can

determine the existence of a tetrahedron with the 6 labels representing values for the edge

lengths.

Arthur Cayley and later Karl Menger showed how could prove the existence of a tetrahedron T

   with a specific collection of edge lengths (x, y, z, X , Y , Z ) or using vertex coordinates by

computing a 5 x 5 determinant to find the “volume “ of T. 16

Later William H. McCrea showed how instead a 3 x 3 determinant could be evaluated to achieve

the same goal. Details can be found in Wirth and Drieiding , 2009. The form of the McCrea determinant is shown below with the letters representing lengths as in Figure 2.5a and Figure

2.5b.

2 2 2 2 2 2 2  2d d + e  c f + d  b 

 2 2 2 2 2 2 2  D(S ) =  d + e  c 2e e + f  a  2 2 2 2 2 2 2  f + d  b e + f  a 2 f  

By using the McCrea determinant, a tetrahedron representing a facial (faces obey the strict

triangle inequality) sextuple S = (a, b, c, d, e, f) exists if and only if D(S), the determinant of the

McCrea matrix, is positive. 17

Thus, treating the McCrea determinant as a "black box" it is easy to tell when a tetrahedron exists with 6 particular length sticks.

Wirth and Drieiding, 2009 show that a tetrahedron corresponding to a facial sextuple

S = (a, b, c, d, e, f) exists if and only if the star net (Figure 2.6 ) has an acute vertex triple obeying the A-angle inequality (it has at least one acute vertex triple). An acute vertex triple is an

angle whose triple of angles at a vertex each has measure greater than zero and less than 90. The

A-angle inequality refers to requiring that the three angles A, B, C in a triple (A, B, C) obey the

inequalities: A + B < C, B + C < A, A + C < B. Facial means that the lengths corresponding to

the four faces of the proposed tetrahedron obey the strict triangle inequality.

Figure 2.6: Star net

If the McCrea determinant is zero then there is a degenerate tetrahedron whose 4 vertices lie in a

plane. If the McCrea determinant is negative there is no tetrahedron geometrically even when the

facial condition holds. The significance of a negative value of the McCrea determinant of

a tetrahedron with a given set of edge lengths is that one can be certain that a tetrahedron cannot 18

exist with the given edge lengths. Having a positive McCrea determinant for 6 stick lengths is a

necessary but not sufficient condition for the existence of a tetrahedron with these 6 stick lengths

as edge lengths.

The second form of labeling a tetrahedron (Figure 2.5b) enables us, using the McCrea determinant as a "black box" to check relatively easily if a given set of six edge lengths can be used for “making “ a tetrahedron.

Nets of Tetrahedra

Nets for a tetrahedron are obtained by cutting three edges at a vertex of the tetrahedron or along a

sequence (path) of three edges that visit each vertex exactly once. Thus, there are two types of nets: star nets and path nets. Star nets are obtained by cutting three edges of tetrahedron at one vertex of the tetrahedron. Path nets are obtained by cutting three edges of a tetrahedron along a

sequence of three edges of a tetrahedron - in graph theory terms the edges form a hamiltonian path. More precisely, define an edge unfolding as a "development" of the surface of a polyhedron to a plane such that the surface becomes a flat polygon bounded by segments that derive from edges of the polyhedron (Demaine and O’Rourke, 2007).

Figure 2.7 shows examples of the two ways that one can cut edges of the same (physical) tetrahedron to obtain a net. Figure 2.8a shows a star net at vertex A Figure 2.7b and Figure 2.8b shows the path net that arises from cutting the edges in the path CABD (or DBAC). 19

(a) (b)

Figure 2.7: Edges of a spanning tree

The three edges of a spanning tree which are to be cut to obtain a net are shown as bold lines.

The nets which arise can be either a convex or non-convex polygons.

(a) 20

(b)

Figure 2.8: Star net and Path net

Note that in a star net one of the vertices of the tetrahedron appears in the net in three positions.

As polygons the nets can be written: A ' B A ' C A ' D and A ' BDB ' AC (Figure 2.8a and Figure

2.8b). The net also includes (metrically) the three diagonals of the polygonal 6-gon that bounds

the net as shown in Figure 2.8. Does every spanning tree of a tetrahedron lead to a net?

How many nets can we find for a tetrahedron?

This thesis addresses these mathematical questions. As we have seen from Figure 2.7 and Figure

2.8 there are nine edges and six vertices in a net; this is because the boundary of the unfolding consists of twice the number of cut edges. Each cut edge appears exactly twice on the boundary which has six vertices. If takes the star net which arose from a physical tetrahedron, one can

always fold it up to a tetrahedron. However, for a path net there is no guarantee that one 21

can unfold to a non-overlapping polygon. This does not contradict Shephard's Conjecture. It just means that some spanning trees may not lead to an unfolding. There can be a spanning tree (in the form of a path) for a tetrahedron which when cut will lead to a way to open up the tetrahedron which will overlap (Figure 2.9).

Figure 2.9: Overlapping tetrahedron

In response to a conjecture by K.Fukuda that overlap couldn’t occur for a tetrahedron, M.Namiki

found an example where overlap does occur. Figure 2.10 shows an explicit example of this

phenomenon that was published by Branko Grünbaum (Figure 2.10) inspired by Namiki’s

example. 22

Figure 2.10: Grünbaum overlapping tetrahedron

This thesis also found an overlapping unfolding for a tetrahedron which is not scalene (not all edge lengths different). Previously known examples all produced scalene (Figure 2.10). 23

Figure 2.11 :{ 2, 1, 1, 1, 1} overlapping tetrahedron

How can tell when two tetrahedra are “different” or not congruent?

If two figures are called congruent when there exists a one - to - one distance preserving correspondence between their point sets. Recall that two triangles are congruent if the sides of one are congruent (equal in length) to the sides of the other. This is the SSS congruence condition for triangles. SAS and AAS also suffice to show congruence where A refers to an angle and S refers to a side. 24

D(S) = 10280448 D(S) = 10870398

Figure 2.12: Incongruent tetrahedra

The faces of the two tetrahedra in Figure 2.12 are not congruent even though the six edges of T1

(Tetrahedron 1) are congruent or equal to the six edges of T2 (Tetrahedron 2). Figure 2.12a has

two isosceles and two scalene triangles and Figure 2.12b has three isosceles and one scalene

triangles. They are incongruent tetrahedra. Also, note that to check if there are actually physical tetrahedra that correspond to these two diagrams one has to check that the facial condition holds as well as check that the McCrea determinant is positive. In general when one moves the positions of the lengths in Figure 2.5b, even keeping the set of lengths the same, there is no guarantee that if a physical tetrahedron exists for the initial drawing that a physical tetrahedron will exist for the altered drawing!

One Goal of this thesis s to investigates the number of different partition types (see below) in

which tetrahedra can be classified. It is shown that for the 11 different partitions of 6 (the number

of edges of a tetrahedron) some of these partitions allow a “finer“way to tell tetrahedra apart. For 25

example given the partition {3, 3} one can tell apart tetrahedra where the two pairs of three equal

length edges form a) two paths b) three edges meeting at a vertex and three edges forming an equilateral triangle.

Figure 2.13: {3, 3}Tetrahedra

Partition Types of Tetrahedra

A partition of a positive integer n is a way of writing n as the sum of positive integers. For

example {2, 1} is a partition of 3. Note that {1, 2} represents the same partition as {2, 1}.

The partitions of 4 are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}.

The tetrahedra (6 edges) can be classified into eleven classes of partition type in terms of their

edges: {6}, {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1, 1}, {2, 2, 2}, {2, 2, 1, 1},

{2, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}. If we take {2, 2, 1, 1} one can interpret the partition as a

tetrahedron with 4 different edge lengths, two different sets of two edges having the same length

but different from each other, and different from the other two lengths of edges. Both tetrahedra 26

in Figure 2.12 represent the partition {2, 2, 1, 1}. For each partition of 6 one can find

incongruent tetrahedra corresponding to this partition. Note that one way of being certain that the two physical tetrahedra shown in Figure 2.14 are incongruent is that their McCrea determinants

(shown) are not equal. Eewven without invoking the McCrea determinant, one can see using simpler ideas that the tetrahedra are not the same.

D(S) = 10723552 D(S) = 11941600

Figure 2.14 :{ 2, 2, 1, 1}Tetrahedra

For the partition {2, 2, 1, 1} one can find two incongruent tetrahedra using the same edge

lengths. The refined approach, taking some geometric information (but not relative size of edges) into account, leads to different potential types as has been determined in this thesis. This thesis not only enumerates the refined number partition classes but also determines for each of these classes if there is an integer collection of lengths for a tetrahedron in this class. There are actually 27

several issues here which have some pedagogical implications in addition to the new

mathematical results that have been obtained.

Integer Lengths

For pedagogical purposes and making physical models of tetrahedra, working with integer lengths instead of using general real numbers is preferable. The discussion here is valid for real numbers as long as the lengths of each of the four faces of the tetrahedron obeys the triangle inequality and the McCrea determinant is positive. If the tetrahedron exists with integer lengths then one can design (for the benefit of students) a simple way of labeling a tetrahedron to determine the number of distinct ways of using those integer lengths for different partition classes of tetrahedra. With six distinct lengths, there are at most 30 incongruent tetrahedra. Since

the labels in Figure 2.5b can be assigned in 720 (6!) different ways (the first choice can be made in 6 ways, the next choice in 5 ways, etc.) this means that for each of the 30 different tetrahedra there are 24 drawings which lead to a congruent version of a given tetrahedron. Another way of thinking of this that the tetrahedron, whose graph is the complete graph on 4 vertices, as a graph, can have 24 symmetries (automorphisms).

From a given tetrahedron one can find a “dual tetrahedron” taking into a account the triples that are the edges at a vertex and the triples that form the faces of a tetrahedron. For example the tetrahedron in the Figure 17a has faces 28, 30, 45; 30, 35, 42; 28, 36, 42; and 35, 36, 42. The dual has faces with sides corresponding to the numbers (edge lengths) at the vertices in Figure

2.15b. 28

Since the tetrahedron exists, which is facial and the McCrea determinant is positive then we can try to find a dual tetrahedron and that has faces with sides that correspond to the number at the vertices of the original tetrahedron.

D(S) = 2250079808

Faces 28,30,45 ; 30,35,42 ;28,36,42 ; and 35,36,42

(a) 29

D(S) = 2341239800

Vertex 28, 30, 45; 30, 35, 42; 28, 36, 42; and 35, 36, 42

(b)

Figure 2.15: Dual Tetrahedra

Degenerate Tetrahedra

When the McCrea determinant of the tetrahedron is zero the four points of the "potential" tetrahedron will lie in a single plane or lie on a single line so the tetrahedron will become

degenerate. 30

Figure 2.16: Degenerate tetrahedra

Note that the parallelogram shown in Figure 2.16 cannot be made metrically while the trapezoid realization in the plane is metrically possible. 31

fig1

(a) (b)

Figure 2.17: Degenerate tetrahedron

Figure 2.17b shows the degenerate tetrahedron that is determined by four vertices with six edges and four faces such that three vertices are collinear in Figure 2.17a. 32

CHAPTER III

TETRAHEDRA

Tetrahedron: A tetrahedron (plural: tetrahedra) is a three-dimensional solid having four vertices,

four triangular faces and six edges which don’t lie in a single plane. Figure 1 may help with following the development adopted here. 33

Figure 3.1: Potential Tetrahedron

Figure 3.2 below is a diagram of a "potential" tetrahedron with edges of length

S = (a, b, c, d, e, f) positioned as shown. 34

Figure 3.2: Tetrahedron

Using the McCrea determinant, a sextuple S = (a, b, c, d, e, f) will allow the construction of a

tetrahedron if and only if D(S) is positive and S is facial. D(s) refers to the determinant of the

matrix shown below. If M is a matrix then |M| will denote the determinant of M.

2 2 2 2 2 2 2  2d d + e  c f + d  b 

 2 2 2 2 2 2 2  D(S ) =  d + e  c 2e e + f  a  2 2 2 2 2 2 2  f + d  b e + f  a 2 f  

If all four face triples (Figure 3.2) obey the strict triangle inequality then S is facial. In order to ensure that a sextuple S exists as a tetrahedron we must test if the tetrahedron is facial and that 35

the McCrea determinant is positive. If one takes a sextuple of a fixed tetrahedron S and one with

6 distinct edge lengths, the lengths can be placed on the edges in 720 different ways and at most

30 of these can exist as non-congruent tetrahedra. Thus, for any particular legal tetrahedron, there are in fact 24 other ways to label that the edges which gives rise to the same tetrahedron.

To study these ideas consider the sextuple S = (35, 30, 42, 28, 36, 45)

D(S) = 2250079808

Figure 3.3: Tetrhedron in terms of faces

We can check that this tetrahedron Figure 3 exists, which involves if D(S) positive and S facial.

The faces will have the side lengths: 28, 30, 45; 28, 36, 42; 30, 35, 42; and 35, 36, 45 and they

obey the strict triangle inequality.

Now, suppose we list the numbers of S in reverse order, giving the sextuple:

S = (45, 36, 28, 42, 30, 35) 36

D(S) = 2341239800

Figure 3.4: Tetrahedron in terms of vertices

In the same way using the McCrea determinant one can check also that this tetrahedron exists.

Thus D(S) is positive and S is facial. Its faces will have the edge lengths: 28, 30, 42; 28, 36, 45;

30, 35, 45; and 35, 36, 42 (see Figure 3.3 and 4.4 ). However these two tetrahedron are not congruent. This raises to the question of what is the largest number of tetrahedron that a particular collection of 6 distinct edge lengths might give. In fact, for the partition of the sextuples of S = (35, 30, 42, 28, 36, 45) one can find exactly 30 incongruent tetrahedra and they all exist.

One can generalize that for a sextuple S of a legal tetrahedron with six distinct lengths, one can find at most 30 incongruent (scalene) tetrahedra.

For the partition of the sextuple of S = (61, 80, 90, 100, 110, 120) we find 29 of the possible 30

incongruent (scalene) tetrahedra associated with these 6 length can exist. The sextuples which

does not work is S = (110, 61, 90, 120, 100, 80). 37

If one creates other sextuples S there is no guarantee that S is a legal tetrahedron. For example, if

we consider the sextuple S= (5, 4, 3, 5, 3, 4) it is facial but D(S) = - 2450. The fact that the

McCrea determinant is negative means that no tetrahedron exists.

Figure 3.5:Tetrahedron with negetaive determinant

Furthermore the sextuple S = (1, 1, 3, 6, 1, 3) which is not facial but D(s) = 952 (positive), also

doesn’t exist as a tetrahedron.

Figure 3.6: Tetrahedron with non facial 38

Consider a tetrahedron which is facial and the McCrea determinant is positive such as the

tetrahedron (T ' ) in Figure 4 with faces 28, 30 ,42; 28, 36, 45; 30 , 35, 45; and 35, 36, 42.

These numbers correspond to the numbers at the vertices of the original tetrahedron ( T ) in

Figure 3.3.These edges are vertices are 28, 30, 42; 28, 36, 45; 30, 35, 45 ; and 35, 36, 42.We can

think of these as dual tetrahedra .

If S is a sextuple for (potential) tetrahedron T, S = (a, b, c, d, e, f) then T has faces a, b, c; a, e, f;

b, d, f and c, d, e and the edges at the vertices has the pattern a, b, f; a, c, e; b, c, d and d, e, f .

If we have a potential tetrahedron T ' and where the pattern of faces and vertices is interchanged then T ' is called the dual of tetrahedron T.

Theorem: If the potential tetrahedron T has sextuple S = (a, b, c, d, e, f) then the sextuple

( f, e, d, c, b, a ) gives rise to the potential dual tetrahedron.

Figure 3.7a:Dual tetrahedron

Faces : a, c, b; b, d, f; a, e, f and c, d, e

Vertices: a, c, e; b, c, d; d, e, f and a, b, f

Figure 3.7b:Dual Tetrhedron

Vertices : a, c, b; b, d, f; a, e, f and c, d, e

Faces : a, c, e; b, c, d; d, e, f and a, b, f

they are dual tetrahedra.

Standard form of labeling a tetrahedron

There are two standard forms for drawing a labeled tetrahedron in the plane. Both involve

isomorphic copies of K 4 with different symmetry groups as drawings in the plane. 40

(a) (b)

Figure 3.8: labeling of tetrahedra

Figure 3.8a has 8 symmetries and it enables one to see the edges of the perfect matching of a tetrahedron more clearly while Figure 3.8b enables us more easily to assign the given number of six distinct edge lengths in a visually appealing manner.

Partition types of tetrahedra

Using only the partition information tetrahedra would lie in one of the 11 classes. These 11 classes exist as 3D types but not as degenerate 2D types because one type doesn’t exist in the plane.

A new mathematical question is what is the largest number of incongruent tetrahedron with six edge lengths can be formed for each partition type? 41

Suppose a tetrahedron with a given 6 distinct edge lengths S = {a, b, c, d, e, f} for partition

{1, 1, 1, 1, 1, 1}. If I fix c, f, e there are 6 permutations of the remaining 3 edges. Assume choice c as the smallest edge length. Consider the edges of length e and f which are a matching edge of the tetrahedron. Now there are 6 permutation edges of the other 3 edges a, b, d giving rise to the 6 patterns shown in the diagram below. Note, in each of the 6 diagrams each pair of letters from a, b,…, f appear in two triangles .Now having chosen c (top edge) as the smallest and the other edge e as one of the diagonal edges then there are 4 other ways to pair e with another letters as the other “diagonal “edges e and f, e and d, e and b, and e and a one can obtain an additional 24 incongruent tetrahedron. The edges opposite to it can be chosen in 3 ways and each of them admits 2 edges (e and c as matching edges) then there are 6 in congruent tetrahedron.

t t

f e f e

d t a

a d

abt ade bee c d f bdf a de bee a c 1

t t

e f f e d b

a d

a b ad f abe cde btl abe acf b df t de

t t

a b a d

d b

bde adf btl ace ace bd e abf tdf 43

r a

b d

ad e be r ab c t df

f a a

b b d e

e d

a d e bet abc cdr ade bdf abc e el 44

c c

t a d e e d

b b

abe bdf adC eel bet abd edt ace

c c

d b e b

e d

det abe bet acd det abd bet ace 45

f d

b e

ab f ade cer btd

b a

a e

abf ada cer btd bef ade ac l btd 46

c c

f d f d

e b e b

a a

a ef ab d bt l cd e aef abd btl cd e

c c

d f d

b e a a

e b

b et cde act abd aef bde be f acd 47

c c

f b f b e a d

d e

a ct bee del abd det abe act bed

c c

f b b

a e e

d a

adf bde ee l abc adf bed eel abe 49

b d

a t d abr de f 3ce

e e

f r

b d a d

a b

a Cd abt del bee bed abt del ace 50

Figure 3.9: Construction of congruent tetrahedra

In terms of symbols there are at most 30 potentially incongruent tetrahedra with 6 distinct edge lengths, and each tetrahedron is congruent with 24 tetrahedra corresponding to the 24 ways of labeling the four vertices for a total of 720 tetrahedra. The 24 congruent tetrahedra for a given sextuple S = (a, b c, d, e, f) appear in Figure 3.10. 51

(

Figure 3.10: congruent tetrahedra 55

If we Consider a tetrahedron for the partition {2, 1, 1, 1, 1} with edge length

{16, 12, 9, 12, 8, 18 } one can find a “dual tetrahedron” taking into a account the triples that are the edges at a vertex and the triples that form the faces of a tetrahedron for a matching edge however there is no guarantee that the dual tetrahedron exists .

S = (16, 12, 9, 12, 8, 18)

. T 1

D(S) = 844312 56

S = (12, 9, 12, 16, 18, 8)

T 2

D(S) = 1489752 57

S = (16, 18, 12, 8, 9, 12)

T 3

D(S) =- 3675992 58

S = (18, 16, 9, 8, 12, 12)

T 4

D(S) = - 5581152 59

S = (18, 16, 12, 8, 12, 9 )

T 5

. D(S) = - 1912032 60

S = (18, 16, 12, 8, 9, 12)

T 6

D(S) = - 4353408

Figure 3.11: Dual tetrhedra

Tetrahedron 1 and 2 exist and are dual. Tetrahedron 4 and 5 are dual and doesn’t exist;

Tetrahedron 3 and 6 are self-dual and doesn’t exist.

Conjecture: If S is a self-dual {2, 1, 1, 1, 1} tetrahedron then S can’t exist.

If we consider the tetrahedron for the partition of {2, 1, 1, 1, 1} and the sextuple

S= ( 8, 12, 9, 18, 12, 16) where the product of the matching edge lengths equal

12 x 12 = 9 x 16 = 8 x 18 = 144 and all triples are facial then one can generate at most 15 tetrahedron. (Refer to page 115) 61

Figure 3.12: {2, 1, 1, 1, 1} tetrhedron with equal product matching edge

If we consider for the partition { 1, 1, 1, 1, 1, 1 } with edge length (36, 28, 30, 35, 45, 42 ) and the product of the matching are equal, 35 x 36 = 42 x 30 = 28 x 45 = 1260, and all triples are facial then

D(s) = 6441334200

Figure 3.13a :{ 1, 1, 1, 1, 1, 1} tetrahedron with equal product matching edge 62

One can generate 30 scalene tetrahedron and all exists. (Refer to page 108)

Figure 3.13b :{ 1, 1, 1, 1, 1, 1} tetrahedron with equal product matching edge

Conjecture: If ad = be = cf and D (a, b, c, d, e, f ) > 0 and S (a, b, c, d, e, f) is facial then all the

30 assoiciated tetrahedron sextuples exsit.

Partition of Tetrahedron

The partitions of n are the ways of writing n as the sum of positive integers. we classify the tetrahedron according to the edges since the tetrahedron has 6 edges then there are 11 partitions

{6}, {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1}, (2, 2, 2}, {2, 2, 1, 1}, {2, 1, 1, 1, 1},

{1, 1, 1, 1, 1, 1}. Since a tetrahedron has 6 edges, one can interpret the partition {2, 2, 1, 1} as that one has a tetrahedron with 2 edges of the same length, two edges of the same length but

different from the first two edges of equal length and two different other edges whose length

differs from the other 5 edges. For example one might have the lengths a, a, b, b, c, and d for the 63

edges of the tetrahedron, which represents {2, 2, 1, 1}. In the same way the lengths a, a, a, b, b and c for the edges of a tetrahedron, can represent a {3, 2, 1} partition. Using only the partition information, tetrahedra would lie in one of the 11 classes. These 11 classes all exists as 3 D types but not as a degenerate 2D type because a 3D type may not exist in the plane.

Tetrahedra for each of the 11 partition classes

1. { 6 }

{5, 1}

3. {4, 2}

4. {4, 1, 1}

5. {3, 3}

6. {3, 2, 1}

7. {3, 1, 1, 1}

8. {2, 2, 2}

9. {2, 2, 1, 1}

10. {2, 1, 1, 1, 1}

11. {1, 1, 1, 1, 1, 1}

Figure 3.14: 11 partition classes of tetrhedra 69

However with edge lengths that belongs to a particular partition we will consider whether or not

there exists a tetrahedron that takes into account other geometrical information. For example, for

the partition {2, 2, 1, 1} we are looking to differentiate between the cases where the edge lengths

constitute a perfect matching or paths of length 2. So there will be 4 types of tetrahedron that will

be different from each other. Using the partition information any legal tetrahedron would lie in

one of a bigger number partition classes.

We can also consider the partition for a particular tetrahedron based on congruence of triangles.

Thus we might have two congruent triangles of one type, and two congruent triangles of another

type. This system can be further refined to take into account whether the triangles are equilateral,

isosceles or scalene. We will use the notation E for equilateral, I for isosceles and S for scalene

so when we have a 2I E S tetrahedron it means two congruent isosceles triangles and one

equilateral triangle and scalene triangle.

The refined approach taking some geometric information (but not relative size of edges) into

account leads to potentially 25 classes as was determined in this thesis.

Theorem: There are 25 different partition classes of tetrahedra taking into account graph theoretical aspects of the position of the edges, and all 25 types exist. 70

2 5 Tvpes of tetrahedra bv partition tvpe

1. {6}

(6)

Faces 4E: Vertices 4E:

2. {5, 1}

{!J,1}

F:acP.2E 21 Vertices 2C 21 71

3-1. {4, 2}

3-2. {4, 2}

4-1. {4, 1, 1}

4-2. {4, 1, 1}

5-1. {3, 3}

5-2. {3, 3}

6-1. {3, 2, 1}

6-2. {3, 2, 1}

6-3. {3, 2, 1}

6-4. {3, 2, 1}

7-1. {3, 1, 1, 1}

7-2. {3, 1, 1, 1}

7-3. {3, 1, 1, 1}

8-1. {2, 2, 2}

8-2. {2, 2, 2}

8-3. {2, 2, 2}

9-1. {2, 2, 1, 1}

9-2. {2, 2, 1, 1}

9-3. {2,2,1,1}

9-4. {2, 2, 1, 1}

10-1. {2, 1, 1, 1, 1}

10-2. {2, 1, 1, 1, 1}

11. {1, 1, 1, 1, 1, 1}

Figure 3.15: 25 types of tetrahedra by partition type

We can use a partition pair by edges together with a congruence type partition to pin down exactly which of the 25 types of tetrahedra we are looking at. We can classify faces by partition type with regard to congruence.

There are 4 faces, and 4 can be partitioned in 5 ways.

Noting partitions of 4

{4}, {3, 1}, {2, 2}, {2, 1, 1}, and {1, 1, 1, 1}

we can classify tetrahedra using numbers of congruent triangles.

{4} 4 congruent triangles ,

{3, 1} 3 congruent; 1 congruent

{2, 2} 2 congruent; 2 congruent 83

{2, 1, 1} 2 congruent; 1 congruent; 1 congruent

{1, 1, 1, 1} 1congruent ;1congruent ; 1congruent ; 1 congruent ; 1 congruent .

For example, if we consider {3, 1}, there are three triangles congruent to each other, and another triangle which is different, not congruent to the other 3.

If we consider {2, 1, 1}, there are two triangles congruent to each other, and another two

triangles which is different, not congruent to each other and the other 2 triangles.

E, I, 2S {2, 1, 1}

Figure 3.16a: {3, 2, 1} tetrahedron regard to the numbers of congruent triangles

However, one could imagine an additional refinement to this system as in Figure 3.16a.

This is because triangles come in three types: equilateral, isosceles, and scalene.

When you see 2I, 2S, we are looking at a tetrahedron with two congruent isosceles triangles and

two congruent scalene triangles as faces (Figure 3. 16b). 84

2I, 2S

{2, 2}

Figure 3.16b: {3, 2, 1} tetrahedron regard to the numbers of congruent triangles

I use the convention that larger numbers in the partition appear first-thus, 2S, 1I, 1I rather than

1I, 1I , 2S using alphabetical order and also which also goes from" most symmetric"

(equilateral) to least symmetric" (scalene).

Each vertex of a tetrahedron can also be classified as to whether it is equilateral or isosceles or

scalene and by congruence type. This information is recorded in a table below. Thus, the one

type {6} also is labeled faces 4E, vertices 4E. Note that it is possible that we can have for faces:

1I, 1I, 1I, 1S and for vertices 1I, 1I, 1I, 1S but the letters involved can be different. Thus, the

triangles from the face point of view have labels aab, acc, bbc, abc but the vertices have the

labels aac, abb, bcc, abc . 85

Congruence Property by face and vertex

vertex {4} {3,1} {2,2} {2,1,1} {1,1,1,1}

Face

{4} {6}, {4,2}, No No No No

{2,2,2}

{3,1} No {3,3} No No No

{2,2} No No {5,1}, {4,1,1} No No

{3,3}, {3,2,1}

{2,2,1,1}

{2,1,1} No No No {4,2},**{3,2,1) No

{2,2,2},

{2,2,1,1}

{1,1,1,1} No No No No {4,1,1},{3,2,1},

***{3,1,1,1}

{2,2,2},

**{2,2,1,1},

**{2,1,1,1,1},

{1,1,1,1,1,1}

E - Equilateral I – Isosceles S – Scalene

Table 3.1: Congruence Property by face and vertex

The number of stars implies the number of different face-vertex congruence types for this partition, for example for** {2, I, I} we have 2S E I and 2I I I. (Refer to page 87 and 89) 87

Congruent face partition with edge partition behavior

Face Edge {4} {3,1} {2,2} {2,1,1} {1,1,1,1} partition

{6} 4E No No No No 1

{5,1} No No 2E 2I No No 1

{4,2} 4I No No 2I E S No 2

{4,1,1} No No 2I 2I No E I S S 2

{3,3} No 3I E 2I 2I No No 2

{3,2,1} No No 2I 2S 2S E I I III 4

2I I I

{3,1,1,1} No No No No E S SS 3

I I S S

I II S

{2,2,2} 4S No No 2S I I I II S 3

{2,2,1,1} No No No 2S I S I S SS 4 I I S S I I S S

{2,1,1,1,1} No No No No S SSS 2

I S SS

{1,1,1,1,1,1} No No No No S SSS 1

3 1 4 5 12 25

Where E – Equilateral, I – Isosceles and S – Scalene.

Table 3.2: Congruent face partition with edge partition behavior 88

One can interpret the entry in the above chart {3, 3} by {3, 1} using the Figure 3.16b.

(5th row and 2nd column)

Figure 3.17: Congruent face partition with edge partition behavior

We have 3I E as the congruent triangle pattern type here. 89

Congruent vertex partition with edge partition behavior

Vertex {4} {3,1} {2,2} {2,1,1} {1,1,1,1}

Edge partition

{6} 4E No No No No 1

{5,1} No No 2E 2I No No 1

{4,2} 4I No No 2I E S No 2

{4,1,1} No No 2I 2I No E I S S 2

{3,3} No 3I E 2I 2I No No 2

{3,2,1} No No 2I 2S 2I I I I III 4

2S E I

{3,1,1,1} No No No No E S SS 3

I I S S

I II S

{2,2,2} 4S No No 2S I I I II S 3

{2,2,1,1} No No No 2S I S I S SS 4 S S I I I I S S

{2,1,1,1,1} No No No No S SSS 2

I S SS

{1,1,1,1,1,1} No No No No S SSS 1

3 1 4 5 12 25

Where E - Equilateral I – Isosceles S – Scalene.

Table 3.3: Congruent vertex partition with edge partition behavior 90

Theorem1: A tetrahedron T cannot have exactly three congruent scalene triangles and one equilateral triangle. Having (1E, 3S) tetrahedron is impossible.

Figure 3.18a: Congruent face partition with edge partition behavior

Proof:

From the given sides of six edges, three of them must be equal.

Let the three sides AB=AC=BC = a.

For ABD and ACD to be scalene two of AD, BD, and CD must be different from a, say b and c.

By congruence type CD = BD=b and AD = c which is impossible. 91

Theorem 2: A tetrahedron T cannot have two faces which are each equilateral but have different side lengths.

Figure 3.18b: Congruent face partition with edge partition behavior

Proof:

If one triangle T is equilateral (side a), the three other faces have an edge in common with T. If any edge of these triangles is of length b ∑ a, these triangles can’t be equivalent.

Let the three sides of an equilateral triangle be represented by a AC = AD = CD= a and let BC = b. Then by congruence type these triangles can’t be equivalent, which is impossible. 92

Theorem 3: A tetrahedron T cannot have exactly three equilateral triangles as faces.

Figure 3.18c: Congruent face partition with edge partition behavior

Proof:

From the given six edges, five must be equal in length.

Let AB = AC = BC = CD = BD = a

By congruence type AD = b which is impossible 93

Theorem 4: A tetrahedron T cannot have two congruent equilateral triangles and 2 scalene triangles as faces.

Figure 3.18d: Congruent face partition with edge partition behavior

Proof: Having two congruent equivalent triangles means we have the situation in Figure 3.18.

Let the five sides AB = AC = AD = BD = CD = a

And BC = b ∑ a, then by congruence type this is impossible.

Remember if the triangular inequality holds for all triples of six lengths there may be no tetrahedron with the given six edge lengths.

Another question is which integer sextuples can be edge lengths of tetrahedron?

This thesis not only enumerates the refined number of partition classes but also determines for 94

each of these classes of partition types if there is an integer collection of lengths for a tetrahedron in this class exists.

Partitions of a tetrahedron with integer lengths

{6}

D(s) = 400000 95

{5, 1}

D(S) = 47200

{4, 2}

D(S) = 321408 96

D(S) = 329182

{4, 1, 1}

D(S) = 243000 97

D(S) = 270444

{3, 2, 1}

D(S) = 226750 98

D(S) = 218638

D(S) = 103390 99

D(S) = 202752

{3, 3}

D(S) = 657886 100

D(S) = 679936

{3, 1, 1, 1}

D(S) =1726200 101

D(S) =1598292

D(S) = 1795968 102

{2, 2, 2}

D(S) = 1968300

D(S) = 1904992 103

D(S) = 1747980

{2, 2, 1, 1}

D(S) = 285750 104

D(S) =202158

D(S) = 151092 105

D(S) = 564408

{2, 1, 1, 1, 1}

D(S) = 124558 106

D(S) =227200

{1, 1, 1, 1, 1, 1}

D(S) = 301150

Figure 3.19: Partitions of a tetrahedron with integer lengths 107

We see that all 25 partition types exist with integer lengths. Now we show that for the

{1, 1, 1, 1, 1, 1} partition type and the sextuple ( 28, 30, 35, 36, 42, 45), there are a total of 30 incongruent tetrahedra.

(The 30 Incongruent Scalene tetrahedron with integer length).For each case the value of the

McCrea determinant is given.

1. 2.

22500809 22278809 108

3. 4.

2785555008 2314736352

5. 6.

27898509 23412409 109

3. 4.

51468809 51538909

9. 10.

55755809 56182609 110

3. 4.

56667409 57024209

13. 14.

5335041996 5305835052 111

15. 16.

5804836812 6013623852

17. 18.

5717976012 59559709 112

19. 20.

62710809 64419552

21. 22.

63367009 58232709 113

23. 24.

64933209 58067109

25. 26.

64868209 6291509 114

27. 28.

659359 09 5913984312

29. 30.

64413309 59570509

Figure 3.20: Incongruent Scalene tetrahedron with integer length 115

(Ten incongruent tetrahedra with integer length sides of partition type {2, 1, 1, 1, 1} of a possible

15.) For each case the value of the McCrea determinant is given.

1 . 2.

D(S) = 28450506 D(S) = 11274106

3 4.

D(S) =60241706 D(S) = 35838706 116

5. 6.

D(S) = 55811506 D(S) = 36759906

7. 8.

D(S) = -33321306 D(S) = 10224306 117

9. 10.

D(S) = 43534106 D(S) =19630106

11. 12.

D(S) = -27049207 D(S) =-27734707 118

13. 14.

D(S) = -3076996 D(S) = -2298707

15.

D (S) = 1912036

Figure 3.21: {2, 1, 1, 1, 1} incongruent tetrahedra with integer length 119

Examples with negative McCrea determinants don’t exist. Which values from 0 to 15 for this

partition type are obtainable can serve as a project.

The monotone up and monotone down series generate tetrahedra from a given sextuples S by increasing and decreasing the sequence of the given edge lengths as shown in the Figure 3.22.

For example S = (35, 36, 42, 28, 30, 45) goes next to S = (35, 36, 42, 30, 30, 45) in the increasing sequence and S = (35, 36, 42, 28, 30, 42) in the decreasing sequence.

D(S) = 2341239800 120

Monotone up Monotone down

D(S) = 3337587000 D(S) = 3578189792

{2, 1, 1, 1} {2, 1, 1, 1}

D(S) = 8432093950 D(S) = 6098540000

{3, 1, 1, 1} {3, 1, 1, 1} 121

D(S) = 9966159000 D(S) = 46699915550

{4, 1, 1} {4, 1, 1}

D(S) = 23340101400 D(S) = 2703859200

{5, 1} {5, 1} 122

D(S) = 2703859200 D(S) = 1927561216

{6} {6}

Figure 3.22 a: monotone up and monotone down series

D(S) = 2250079808 123

Monotone up Monotone down

D(S) = 3270405600 D(S) = 3578189792

{2, 1, 1, 1, 1} {2, 1, 1, 1, 1}

D(S) = 8432093950 D(S) = 4931436128

{3, 1, 1, 1} {3, 1, 1, 1} 124

D(S) = 99661590000 D(S) = 46699915550

{4, 1, 1} {4, 1, 1}

D(S) = 23340101400 D(S) = 27033859200

{5, 1} {5, 1} 125

D(S) = 33215062500 D(S) = 1927561216

{6} {6}

Figure 3.22 b: Monotone up and Monotone down series

Many interesting research questions and projects grow out of these ideas. 126

Degenerate Tetrahedra

A sextuple tetrahedron S = {a, b, c, d, e, f} is a degenerate tetrahedron if and only if D(s) is zero and S satisfies the triangular inequality (the sum of two sides of a triangle is greater or equal to the third side). From the sextuples of degenerate tetrahedra one can try to generate different types of quadrilaterals with integer lengths.

The partition of {6} does not exist in the plane with integer or real number lengths. For the other partitions of 6, namely {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1, 1}, {2, 2, 2},

{2, 2, 1, 1}, {2, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1} one can find a quadrilateral with real number lengths in the plane which can be interpreted as a degenerate tetrahedron.

The degenerate tetrahedron

(In this section the first diagram shows a degenerate tetrahedron and the second shows a polygon)

{6} don’t exist as a degenerate tetrahedron in the plane. 127

{5, 1}

D(S) = 0

AD = 3 ⊕ 5 = 8.66

The complete quadrilateral above arises from the metric rhombus shown here.

Rhombus 128

Note that it can’t be realized with integer length.

This follows from the fact that a rhombus with equal diagonals must be a square.

{4, 2}

AB = CD = 2 ⊕ 17 = 24.04

Square 129

2-2. {4, 2}

AB = CD = 2 ⊕ 5 = 7.07

Square 130

{4, 1, 1]

Rhombus 131

3-2. {4, 1, 1}

Rhombus 132

4-1. {3,2, 1}

13 13

B A 1 3

24.78

15 15

1 3 1 .

A u

1 !; 1 .

Kite 133

4-2. {3, 2, 1}

This kite type can’t be realized using only integers.

Kite 134

5. {2, 2, 2}

Rectangle 135

6. {2, 2, 1, 1}

Trapezoid 136

7. {1, 1, 1, 1, 1, 1}

Figure 3.23a: Degenerate tetrahedra

To get another perspective on degenerate tetrahedra consider the following.

137

Figure 3.23c shows a "degenerate" tetrahedron because the lengths making up ∆ABC do not obey the strict triangle inequality. This degenerate tetrahedron can be redrawn in the plane as show in Figure 3.23b. In this diagram vertices A, B, C, and E are collinear.

Figure 3.23b: Degenerate tetrahedra Figure 3.23c: Degenerate tetrahedra

If we consider triangle BCD in Figure 30 and flex it by moving vertices B and C towards each other (bending the triangle along line segment AD) we can form a tetrahedron in 3-space.

We can think of this "flexing" as shortening the length of BC in Figure 3.23b. If we don't shorten AD "too much" we will be able to get legal 3-dimensional tetrahedra for a range of values where the length of AD is even an integer. Examples, of such tetrahedra with shorter length values for BC are shown in Figure 3.24.

Labels now are for a “standard “labeled tetrahedron

138

Figure 3.24: labeled tetrahedron

If edge BC is shorted from 5 to 4, then again we will have a degenerate tetrahedron since the triangle inequality again will no longer hold for ∆ABC.

139

CHAPTER IV

NETS OF TETRAHEDRA

Nets are obtained by cutting three edges of the tetrahedron at a vertex of the tetrahedron or along a sequence of three edges that visit each vertex exactly once and flattening out the tetrahedron. Thus, there are two types of nets: star nets and path nets

Star nets:

Star nets are obtained by cutting three edges of a tetrahedron at one vertex of the tetrahedron.

Figure 4.1: spanning cut tree and their nets

Path nets:

Path nets are obtained by cutting three edges of a tetrahedron along a sequence of three edges of a tetrahedron that form a path.

140

Figure 4.2: spanning cut tree and their nets

The dark (Thickened) edges in Figures 4.1and 4.2, indicate the edges of a spanning tree which are to be cut to obtain a net. A spanning tree is a sub graph of a graph which is a tree and includes all the vertices.

Figure 4.3: spanning cut tree

141

There are potentially 16 distinct edges unfolding of a tetrahedron of which four are star nets and twelve are path nets.

For example for the {5, 1} partition, we have 4 spanning cut trees leading to star nets and 12 spanning cut trees for path nets, some of which due to symmetry may yield the same net.

Figure 4.4: {5, 1} tetrahedron

Star nets:

For {5,1}, when we cut along the 4 spanning trees of a tetrahedron, which cut all of the three edges at a single vertex, one will find 2 different star nets.

142

Figure 4.5a: star net for {5, 1}

Path nets

For a {5, 1} partition the 12 possible path nets give rise to three different path types: a, a, a, a, b, a and a, a, b. Note b, a, a gives rise to the same net as a, a, b. Also for a {5, 1} partition there are two types of a edges: an a edge with two a edges at each end and an a edge with two a’s at one end and a and b at the other.

143

c 8

D A

------A-

B c c

Figure 4.5b: path net for {5, 1}

144

Among the 16 spanning cut trees of a tetrahedron for partition type {5, 1} one can find 5 distinct nets (Figure 5a and Figure 5b).

Partition type {2, 1, 1, 1, 1} has two “sub-types “when the two equal length edges form a path or form a matching .We know that there can be a maximum of 4 star nets and 12 path nets for a fixed tetrahedron. Fixed in the sense that one does not move around on the tetrahedron the edge lengths (because when one does that for{2, 1, 1, 1, 1} there may be ten in congruent tetrahedra).

Type 1 Type 2

Figure 4.6a :{2, 1, 1, 1, 1} type1 tetrahedron Figure 4.6b :{2, 1, 1, 1, 1} type 2 tetrahedron

1. For partition {2, 1, 1, 1, 1} where the two equal edges are a matching we have at most 4 spanning cut trees at vertices (star nets) and at most 12 spanning cut trees for a path nets.

145

Type 1

Figure 4.6a :{ 2, 1, 1, 1, 1} type 1 tetrahedron

Star Nets and Path Nets

Star nets

When we cut along the 4 spanning trees of a tetrahedron which cut all of the three edges at a single vertex one can find 4 different star nets.

146

Figure 4.7a: Star net for {2, 1, 1, 1, 1} type 1 tetrahedron

147

Path nets

For ( 2, I, I, I, I } the 12 possible path nets give rise to 10 different path types: we have 10 different types of path nets (a, b, a), (a, c, a), (a, d, a), (a, e, a), (b, a, d), (b, c, d), (b, e, d), (c, a, e), (c, b, e) and (c, d, e).

c B

148

D D 8

C b A

•J

0 149

e b

A 150

Figure 4.7b: Path net for {2, 1, 1, 1, 1} type 1 tetrahedron 151

Among the 16 potential nets for a tetrahedron for partition of type {2, 1, 1, 1, 1} with two equal edges in a matching one can find 14 distinct nets.

2. Partition type {2, 1, 1, 1, 1} where the two equal edges are a path.

We have 4 vertices spanning cut trees and 12 spanning cut trees for a path net.

Type 2

Figure 4.6b :{2, 1, 1, 1, 1} type 2 tetrahedron

Star nets

When we cut along the 4 spanning trees of a tetrahedron which cut all of the three edges at a single vertex one can find 4 different star nets. 152

c 8

153

Figure 4.8a: Star net for {2, 1, 1, 1, 1} type 2 tetrahedron

Path nets

For {2, 1, 1, 1, 1} the 12 possible path nets give rise to 12different path types : ( a, a, d), (a, e, d), (b, a, e), (b, c, e), (b, d, e), (c, a, a), (c, b, a), (c, d, a), (c, e, a), (d, b, a), (e, a, b) and (d, c, a) .

154

C B

D 155

156

A c

157

Figure 4.8b: Path net for {2, 1, 1, 1, 1} type 2 tetrahedron

Among the 16 potential nets for a tetrahedron for partition of type {2, 1, 1, 1, 1} with two equal edges form a path of length two ) one can find 16 distinct nets .

For one tetrahedron with two paths with the same letters, the two paths can give rise to different nets. Paths with two different letter patterns must give rise to different nets.

If we consider a tetrahedron with partition {2, 1, 1, 1, 1} type 2 with the same letters on different paths (path nets b, a, e) they can have different nets. They both have hexagons with the same pattern of edges. 158

Figure 4.9a: Path net for {2, 1, 1, 1, 1} type 2 tetrahedron

Figure 4.9b: Path net for {2, 1, 1, 1, 1} type 2 tetrahedron

Could one argue that these two tetrahedron are the “same “because they had the same path net? These two path nets if congruent would fold to the same tetrahedron but the tetrahedron are not congruent and the complements of the path nets are not the same Type -1 (b, a, e; a, c, d) and Type-2 (b, a, e; a, d, c). On combinatorial grounds these two tetrahedra can’t be the same.

If we consider Type 1and 2 partition of {2, 1, 1, 1, 1} tetrahedron with different letters of the same path have different nets 159

Figure 4.10a: Path net for {2, 1, 1, 1, 1} type 2 tetrahedron

Figure 4.10b: Path net for {2, 1, 1, 1, 1} type 1 tetrahedron

Theorem: If two paths have different letters for the same tetrahedron they can’t give rise to the same net but two paths with the same letters can give rise to different nets. (See Figure 4.9)

If we consider partition type {2, 1, 1, 1, 1} tetrahedra with the same path and switch one pair of edges in a matching edge one gets the dual tetrahedron and then their nets are the same. 160

Figure 4.11: Dual tetrahedron with the same path for {2, 1, 1, 1, 1} type

From the above fact, the dual tetrahedron with the same path has the same net. The faces of their nets are the same with different patterns of cut edges. If we switches edge length bd (instead of edge length ce) in a matching edge one gets the dual tetrahedron and their nets are the same.

161

Figure 4.12: Dual tetrahedra with a matching edge for {2, 1, 1, 1, 1} and their nets

For a tetrahedron if one takes a star net, one can always unfold to a non-overlapping polygon. However for a path net there is no guarantee that one can unfold to a non-overlapping polygon. There might be a spanning tree for a tetrahedron that when cut will lead to a way to open up the tetrahedron which will overlap.

Can any tetrahedron be cut along a path and unfolded into the plane without overlap?

There is a conjecture (Fukuda) that for every tetrahedron the cut tree yields a simple unfolding. In response to this conjecture that overlap couldn’t occur by F. Fukuda, M. Namiki found a counterexample. An explicit example of this phenomenon was published by Branko Grünbaum inspired by Namiki’s work. 162

Recall that, according to Demaine and O’Rourke an edge unfolding of a polyhedron is a cutting of the surface along its edges that unfolds the surface to a single, non-overlapping piece in the plane. It has three characteristics:

* The unfolding is a simply connected piece.

* The boundary of the unfolding is composed of edges of the polyhedron.

* The unfolding doesn’t overlap.

Which partition types for tetrahedra admit spanning tree cuts which lead to overlap? Grünbaum’s example is a type {1, 1, 1, 1, 1, 1}

There exists a {2, 1, 1, 1, 1} tetrahedral which can be cut along its edges DABC and has an overlapping unfolding.

Figure 4.13a: {2, 1, 1, 1, 1} tetrahedron 163

I found an overlapping net for a tetrahedron with partition type {2, 1, 1, 1, 1}

Figure 4.13b :{ 2, 1, 1, 1, 1} overlapping tetrahedron 164

CHAPTER V

SUMMERY, CONCLUSION AND RECOMMENDATION

A polyhedron is a 3-dimensional object that contains flat faces and straight edges. Examples include the Platonic solids and tetrahedra in general, combinatorial cubes, and other convex polyhedra. The tetrahedron is the simplest three dimensional solid, having four vertices, four faces (all triangles) and six edges. This thesis investigates new mathematical properties of tetrahedra motivated by showing novel approaches to traditional questions by mathematics educators interested in geometry and how geometry is connected to other parts of mathematics.

New theorems about tetrahedra are developed with these educational goals as motivation.

At the start of this dissertation a variety of mathematical and mathematics education issues were raised .Here I wish to address how these issues were resolved and offer ways to connect up the new mathematical ideas with the educational issues implicit in them .Referring to the need and purpose of this study, I have developed new mathematical results which connect up also the knowledge base of students with the definitions and ideas of geometry acquired through high schools and middle schools, with the goals:

1. Showing ways to transition from geometrical problems arising in 2-dimensions (2D) to ones in

3-dimensions (3D)

2. Showing connections between different mathematics topics (algebra, geometry,

combinatorics, etc.) that should be treated in a more integrated way

3. Enhancing sense making for the role of definitions in geometry and to design activities containing exploration problems and classroom activities that enable students to study geometry 165

by investigating, exploring, and establishing geometric conjectures, which eventually will

develop into proofs.

4. Showing that there are new mathematical questions about polygons and tetrahedra accessible

at the college and pre-college level.

With regard to the above goal I introduce Partitions of tetrahedron and the notion of a net for a

tetrahedron (polygon).Student rarely see this approach to polyhedra.

I discuss that nets offer an alternative way of approaching the kinds of geometrical question raised about polyhedr in 3D, in the plane 2D. Sample ideas for implementing this appear in the

Appendix .Partition type of tetrahedra offers a way to classify tetrahedra, and to address questions of research interest to mathematics education researchers and mathematicians while offering K-12 students a “laboratory” to make models, and get insight into representation of geometrical objects, visualizing these objects, and telling when the objects are the same or different. New theorems about tetrahedra are developed with these educational goals as motivation.

The following answers have been given to the questions raised at the start

1. How to classify tetrahedron into different types based on the notion of a partition.

I have shown the following new results. There are 11 types of tetrahedra based on the partition of

6 (Figure 3.14) and the refined approach taking some geometric information (but not relative size

of edges) into account leads to potentially 25 classes (Figure 3.15)

Theorem: There are 25 different partition classes of tetrahedra taking into account graph

theoretical aspects of the position of the edges, and all 25 types exist. 166

We can use partition notion based on the congruence property by face and vertex and congruent

face / vertex partitions with edge partition behavior (Table3.1, Table 3.2, and Table 3.3).

Theorem1: A tetrahedron T cannot have exactly three congruent scalene triangles and one

equilateral triangle. Having (1E, 3S) tetrahedron is impossible.

Theorem 2: A tetrahedron T cannot have two faces which are each equilateral but have different

side lengths.

Theorem 3: A tetrahedron T cannot have exactly three equilateral triangles as faces.

Theorem 4: A tetrahedron T cannot have two congruent equilateral triangles and 2 scalene

triangles as faces.

2. Existence questions for tetrahedron with integer lengths.

For all 11 types these types can be realized with integers. This extends to the refined 25 classes.

If S is a sextuple for (a potential) tetrahedron T, S = (a, b, c, d, e, f) then T has faces a, b, c; a, e,

f; b, d, f and c, d, e and the edges at the vertices has the pattern a, b, f; a, c, e; b, c, d and d, e, f .

If we have a potential tetrahedron T ' and where the pattern of faces and vertices is interchanged then T ' is called the dual of tetrahedron T.

Theorem: If the potential tetrahedron T has sextuple S = (a, b, c, d, e, f) then the sextuple ( f, e, d,

c, b, a ) gives rise to the potential dual tetrahedron.

3.Relating the existence of nets of tetrahedra to the type of spanning tree that is used to generate the net, and under what circumstances attempts to generate a net for a tetrahedra fail due to overlap of the faces when the tetrahedron is flattened into the plane. 167

Nets are obtained by cutting three edges of the tetrahedron at a vertex of the

tetrahedron or along a sequence of three edges that visit each vertex exactly once

and flattening out the tetrahedron. Star nets can’t result in overlaps; only for path

nets can this phenomenon occur.

4. Which partition types of tetrahedra admit spanning tree with cuts that lead

overlaps?

Grunbaum, in my notation showed there is a {1, 1, 1, 1, 1, 1} tetrahedron that leads to an overlapping “net “when a path is cut. I have shown there also exists a

{2, 1, 1, 1, 1} tetrahedron .

The {6} partition can’t lead to an overlap.

There exists a {2, 1, 1, 1, 1} tetrahedra (Figure 4.13a) which can be cut along its edges and has an overlapping unfolding (Figure 4.13b).

Recommendation

There is still the need to extend the pedagogical examples to support the evolving changes in

geometry occasioned by the Common Core State Standards in Mathematics. Additional ideas

related to those developed in the appendix can be used to show that mathematics, geometry in

particular, is a dynamic subject where one can get new insight into well studied objects

( tetrahedra ) as well as directions for further insights (nets). Thus educators can use the study

using the mathematics provided and the associated material in the Appendix.

An important open problem that is accessible to researchers, mathematics educators,

mathematician, and undergraduates or high school students is to determine for each of the 25

partition classes what are the admissible numbers of incongruent tetrahedra for the given

class? 168 For {1, 1, 1, 1, 1, 1} the answer includes 0 and 30 but is there a set of lengths with exactly say,

23 incongruent tetrahedra?

An interesting research question is to see what other partition types of tetrahedra allow cut trees

that lead to overlapping faces. 169

REFERENCES

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273-276).

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Grünbaum, B. (2009). Configurations of points and lines (vol. 103).American Mathematical

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Tucker, A. (2007). Applied Combinatories , Danvers, MA .John Wiley & Sons, Inc.

Usiskin, Z & Griffin, J. (2007). The classification of Quadrilaterials: A study of Definition (

Research in Mathematics education ).Information Age Publishing ,Inc.

Wirth, K., & Dreiding, S. (2009). Edge length determining tetrahedron. Elementeder

Mathematik, 64(4), 160-170.

William H. McCrea. (2006). Analytical geometry of three dimensions.(Courier Dover Publications). 171

Zalgaller, V. A. (1969). Convex polyhedra with regular faces. Zapiski Nauchnykh Seminarov

POMJ, 2, 5-221. 172

APPENDICES

NEW IDEAS TO AUGMENT TRADIONAL TOPICS AND PEDAGOGY RELATED TO POLYGONS AND POLYHEDRA (TETRAHEDRA)

Appendix A

Geometry in School

The new mathematics developed in chapters 1-4 has implications for teaching geometry in the era

of the common core state standards in Mathematics –CCSS-M. I will be concerned with the

important ideas touched on in the CCSS-M that involve geometry and connections between geometry and other parts of the middle and high school curriculum. In particular, I will also treat

the way one moves from the simplest polygon, a triangle, to more complex polygons like

quadrilaterals and n-gons. This is a change within the dimension of the object (2D). I will also be

concerned with the transition between 2D and 3D. Thus, a triangle is a 2D idea, while its natural

generalization to 3D is the tetrahedron. Tetrahedra are a special case of a more general kind of

3D object, namely polyhedra. Let me give background for these issues by giving a brief account

of the history of geometry, in particular, as the history relates to the concepts of polygon and

polyhedra and what students are encouraged to learn about these geometrical objects.

For thousands of years civilized humanity has needed to know how to work with the size,

shapes, or position of things in order to help him solve many of the practical problems of the day.

We devised methods of measuring line segments, angles and surfaces. (Dressler)

Geometry begins with a practical need to measure shapes. The word geometry means to measure the earth

and has come to concern itself with the science of shape and size of things, as well as visual phenomena.

Around 2000 BC the first Egyptian pyramid was constructed. Knowledge of geometry was essential for 173

building pyramids which consisted of a square base and triangular faces. Actually, these pyramids have

steps. The triangular faces are not flat. They only look that way from a distance. The earliest record of a

formula for calculating the area of a triangle also dates back to 2000 BC.

The Egyptians (5000-500 BC) and the Babylonians (4000-500 BC) developed practical geometry to solve

everyday problems but there is no evidence that they logically deduced geometric facts from basic

principles.

It was the early Greeks (600 BC – 400AD) who developed the principles of modern geometry.

Noteworthy, is the work of Thales of Miletus (624-547BC). Thales is credited with bringing the science of geometry from Egypt to Greece. Pythagoras (569-475BC) was a Greek geometer and is regarded as the first pure mathematician to logically deduce geometric facts from basic principles.

Euclid of Alexandria (325-265BC) was one of the greatest of all the Greek geometers and is considered by many to be the father of modern geometry. His book Elements is one of the most important works in history and had a profound impact on the development western civilization.

Euclidean geometry today is a study of geometry based on definitions, undefined terms (points, lines and planes) and the assumptions (axioms) stated. However, it is widely viewed that rather

than being a fully axiomatic system in the modern sense, that Euclid was codifying the geometry of the space that one sees around one in daily life. One can view geometry as either a branch of

mathematics or a branch of physics. Euclidean geometry as presented in the Elements is based on

five basic notations and five fundamental axioms. However, contrary to what would be done

today, Euclid defines such words as point and line. His axioms are an attempt to assert as little as

possible without proof and everything else results by deductive reasoning from the rules of logic.

Euclidean Geometry enables us to visualize space and provide the tools often used to understand 174

the space we live in. However, physicists are still unsure of the exact nature of the space we live

in. Thus, we may, in fact, live in a non-Euclidean space even though in most applied settings we

can use results from Euclidean geometry to build skyscrapers and bridges.

The next great development in geometry came with the development of non-Euclidean Geometry

however almost nothing is done with non-Euclidean Geometry in K-12.

The importance of geometry, specifically three dimensional / space geometry, will give an

enormous amount of attentions in the school system to graphical representation and mental visualization in terms of developing an understanding of geometric relationships in a plane and in space and the ability to think creatively and critically in both mathematical and non- mathematical situations.

Carl Fredrick Gauss (1777-1855) who along with Archimedes and Newton is considered to be

one of the three greatest mathematicians of all time, invented non-Euclidean geometry prior to

the independent work of Janos Bolyai (1802-1860) and Nikolai Lobachevski (1792- 1858) .

Geometry in the Era of the CCSS-M

From the NCTM and of Regents of the University of the State of New York, students need to develop a better way of understanding of the definitions, properties and relations of geometrical

objects in a plane and in space and integrate geometry with arithmetic, algebra, and

trigonometry. The current New York State curriculum documents emphasize the use of terms

such as investigate, explore, discover, conjecture, reasoning, arguments, justify, explain, proof,

and apply as performance indicators in developing students mathematical reasoning ability in

connection with the necessity of the development of students skill in relation with the core 175

curriculum process strands: representation, connections, communication, reasoning and proof,

and problem solving.

The study of geometry is categorized in terms of shapes (circles, triangles, and squares), figures and positions in space and measurements and comparison of lines, angles, points, planes and

surface, and solids composed of combinations of these. A shape is the form of an object or figure

such as a circle, triangles, squares, rectangles, parallelogram, trapezoid, rhombus, octagon, pentagon and hexagon. For the most part polygons will be thought of as "rod structures" and when drawn in the plane. A polygon will be the boundary between two regions, the inside of the

polygon (a bounded region) and the exterior of the polygon which is an unbounded region. (This

is in essence what is known as the Jordan Curve Theorem for polygons.) A solid is a three dimensional figure such as cube, cylinder, cone, prism or pyramid; other solid shapes include the tetrahedron, octahedron and dodecahedron.

Below are indicated pedagogical and mathematical approaches to the study of geometrical objects and their classifications using the tetrahedron as a specific example in the school . The activities are organized within four units. Most of the activities are preceded with the background information to guarantee students can have active and successful participation in the solution of problems with different levels of difficulty through cooperative learning work. 176

Appendix B

Background information

Unit I: Points, lines and polygons

A point is represented by a dot and capital letters will be used to denote points. A point has no length, width or thickness. It only has location.

Figure 1.1: A point

A line is set points which is “straight “rather than curve. A line has infinite length.

Figure1. 2: A line 177

A line segment is a set of all points on the line between two end points.

Figure 1.3: Line segments

A plane is a set of points that form a completely flat surface extending indefinitely in all directions. Planes are denoted by Greek letters.

Figure 1.4: A plane

Collinear points: a set of points which lies on the same line.

Figure 1.5: Three collinear points 178

If A, B, and C are collinear then AC + BC = AB.

XY Will denote both the segment XY and its length.

Coplanar points: a set of points which lies on the same plane

Figure1. 6: Coplanar points

A Ray is a set of all points in a half line. It begins at a point and extends infinitely in one direction. AB and AC are examples of a ray.

Figure 1.7: Rays 179

Congruent segments are segments that have equal measure length.

AB 〈 CD

Figure 1.8: Two congruent segments

The midpoint of a line segment is the point of a line segment which divides the segment into two congruent segments.

AC 〈 AB

Figure 1.9: Mid point of line segment 180

An angle is a set of points which is formed by two rays that share a common end point.

Figure 1.10: Angle

A right angle is an angle whose measure is 90 degrees.

B is a right angle.

Figure1.11: Right angle 181

An acute angle is an angle whose measure is greater than 0 and less than 90 degrees. Figure 1.12 has acute angles at A, B and C.

Figure 1.12: Acute angle

An obtuse angle is an angle whose measure is greater than 90 and less than 180. Figure 1.13 has an obtuse angle at B.

Figure 1.13: obtuse angle 182

A reflex angle is an angle whose measure is greater than 180 and less than 360 degree. Figure

1.14 has a reflex angle at B.

Figure 1.14: Reflex angle

Two angles are adjacent if they have a common vertex and common side but don’t have any interior points in common. In Figure 1.15

Figure 1.15:Adjacent 183

Two angles which have vertical angles are a common vertex and whose sides are two pairs of straight lines. EA, ED, EC and EB are rays. AB and CD are two straight lines.

Figure 1.16: Vertical angles

Complementary angles are two angles whose measure sum to 90 degree.

Figure 1.17: Complementary angles 184

Supplementary angles are two angles whose measures sum to 180 degrees.

Figure 1.18: Supplementary angles

Activity:

1. Which one represents a point, a line and a plane?

Figure 1.19: Point, line, and Plane 185

2. Use your ruler to find the length of each line segment in Figure 20 and write your answer

Figure 1.20: Line segment

3. Are segments AB and CD in Figure 20 congruent?

4. Draw a line segment AB and mark C as a midpoint of AB such that AC 〈 CB

5. Draw a plane containing five points A, B, C, D and E with exactly three of the points A, C and

D on a line. What is relative position of the these points you need a diagram

6. List all the different segments with a) 3 collinear points b) 4 collinear points c) 5 collinear points e) 7 collinear points f) Can you create a formula for the number of different segments?

7. If

8. Given the polygon

Figure 1.21a: Polygons

List the vertices of the polygon in clockwise consecutive order starting at vertex B.

9. Which angles in Figure 21 are acute? Obtuse? Reflex?

A polygon is a closed figure in a plane formed by connecting line segments.

The line segments are the sides of the polygon. The end points of the line segments are vertices of the polygon. 187

Figure 1.21b: Polygons

A diagonal of a polygon is a line segment that connects two nonconsecutive vertices of the polygon. AC and BD are diagonals of the polygon ABCD.

Figure 1.22: Diagonal of Polygons 188

Note that a convex polygon is a polygon each of whose interior angle measure less than 180 degrees.

A non-convex polygon is a polygon which has at least one interior angle that measure more than

180 degree. Note there is a reflex angle at A.

Figure 1.23: Non Convex polygon

An equilateral polygon is a polygon that has congruent sides.

Figure 1.24: Equilaterial Polygon 189

An equiangular polygon is a polygon that has congruent angles.

Figure 1.25: Equiangular polygon

A regular polygon is a polygon that has congruent angles and congruent sides.

Figure 1.26: Regular polygon 190

A triangle is a polygon that has three sides.

Figure 1.27: Triangle

A, B, C are vertices of a triangle. AB, BC, AC are sides of a triangle.

The altitude of a triangle is a line segment which is drawn from any vertex of the triangle perpendicular to one of the opposite sides. 191

Properties of Quadrilateral

Kite

Figure 1.28: Kite

DC=BC, AB=AD, AO ∑ OC if BD ) AC = O then the quadrilateral is convex .

d 1 d 2 A = and are the diagonals. 2 d 1 d 2 192

Isosceles trapezoid

Figure 1.29: Isosceles trapezoid

AB CD,

(M and N are a line segment between AB and CD), C and D, and A and B are reflection images each other.

A= ½ h ( b1 + b2 ) where AB = b1 and CD = b2

Rhombus

Figure 1.30: Rhombus 193

AB CD and AD BC,

AD=BC=AB=DC

A= bh

(AC=BD for a square only).

Parallelogram

Figure 1.31: Parallelogram

AB CD and AD BC ABD 〈 CDB,

A=bh 194

Rectangle

Figure 1.32: Rectangle

AB CD and AD BC, AB=CD and AD=BC,

A = LW where L is length and w is width

Square

Figure 1.33: Square

AB=DC=AD=BC,

A= S x S where S is a side of the square. 195

Students can look at it the issue of the existence and properties of triangles with specified edge lengths, area formulas for triangles and quadrilaterals and integer realization of triangles (including Pythagorean triples) and quadrilaterals.

If A, B, C are distinct and non collinear points the set union AB  BC  AC  A  B  C is called triangle ABC ( ABC ) . A, B, C are the vertices of a triangle ABC and, AB, BC, AC are sides of the triangle. Furthermore

Figure 1.34: Triangle

Triangles are classified into three types in terms of sides and angles (Figure 1. 35)

Scalene triangle: a triangle that has no congruent sides.

Isosceles triangle: a triangle that has at least two congruent sides.

Equilateral triangle: a triangle that has three congruent sides. 196

Figure 1.35: Equilateral, Isosceles, and Scalene triangles

Note that under this definition a triangle that is equilateral is also isosceles.

Inequalities in a triangle:

Theorem: The sum of the lengths of two sides of a triangle is greater than the length of the third side.

AB +BC > AC, BC+AC>AB and AC+AB>BC

Theorem: If two sides of a triangle are unequal, the angles opposite these are unequal and the greater angle lies opposite the greater side.

Figure 1.36: Triangle

If AC is greater than BC then angle ABC > angle BAC

Theorem: If two angles of a triangle are unequal, the sides opposite these angles are unequal and the greater side lies opposite the greater angle. 197

In ABC, if BAC > ABC ABC then the measure of the side opposite < BAC is greater than the measure of the side opposite AC

Congruent Triangles: Two polygons are congruent if there is a one to one correspondence between their angles and sides.

. All pairs of corresponding angles are congruent

. All pairs of corresponding sides are congruent

Note: Triangles that have equal angles need not be congruent.

In ABC and DEF if

Figure 1.37a: Congruent triangles

1)  ⊕ D , B ⊕ E and C ⊕ F

2)  ⊕ DE, BC ⊕ EF and AC ⊕ DF , then ABC ⊕ DEF

Postulate (SAS): Two triangles are congruent if two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of the other. 198

Figure 1.37b: Congruent triangles

AB ⊕ AC, AD ⊕ AD, BAD ⊕ CAD and then ABD ⊕ ACD

Postulate (ASA): Two triangles are congruent if two angles and the included side of one triangle are congruent respectively to two angles and the included side of the other.

Figure 1.38a: Congruent triangles

ACD ⊕ BCD, CD ⊕ CD and ADC ⊕ BDC then ACD ⊕ BCD

Postulate (SSS): Two triangles are congruent if the three sides of one triangle are congruent respectively to the three sides of the other. 199

Figure 1.38b: Congruent triangles

AD ⊕ BD, AC ⊕ AB and CD ⊕ CD then ACD ⊕ BCD

Note SSA does not guarantee congruence.

1 The area of a triangle (figure 1.39) A= bh where are A is the area, b is base of the triangle and 2 h is the height of the triangle

Figure 1.39: Right angle triangle

A Heron’s triangle is a triangle whose side lengths and area are all rational numbers. The Greek mathematician named Hero found a formula for the area of a triangle: 200

a + b + c A = s(s a)(s b)(s c) and s= where A is the area of the triangle a, b, and c are 2 the lengths of the three sides of the triangle, and s is the semi-perimeter.

Theorem: An equilateral triangle with rational side length can’t have rational area. No equilateral triangle is Heronian.

Figure 1.40: Heronian triangles

x2 + y 2 = (x a )2 + y 2 = a

A = s(s a)(s b)(s c)

=3a  a  a  a  a 2 3 = 〉 〉 〉 〉  = ∫ 2 ∫ 2 ∫ 2 ∫ 2  4

Pythagorean Theorem: For the three sides of a right triangle the square of the hypotenuse is equal to the sum of the squares of the two sides (legs). The theorem can be written in equation form a2 +b2  c 2 where a and b are the length of the two sides and c is the length of the hypotenuse.

Any triangle whose side lengths for a Pythagorean triple is Heronian. 201

Pythagorean Triple: Suppose that m and n are two positive integers with m < n then a = n 2 m 2 , b = 2mn, c = n 2 + m 2 is a Pythagorean triple. One can simply generate the

Pythagorean triple.

For example n= 4 and m =3 then n 2 m 2 = 16- 9 = 7, 2mn = 24 and n 2 + m 2 = 25 and

2 2 2 7 + 24 = 25

Activity

1. Which of the following number triples may be used as the lengths of the sides of a triangle a) (7, 4, 2) b) (8, 5, 3) c) (9, 6, 4) d) (16, 12, 3) e) (16, 14, 12)

2. Find the area of a triangle whose base and altitude are 12 inches and 8 inches.

3. If the area of a triangle is 12 sq inches and if one side of a triangle is 6 inches then find the length of the altitude to that side.

4. Find the area of an isosceles right triangle whose hypotenuse is 4 2 .

5. If the sides of a triangle have lengths of 3, 4, and 5, find the area of the triangle.

6. In ABC,

7. In ABC,

Here I develop some ideas to set forth the properties of quadrilaterals in order to set the stage for comparing and contrasting the behavior of polygons (2D) with polyhedra (3D).

A quadrilateral (quad-four and lateral – side) is a polygon with four sides (or edges) and four vertices or corners. If we consider a quadrilateral ABCD (Figure 44 ) we can (sometimes ) flex it 202

to a tetrahedron along AD or BC or sometimes it becomes a degenerate tetrahedron , when we add the segments AD and BC .

Figure 1.41a: Quadrilaterals

Intuitively, a quadrilateral is just any four sided, shape. There are many ways to classify quadrilaterals. One approach due for Miller is: Plane quadrilaterals can be classified into types as follows:

Convex in which the two diagonals are internal and intersect

Non-Convex and not self-Intersecting (‘dart’) in which one diagonal is internal and one external, the two not intersecting the external diagonal spanning the “concavity.” Self-intersecting (Zigzag) in which one pair of opposite sides intersect and both diagonals are” external “and don’t intersect.

* (Partially) degenerate, in which a particular two adjacent side lie in the same line.

- A flag, where one vertex is an internal point of a side and two sides overlap.

- A triangle where two adjacent sides are in one straight line but not overlapping.

- A bent line where two opposite vertices coincides.

Fully degenerate, in which the whole figure is contained in one dimension. 203

Convex dart

Zigzag

Flag bentline 204

fullydegenerate

Figure 1.41b:Quadrilaterials

Quadrilaterals with self-intersection are called complex while those with no self-intersection are called simple. Simple quadrilaterals are either convex or concave and their interior angles are add up to 360 0 (angle sum formula (n-2)180).

Quadrilaterals are classified into eight types in school geometry by their properties:

Parallelogram Area=base x height 205

Rectangle Area = Length X width

Square Area = Side X Side 206

Rhombus Area = base X height

d d Kite Area = 1 2 2 207

Isosceles trapezoid Area = ½ h ( b1 + b2 ) where b1 = BV and b2 = ZW

Figure 1.41c: Quadrilaterials

According to Usiskin & Griffin : Hierarchies of Quadrilaterals .

Figure 1.42: Hierarchies of quadrilaterals 208

A quadrilateral is a parallelogram if and only if both pairs of opposite sides have the same length and are parallel

– Both pairs of opposite angles have the same measure

–Its diagonals have the same midpoint (bisect each other)

- It possess rotation symmetry

-Two consecutive angles are supplementary

A quadrilateral is an trapezoid if one pair of opposite sides is parallel

A quadrilateral is Isosceles Trapezoid if the non parallel sides are congruent and their base angles are equal measure

A quadrilateral is rectangle if it has all the properties of a parallelogram

–If it contains four right angles

–The diagonals of a rectangle are congruent

A quadrilateral is rhombus if it has all the properties of a parallelogram,

It is an equilateral

- The diagonals are perpendicular to each other

-Their diagonals are bisecting its angle.

A quadrilateral is Square if it has all the properties of a Rectangle

– If it has all the properties of a rhombus

A quadrilateral is kite if two pairs of adjacent sides are of equal length.

Theorem: a quadrilateral is cyclic If and only if the perpendicular bisectors of all its sides are concurrent. 209

Cyclic Quadrilateral is a cyclic Quadrilateral is a Quadrilateral whose vertices all lie on a single circle

Figure 1.43a: Cyclic quadrilaterals

According to Usiskin and Griffin: A cyclic quadrilateral possesses two important properties, Ptolemy’s theorem and Brahmagupta’s Theorem.

Ptolemy’s theorem: In the cyclic quadrilateral ABCD, AC x BD=AB.CD+AD.BC (In a cyclic quadrilateral, the product of the lengths of the diagonals equals the sum of the products of the lengths of its opposite sides)

Brahmagupta’s Theorem: The area of a cyclic quadrilateral with sides a, b, c, and d and semi

a + b + c + d perimeter s is ( s a)(s b)(s c)(s d ) where s = S is the semi perimeter. 2

According to Miller’s for a given ordered tuple ( a, b, c, d ) of four positive numbers a necessary and sufficient condition for the existence of a convex Quadrilaterals having those side lengths is each variable is less than the sum of the remaining three. 210

Brahmagupta’s Theorem is a generalization of the better known Heron’s formula for the area of a triangle that is often found in high school geometry texts. Let d=0 in Brahmagupta’s formula by bringing two vertices of the cyclic quadrilateral together and Heron’s formula appears. So a triangle can be thought of as a degenerate cyclic quadrilateral.

Figure 1.43b: Cyclic quadrilaterals

Using Ptolemy’s theorem, Pythagorean Theorem and Heron’s formula one can generate the length of the quadrilateral and be able to find the area and perimeter of the quadrilateral.

Activity:

1. Find the area of a rectangle whose width is 4inch and whose length 5 inch.

2. Find the base of a rectangle whose area is 20 square inch and whose altitude is 5 inch.

3. The perimeter of a rectangle is 24 inch. Find the area of rectangle if one of the sides is 6 inch.

4. Find the area of a square whose side is 8 inches.

5. Find the area of a square whose perimeter is 36 inches.

6. Find the area of the parallelogram whose base and altitude are 4 inches and 6 inches.

7. The area of a parallelogram is 30 square inches and the base is 10 inches. Find the altitude. 211

8. Find the area of a rhombus one of whose angles is 60 degree and whose shorter diagonal is 6 inches.

9. Find the area of trapezoid whose bases 8 inch and 4 inches and height is 6 inches 212

Appendix C

Unit II: Classification of degenerate tetrahedra in the Euclidean plane

A partition of a positive integer n is a way of writing n as a sum of positive integers and is denoted by the list of positive integers in the sum.

For example {2, 1} is a partition of 3.The partitions of 4 are {4}, {3, 1}, {2, 2}, {2, 1, 1},

{1, 1, 1, 1}.

Activities

1. List the partition of 3.

2. If one interprets the partition {1, 1, 1} as a triangle with unequal length sides (scalene triangles) what type of triangle do the other partitions of 3 represent?

Nets of a tetrahedron are intuitively two dimensional patterns that can be folded to form a three dimensional figure. We can study nets of polygons, one dimension down from the usual discussion.

Figure 1.44:Traiangle

If i cut at A we will get ABCA 3, 2, 4 = 4, 2, 3 = ACBA If i cut at B we will get BCAB 2, 4, 3 = 3, 4, 2 = BACB if If i cut at C we will get CBAC 2, 3, 4 = 4, 3, 2 = CABC

Thus , a triangle which is scalene will have three different “segment” nets . 213

3. Is it possible to have six stick lengths each triple of which obeys the strict triangles inequality but for which one can’t make a tetrahedron?

4. If we cut ABC at each vertex, how many “nets” of a triangle are there? If the length of three sides are different, if the length of two sides are equal and the third side is different, if the length of three sides are equal. List all of them.

A sextuple S= (a, b, c, d, e, f) is a degenerate tetrahedron if and only if the McCrean determinant of S is zero and S satisfies the triangular inequality (the sum of the two sides of a triangle is greater or equal to the third side). Degenerate tetrahedra exist for each of the partition {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1, 1}, {2, 2, 2}, {2, 2, 1, 1}, {2, 1, 1, 1, 1},

{1, 1, 1, 1, 1, 1}.

Activities

1. How many partitions are there?

a) For {6}

b) For {4}

c) List all partitions of 6 and 4 numbers

2. How many different subsets of size three are there for six different numbers? What happen when repeats are allowed?

3. If one has four six sticks of different lengths and two lengths the same, how many different triples can one form?

4. If one has three of six sticks of different lengths and three lengths are the same, how many triples can one form? 214

5. Four points determine six different Euclidean distances. For partitions of six can we find a collection of four points (no three of them are collinear) forming a quadrilateral in a plane which corresponds to the partitions. Draw a quadrilateral for each partition which can be exists?

6. Explain how one can interpret a plane quadrilateral as a degenerate tetrahedron.

7. Give example of convex quadrilateral for each partition type of with integer lengths are there.

8. What are the non-congruent two distinct distances sets formed by 4 points in the Euclidean plane?

Examples:

2 distance set 2 distance set

2 distance set 215

3 distance set

3 distance

Figure 1.45: Degenerate tetrahedra 216

Appendix D

Unit III: Nets of tetrahedra

Nets of tetrahedron are obtained by cutting three edges of the tetrahedron at a vertex of the tetrahedron or along a sequence of three edges that visit each vertex exactly once and flattening out the tetrahedron.

Activities

1. How many types of nets can one find for a given tetrahedron?

2. How many nets can one find for partition {5, 1}? List the nets.

3. How many nets of tetrahedron are there for partition (4, 2)? List all the nets?

4. If we switch a pair of edges in a matching edge one gets a dual tetrahedron. Do these tetrahedra have the same nets? List the net?

5. If two paths of a tetrahedron have the same letters then do they have the same /different path? Can you give an example for partition {2, 1, 1, 1, 1}. 217

Appendix E

Unit IV: Classification of tetrahedron in 3D

Tetrahedron lie in one of 11 classes and these classes exist as 3D type but not all as degenerate

2D types. Tetrahedra (6 edges ) can be classified into eleven classes by partition type in terms of their edges {6}, {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1, 1}, {2, 2, 2}, {2, 2, 1, 1}, {2, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}.

Activity

1. List all partitions of the number six.

2. Draw a diagram to represent a tetrahedron in a plane.

3. How many faces, edges and vertices are there for a given tetrahedron? List all the edges faces and vertices of the tetrahedron.

4. a) What is the smallest series of consecutive positive integers where every triple satisfies the strict triangle inequality interpreting the integers as side lengths? b) Does every triple from {1, 2, 3, 4} obey strict triangle inequality?

c) Which triple from {3, 4, 5, 6, 7, 8} don’t obey the strict triangle inequality ?

5. How can one tell if two tetrahedron with the same length of six edges are congruent or different?

6. Can one may be a tetrahedron with the edge lengths {1, 2, 3, 4, 5, 6} for partition

{1, 1, 1, 1, 1, 1}? Can one label a plane drawing of a tetrahedron with these numbers so that the faces obey the triangular inequality?

7. What is the significance of a negative value of the McCrea determinant of a tetrahedron with a given set of edge lengths?

8. For each partition type of tetrahedron how many equilateral, isosceles and scalene triangles are there? List all the partitions and types triangles. 218

9. If S is a sextuple for which a tetrahedron does exist when does the monotone up and monotone down series exist with integer length? Take one tetrahedron with integer length and list all the monotone up and monotone down series.

10. Does the set S = {6, 7, 8, 9, 10, 11} generate all 30 scalene tetrahedra?