ON the ARCHIMEDEAN SOLIDS a THESIS Presented to the University

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ON THE ARCHIMEDEAN SOLIDS

A THESIS

Presented to the University Honors Program

California State University, Long Beach

In Partial Fulfillment

of the Requirements for the

University Honors Program Certificate

Vivian Tran

Fall 2017 I, THE UNDERSIGNED MEMBER OF THE COMMITTEE,

HAVE APPROVED THIS THESIS

ON THE ARCHIMEDEAN SOLIDS

Vivian Tran

______

Scott Crass, Ph.D. Mathematics

California State University, Long Beach

Fall 2017

ABSTRACT

ON THE ARCHIMEDEAN SOLIDS

Vivian Tran

December 2017

Some of the Archimdean solids were initially discovered by Archimedes, and others were discovered by thinkers including Plato and Pappus. Johannes Kepler rediscovered the complete set of the Archimedean solids, and provided a geometric proof that there is a limited number of the solids. The Archimedean solids are also known as semiregular polyhedra because each of the consists of more than one type of regular polygon face, so that each face is equiangular and equilateral. Furthermore, these polyhedra have the same pattern of faces at every vertex. Due to their unique characteristics, there can be only thirteen of this type of polyhedron. Currently, the only account of the Archimedean solids is Kepler’s geometric proof. So, there are very few proofs of why there are only thirteen of the Archimedean solids, especially a computational one, which is the objective of this thesis.

ACKNOWLEDGEMENTS

I would like to thank Dr. Scott Crass for his enormous amount of help and guidance during this past year. Thank you for teaching me how to use Mathematica to make this thesis possible. I really would not have been able to create this without his support.

iii TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ...... iii

CHAPTER

1. INTRODUCTION ...... 1

Basic Terminology ...... 1 Types of Regular Polyhedra ...... 1 Previous Rediscovery of the Archimedean Solids ...... 4

2. LITERATURE REVIEW ...... 5

Kepler’s Theorems and Lemmas ...... 5 Geometric Rediscovery of the Archimedean Solids ...... 6

3. METHODOLOGY ...... 8

Introduction ...... 8 Filter 1: Euler Characteristic ...... 8 Filter 2: Gauss-Bonnet Theorem ...... 9 Algorithm Explanation ...... 10

4. Results ...... 12

5. Conclusion ...... 13

WORKS CITED ...... 14

CHAPTER 1

INTRODUCTION

Basic Terminology

Before exploring the topic, there are a few terms that need to be explained. First, a polyhedron is a closed three-dimensional shape made of flat polygonal faces. For example, the cube is considered a polyhedron; whereas, the cylinder is not one because it is constructed with a curved face. There are three main characteristics of each polyhedron: vertices, edges, and faces.

An edge is the line segment that two faces share while a vertex is the point where edges coincide.

There are also two types of angle associated with a polyhedron: plane angle and solid angle. The plane angle is the interior angle of a face and is used to describe two-dimensional shapes. The solid angle is the sum of all plane angles at a vertex. To form a solid angle, there must be at least three faces and it is known that the solid angle must have a sum less than 360°. The polyhedra that are the focus of this paper have faces that are regular polygons. Regular means that the polygons are equiangular—interior angles are equal—and equilateral—all sides are the same length. A semiregular polyhedron has regular faces, but there are more than one polygon type of faces. Although regularity is already a restricting characteristic of polyhedra, there are different types of regular polyhedra.

Types of Regular Polyhedra

There are several classes of polyhedra but the most familiar of which are pyramids, prisms, antiprisms, the Platonic solids, and the Archimedean solids. Pyramids are constructed by one polygonal base that is surrounded triangles. The triangles connect at a vertex called the apex.

Pyramids are named after the base polygon. So, a pyramid with a square base is called a square pyramid. With these solids, there are infinitely many pyramids, because a polygon can have an

arbitrary number of sides. Pyramids with equilateral triangles and regular bases are not considered as Archimedean solids, because they do not hold the same pattern of face type at every vertex. The apex is surrounded by only triangle faces, but the vertices connected to the base are surrounded by two triangles and the base face. So, semiregular pyramids are not

Archimedean solids.

Two more families of solids are prisms and anti-prisms. Instead of having one base, prisms and anti-prisms contain two bases of the same polygon type. In a prism, the two n-gon bases are connected by quadrilaterals, while in an antiprism, two n-gon bases are connected by triangles. So, prisms consist of two n-sided polygons and n rectangles. Antiprisms also have two n-gon bases, but have 2n triangles. As with the pyramids, prisms and anti-prisms are named after the base polygons. So, a prism with pentagonal bases is called a pentagonal prism. The antiprism with the same base would be called the pentagonal antiprism. Similar to pyramids, antiprisms and prisms are an infinite set since the polygonal base can have an arbitrary number of sides and can be regular by using regular polygon bases and equilateral triangles and squares.

The Platonic solids form a very special family of polyhedra. These polyhedra are constructed with only one type of regular polygon face. So, all the faces on a Platonic solid are congruent. Due to their special properties, there are only five of these solids: tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron. Euclid’s Elements provides a proof that there are only five Platonic solids. Since the Platonic solids have only one type of regular polygon face, the plane angles of each face are equal. Starting with equilateral triangles, which have a plane angle of 60°, there can only be three, four, or five of these at a solid angle. There cannot be more than six triangles at a vertex, because the sum of six equilateral triangles is already 360°. Three, four, and five equilateral triangles at a vertex, create the tetrahedron,

octahedron, and the icosahedron respectively. The tetrahedron is comprised of a total of four equilateral triangles and is also classified as a triangular pyramid. The octahedron has eight equilateral triangles and the icosahedron has 20 equilateral triangles. By increasing the number of sides by one, the next regular polygon would be a square, which has an interior angle of 90°.

So, there can only be three squares at a vertex because four squares create a flat plane. The result is the hexahedron, more familiarly known as the cube. It consists of six square faces and is also identified as a prism. Following the square would be a regular pentagon with an intended angle of 108°. This means that there can be only one solid made of regular pentagons since using four or more regular pentagons results in an angle sum larger than 360°. This is the dodecahedron, containing twelve regular pentagonal faces. When using three regular hexagons, the angle sum is exactly 360 degrees; so, there cannot be a regular polyhedron with only regular hexagons. More than six sides on a face will produce an angle sum for three faces that is greater than 360° since the interior angle increases as the number of sides increases. Thus, there can be only five

Platonic solids. One major difference between this class of solids and the ones mentioned earlier is that the Platonic solids are classified by one regular polygon type of face and every vertex and edge are identical in face pattern. Prisms and antiprisms are not Platonic solids because they contain more than one type of polygon face.

We now turn to the focus of this paper: the Archimedean solids. There are two differentiating characteristics possessed by Archimedean solids: they have more than one type of regular polygon face and every vertex has the same arrangement of faces. These solids have many similarities with the Platonic solids, prisms, and antiprisms. However, Platonic solids are not considered Archimedean polyhedra since Platonic solids have only one type of regular polygon face. Also, according to Kepler, the Archimedean solids differ from prisms and

antiprisms because prisms and antiprisms are more disk-like even though both families of regular polyhedra can be circumscribed in a sphere. So, excluding the prisms and antiprisms, there are only thirteen Archimedean solids. As in the case of the Platonic solids, there are several geometric methods used to discover the Archimedean solids.

Previous Rediscovery of the Archimedean Solids

During the Renaissance, a number of mathematicians and artists rediscovered the

Archimedean solids. They used two techniques: truncating and chamfering the Platonic solids and nets of the Archimedean solids. Truncation is the process of cutting off all the vertices to create another regular polygonal face while chamfering involves slicing off the edges to create regular polygonal faces. Combining these methods, Jesus Suarez, Enrique Gancedo, Jose Manual

Alvarez, and Antonio Moran produced all of the Archimedean solids except the snub cube and the snub dodecahedron. Another way the Archimedean solids were discovered was through nets.

A net results from cutting along some of the edges to flatten a solid into a two-dimension map.

This method was used during the Renaissance by one of Kepler’s students whose identity is unknown. Using nets, the unknown student was able to rediscover all the Archimdean solids that truncating and chamfering found, but also found the snub cube. So, in both methods, the snub dodecahedron remains unfound. In Polyhedra, Peter Cromwell constructs all of the Archimedean solids using a geometric method—which will be explained in more detail later. The question here is: can all of the Archimedean solids be found using a computational algorithm and can it show that no other polyhedra can be Archimedean?

CHAPTER 2

LITERATURE REVIEW

Kepler’s Theorems and Lemmas

Using Kepler’s theorems and lemmas, Cromwell provides a geometric method to rediscover all thirteen of the Archimedean solids. Kepler’s approach draw on the idea that all

Archimedean solids have the same pattern of polygonal faces at every vertex and that solid angles must be less than 360°. The condition that every vertex has the same pattern of faces refers to the fact that the different polygon faces are organized around a vertex in a way that is identical for all vertices. So, the vertices cannot be differentiated from each other. Moreover, the solid angle at every vertex must be less than 360°. We start with Kepler’s first lemma.

Lemma 1: At most three different types of faces can surround a vertex of a spherical

polyhedron.

Proof of Lemma 1: Suppose there are four different regular polygonal faces at a vertex.

The minimal case of this would be one triangle, one square, one pentagon, and one hexagon at a vertex. Their interior angle measurements are 60°, 90°, 108°, and 120°. This results in a solid angle of 378°. But this contradicts the condition that a solid angle must be less than 360°.

Therefore, an Archimedean polyhedron can have at most three different types of regular polygonal faces. (footnote: this symbol marks the end of a proof)

Now, we consider Kepler’s second lemma.

Lemma 2: A polyhedron that has the same pattern of polygon faces at every vertex

cannot have these types of solid angles:

i) A solid angle surrounded by three faces—a-gon, b-gon, and c-gon—where a is

odd and b ≠ c. See Figure 2.1.

ii) A solid angle surrounded by four faces with at least one 3-gon and a b-gon

opposite of it where a ≠ c.” (Cromwell 159). See Figure 2.2 b

b c a b a c

a 3 c b ? b

Figure 2.1 Figure 2.2 Figure 2.3

Proof of Lemma 2: For part the first part, suppose a is odd. Since every solid angle must have the same pattern, b-gons and c-gons must alternate around a. But this is a contradiction because a is odd, b-gons and c-gons cannot alternate around a. Thus, a polyhedron with the same pattern of polygon faces at every vertex cannot have a solid angle where b ≠c when a is odd.

For the second case, suppose at each solid angle we have a 3-gon and a b-gon directly opposite of each other and every vertex has the same pattern. (See Figure 2.3) Between the 3-gon and b-gon, a-gons and c-gons must alternate. But this leads to a contradiction. Thus, in a polyhedron with the same pattern at every vertex cannot have a solid angle where a ≠ c there is a

3-gon and a b-gon opposite of each other. þ

Geometric Rediscovery of the Archimedean Solids

Based on Kepler’s original proof, Cromwell provides a similar geometric argument for the claim that there are only thirteen Archimedean solids. Using the two lemmas above,

Cromwell proves that there are only thirteen polyhedra—other than prisms and antiprisms—with the same pattern of faces at every vertex. Essentially, Cromwell tests every possible pattern around a vertex to test whether or not it can be repeated at every vertex. Since Archimedean solids must have at least two different faces at a vertex and each polygon face has at least three sides, the starting point of the procedure is the case with only 3-gons and 4-gons at a vertex. It

begins with one 4-gon and the maximum number of 3-gons that can be around a vertex without violating the solid angle condition—the angle sum must be less than 360°. The number of 3-gons decreases with every case until all possible cases with only one 4-gon is tested. Each case is tested using Lemma 1 and 2. If any case contradicts the lemmas, then there is no possible solid with the particular pattern around a vertex.

Next, the number of 4-gons is increased by one and the same process of decreasing the number of 3-gons is repeated. This process is repeated until there are no more cases of 3-gons and 4-gons. Then, the 4-gon is changed to a 5-gon and the lemma tests repeat for all the cases with only 3-gons and 5-gons. This process continues until all the combinations of only 3-gons and n-gons are tested and n stops increasing when the solid angle condition, Lemma 1, or

Lemma 2 is no longer met for all future cases after a certain n—in the case with only 3-gons and one other polygon, n stops increasing after 12. The procedure continues until all combinations of two types of face are tested. The analogous process is applied to all possible combinations of three different types of polygon faces. In the end, there are only thirteen cases that satisfy the lemmas. Therefore, there can be only thirteen Archimedean solids.

CHAPTER 3

METHODOLOGY

Introduction

Similar to the structure of Cromwell’s geometric proof that there are only thirteen

Archimedean solids, my approach is to list possible numerical conditions associated with polyhedra in order to test two computational conditions that any polyhedron must satisfy: the

Euler Characteristic and the Gauss-Bonnet theorem. The goal is to determine which numerical conditions can result in an Archimedean solid. The numerical data set associated with a prospective polyhedron has the form (X, Y, Z, a, b, c, v) where a, b, and c indicate which type of polygon face the solid will have. For example, if a, b, or c is assumed to be three, then one of the polygon face-types would be a triangle. X, Y, and Z refer the total of number of the polygon faces of each respective type. In other words, the number of a-gons is X, the number of b-gons is Y, and the number of c-gons is Z. Lastly, v is the number of vertices. Since there are thousands of possible combinations, a mathematics coding program, Mathematica, is used to apply the two numerical tests to every combination. I will refer to these tests as filters. Only the combinations that pass through both filters can correspond to an Archimedean polyhedron.

Filter 1: Euler Characteristic

A basic characteristic of all polyhedra that close up like a sphere is the Euler

Characteristic. Euler proved that for all spherical polyhedra, � − � + � = 2 where V is the number of vertices, E is the number of edges, and F is the number of faces. As an example, the icosidodecahedron, one of the Archimedean Solids, has 30 vertices, 60 edges, and 32 faces.

Applying the Euler Characteristic results in � − � + � = 30 − 60 + 32 = 2. If any combination

(V, E, F) does not satisfy the Euler Characteristic, then there is no polyhedron with that specific

number of vertices, edges, and faces. For example, say the assumed values are � = 22, � = 42, and � = 20. Plugging into Euler’s Characteristic produces � − � + � = 22 − 42 + 20 = 0.

Thus, there is no polyhedron with 22 vertices, 42 edges, and 20 faces. So, every possible data set will go through the Euler Characteristic as the first test.

From the numerical conditions, the number of edges and faces can be found using a formula, which is implemented into the Mathematica code and incorporated into the Euler

Characteristic test. To find the number of faces, the number of each polygon face-type is summed. So, the formula is � = � + � + �. Finding the number of edges is somewhat more complex. For a-gons, b-gons, and c-gons, there are a number of sides, b number of sides, and c number of sides respectively. Since there are X of a-gons, Y of b-gons, and Z of c-gons, then there are �� + �� + �� number of sides. But, one edge is shared by two faces. Thus, �� + �� +

��, over-counts the number of edges by twice the amount. So, the number of edges is given by the formula � = !"!!"!!" . V is already given in the numerical condition, so the Euler !

Characteristic can finally be used. After all the possible numerical conditions are subjected to the first test, only the numerical conditions that resulted in two for � − � + � can move to the next test, the Gauss-Bonnet theorem.

Gauss-Bonnet Theorem

As with the Euler Characteristic, the Gauss-Bonnet theorem holds for all polyhedra. This theorem states that the sum of all the angle deficiencies is equal to the surface area of the unit sphere, which equals 4π. Angle deficiency at one vertex is the difference between 360° or 2π in radians and the sum of the plane angle measurements, or the solid angle measurement at a single

! vertex. So, the formula is 4� = !!!(2� − �!), where v is the number of vertices and �! is the solid angle measurement at the ith vertex. Since the Archimedean solids are the goal of this

algorithm, it is assumed that the angle sum of every vertex is the same. This is true because every vertex in an Archimedean solid is constructed with the same pattern of regular polygon faces. So, take the difference of 2π and the solid angle measurement and multiply by v to find the angle deficiency. The Gauss-Bonnet theorem is simplified to 4� = � 2� − �� , where AS is the solid angle measurement.

Now, the crucial part is to find the number of each polygon face type that appears at any individual vertex from the given assumptions in the numerical conditions. This is found by multiplying the number of vertices of one polygon—represented by a, b, and c in the numerical condition—by the total number of each polygon face type in the polyhedra—represented by X, Y, and Z—and dividing by the number of vertices, v, of the whole polyhedra. So the number of a- gons at one vertex, expressed by Q, is � = !". Similarly for b-gons and c-gons, the formulas are !

� = !" and � = !" respectively. The measurement of the interior angle of the regular polygon is ! ! found by !!! � where n is the number of sides of the polygon. So the interior angle ! measurements for a-gons, b-gons, and c-gons are found and represented by � = !!! �, !

� = !!! �, and � = !!! �, respectively. Combining the new equations creates the formula to ! ! find the solid angle measurement: �� = �×� + �×� + (�×�). These formulas are integrated in the code on Mathematica to create the second test for the data sets that passed the

Euler Characteristic test. In the end, the code creates a list of candidates for Archimedean polyhedra that satisfy the Euler Characteristic and the Gauss-Bonnet conditions.

Algorithm Explanation

The approach taken by this algorithm is to input the possible data sets

(X, Y, Z, a, b, c, v) into the filters to produce a list that corresponds to Archimedean solids. The smallest number of faces to form a polyhedron is four because three faces are needed to form a solid angle and the last face is to connect the bottom of the three faces. This can be formed with only one type of face. So, since X, Y, and Z refer to the number of faces of each polygon type, only one has to have a lower bound of four. The rest have a lower bound of zero, since the minimal case only uses one type of polygon face. So, let the lower bound of X be four, and let the lower bound of Y and Z be 0. In the data set, a, b, and c refer to the type of faces, and express the number of sides or angles. Since the smallest number of sides a polygon can have is three, the lower bound of one must be three. Let three be the lower bound of a. We do not want to repeat the same number of sides so, the lower bound for b is a+1 and the lower bound of c is a+2. This ensures that the number of sides are not repeated for a, b, and c. For the number of vertices, v, the smallest polyhedron formed is a polyhedron with four triangle faces. Counting the number of vertices, there are only four. So, the lower bound for the number of vertices is four. With all the lower bounds, the smallest data set which is (4, 0, 0, 3, 4, 5, 6, 4). The procedure starts here. The algorithm passes through a loop in which each parameter changes one at a time, increasing by one until a list all possible data sets are created. Then, list of data sets passes through the Euler characteristic filter. Only those that pass the test will move onto a new list that will be tested by the Gauss-Bonnet condition. After the second list goes through the second filter, the code creates a final list of data set that pass through the Gauss-Bonnet and the Euler characteristic.

CHAPTER 4

RESULTS

After all the possible data sets have been input into the two filters, the code produces a list of data sets that pass the Euler Characterisitc and Gauss-Bonnet theorem tests. Ideally, this would produce a list of only Archimedean solids. Unfortunately, the code does not take into account that the Archimedean solids must have the same pattern at every vertex. So, some of the sets that pass through both the tests are not those of Archimedean solids. Even though the lists of numerical conditions are not all Archimedean solids, the initial list of possible numerical conditions for Archimedean solids decreases significantly.

CHAPTER 5

CONCLUSION

Even though the desired goal for this thesis was not reached, the list of possible combinations of numerical parameters for Archimedean solids greatly decreased. This thesis begins a computational approach to the determination of the Archimedean solids as opposed to

Kepler’s more geometric reasoning. The code was able to decrease the amount of possible numerical conditions using the Euler Characteristic and the Gauss-Bonnet theorem. If a data set does not satisfy both tests, then they do not correspond to a polyhedron. Additionally, the Gauss-

Bonnet theorem was applied using the assumption that all the vertices were constructed by the same amount of polygon face types, but did not specify the pattern in which the polygon faces appeared, a crucial characteristic of the Archimedean solids. Thus, a data set that passes the tests can correspond to more than one Archimedean solid. More research can be done to finish the computational approach to rediscover the Archimedean Solids.

WORKS CITED

Cromwell, Peter R. Polyhedra. Cambridge University Press. 1997.

Schreiber, Peter, et. al. “New Light on the Rediscovery of the Archimedean solids during

the Renaissance.” Archive for History of Exact Sciences, vol. 62, no. 4, July 2008, pp.

457-467.

Suárez, Jesús, et. al. “Truncating and Chamfering Diagrams of Regular Polyhedra.” Journal

of Mathematical Chemistry, vol. 46, no. 1, July 2009, pp. 155-163.

Sutton, Daud. Platonic & Archimedean Solids. Walker Publishing Company, Inc. 2002.