Some Additional Notes on Polyhedra Platonic Solids a Platonic Or Regular
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Equivelar Octahedron of Genus 3 in 3-Space
Equivelar octahedron of genus 3 in 3-space Ruslan Mizhaev ([email protected]) Apr. 2020 ABSTRACT . Building up a toroidal polyhedron of genus 3, consisting of 8 nine-sided faces, is given. From the point of view of topology, a polyhedron can be considered as an embedding of a cubic graph with 24 vertices and 36 edges in a surface of genus 3. This polyhedron is a contender for the maximal genus among octahedrons in 3-space. 1. Introduction This solution can be attributed to the problem of determining the maximal genus of polyhedra with the number of faces - . As is known, at least 7 faces are required for a polyhedron of genus = 1 . For cases ≥ 8 , there are currently few examples. If all faces of the toroidal polyhedron are – gons and all vertices are q-valence ( ≥ 3), such polyhedral are called either locally regular or simply equivelar [1]. The characteristics of polyhedra are abbreviated as , ; [1]. 2. Polyhedron {9, 3; 3} V1. The paper considers building up a polyhedron 9, 3; 3 in 3-space. The faces of a polyhedron are non- convex flat 9-gons without self-intersections. The polyhedron is symmetric when rotated through 1804 around the axis (Fig. 3). One of the features of this polyhedron is that any face has two pairs with which it borders two edges. The polyhedron also has a ratio of angles and faces - = − 1. To describe polyhedra with similar characteristics ( = − 1) we use the Euler formula − + = = 2 − 2, where is the Euler characteristic. Since = 3, the equality 3 = 2 holds true. -
Archimedean Solids
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat). -
Regular Polyhedra of Index 2
REGULAR POLYHEDRA OF INDEX 2 A dissertation presented by Anthony Cutler to The Department of Mathematics In partial fulfillment of the requirements for the degree of Doctor of Philosophy in the field of Mathematics Northeastern University Boston, Massachusetts April, 2009 1 REGULAR POLYHEDRA OF INDEX 2 by Anthony Cutler ABSTRACT OF DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate School of Arts and Sciences of Northeastern University, April 2009 2 We classify all finite regular polyhedra of index 2, as defined in Section 2 herein. The definition requires the polyhedra to be combinatorially flag transitive, but does not require them to have planar or convex faces or vertex-figures, and neither does it require the polyhedra to be orientable. We find there are 10 combinatorially regular polyhedra of index 2 with vertices on one orbit, and 22 infinite families of combinatorially regular polyhedra of index 2 with vertices on two orbits, where polyhedra in the same family differ only in the relative diameters of their vertex orbits. For each such polyhedron, or family of polyhedra, we provide the underlying map, as well as a geometric diagram showing a representative face for each face orbit, and a verification of the polyhedron’s combinatorial regularity. A self-contained completeness proof is given. Exactly five of the polyhedra have planar faces, which is consistent with a previously known result. We conclude by describing a non-Petrie duality relation among regular polyhedra of index 2, and suggest how it can be extended to other combinatorially regular polyhedra. -
Shape Skeletons Creating Polyhedra with Straws
Shape Skeletons Creating Polyhedra with Straws Topics: 3-Dimensional Shapes, Regular Solids, Geometry Materials List Drinking straws or stir straws, cut in Use simple materials to investigate regular or advanced 3-dimensional shapes. half Fun to create, these shapes make wonderful showpieces and learning tools! Paperclips to use with the drinking Assembly straws or chenille 1. Choose which shape to construct. Note: the 4-sided tetrahedron, 8-sided stems to use with octahedron, and 20-sided icosahedron have triangular faces and will form sturdier the stir straws skeletal shapes. The 6-sided cube with square faces and the 12-sided Scissors dodecahedron with pentagonal faces will be less sturdy. See the Taking it Appropriate tool for Further section. cutting the wire in the chenille stems, Platonic Solids if used This activity can be used to teach: Common Core Math Tetrahedron Cube Octahedron Dodecahedron Icosahedron Standards: Angles and volume Polyhedron Faces Shape of Face Edges Vertices and measurement Tetrahedron 4 Triangles 6 4 (Measurement & Cube 6 Squares 12 8 Data, Grade 4, 5, 6, & Octahedron 8 Triangles 12 6 7; Grade 5, 3, 4, & 5) Dodecahedron 12 Pentagons 30 20 2-Dimensional and 3- Dimensional Shapes Icosahedron 20 Triangles 30 12 (Geometry, Grades 2- 12) 2. Use the table and images above to construct the selected shape by creating one or Problem Solving and more face shapes and then add straws or join shapes at each of the vertices: Reasoning a. For drinking straws and paperclips: Bend the (Mathematical paperclips so that the 2 loops form a “V” or “L” Practices Grades 2- shape as needed, widen the narrower loop and insert 12) one loop into the end of one straw half, and the other loop into another straw half. -
VOLUME of POLYHEDRA USING a TETRAHEDRON BREAKUP We
VOLUME OF POLYHEDRA USING A TETRAHEDRON BREAKUP We have shown in an earlier note that any two dimensional polygon of N sides may be broken up into N-2 triangles T by drawing N-3 lines L connecting every second vertex. Thus the irregular pentagon shown has N=5,T=3, and L=2- With this information, one is at once led to the question-“ How can the volume of any polyhedron in 3D be determined using a set of smaller 3D volume elements”. These smaller 3D eelements are likely to be tetrahedra . This leads one to the conjecture that – A polyhedron with more four faces can have its volume represented by the sum of a certain number of sub-tetrahedra. The volume of any tetrahedron is given by the scalar triple product |V1xV2∙V3|/6, where the three Vs are vector representations of the three edges of the tetrahedron emanating from the same vertex. Here is a picture of one of these tetrahedra- Note that the base area of such a tetrahedron is given by |V1xV2]/2. When this area is multiplied by 1/3 of the height related to the third vector one finds the volume of any tetrahedron given by- x1 y1 z1 (V1xV2 ) V3 Abs Vol = x y z 6 6 2 2 2 x3 y3 z3 , where x,y, and z are the vector components. The next question which arises is how many tetrahedra are required to completely fill a polyhedron? We can arrive at an answer by looking at several different examples. Starting with one of the simplest examples consider the double-tetrahedron shown- It is clear that the entire volume can be generated by two equal volume tetrahedra whose vertexes are placed at [0,0,sqrt(2/3)] and [0,0,-sqrt(2/3)]. -
Binary Icosahedral Group and 600-Cell
Article Binary Icosahedral Group and 600-Cell Jihyun Choi and Jae-Hyouk Lee * Department of Mathematics, Ewha Womans University 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-3277-3346 Received: 10 July 2018; Accepted: 26 July 2018; Published: 7 August 2018 Abstract: In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions H via spin map of SO(3). In addition, we show that the binary icosahedral group in H is the set of vertices of a 600-cell by applying the Coxeter–Dynkin diagram of H4. Keywords: binary polyhedral group; icosahedron; dodecahedron; 600-cell MSC: 52B10, 52B11, 52B15 1. Introduction The classification of finite subgroups in SLn(C) derives attention from various research areas in mathematics. Especially when n = 2, it is related to McKay correspondence and ADE singularity theory [1]. The list of finite subgroups of SL2(C) consists of cyclic groups (Zn), binary dihedral groups corresponded to the symmetry group of regular 2n-gons, and binary polyhedral groups related to regular polyhedra. These are related to the classification of regular polyhedrons known as Platonic solids. There are five platonic solids (tetrahedron, cubic, octahedron, dodecahedron, icosahedron), but, as a regular polyhedron and its dual polyhedron are associated with the same symmetry groups, there are only three binary polyhedral groups(binary tetrahedral group 2T, binary octahedral group 2O, binary icosahedral group 2I) related to regular polyhedrons. -
Polyhedral Harmonics
value is uncertain; 4 Gr a y d o n has suggested values of £estrap + A0 for the first three excited 4,4 + 0,1 v. e. states indicate 11,1 v.e. för D0. The value 6,34 v.e. for Dextrap for SO leads on A rough correlation between Dextrap and bond correction by 0,37 + 0,66 for the'valence states of type is evident for the more stable states of the the two atoms to D0 ^ 5,31 v.e., in approximate agreement with the precisely known value 5,184 v.e. diatomic molecules. Thus the bonds A = A and The valence state for nitrogen, at 27/100 F2 A = A between elements of the first short period (with 2D at 9/25 F2 and 2P at 3/5 F2), is calculated tend to have dissociation energy to the atomic to lie about 1,67 v.e. above the normal state, 4S, valence state equal to about 6,6 v.e. Examples that for the iso-electronic oxygen ion 0+ is 2,34 are 0+ X, 6,51; N2 B, 6,68; N2 a, 6,56; C2 A, v.e., and that for phosphorus is 1,05 v.e. above 7,05; C2 b, 6,55 v.e. An increase, presumably due their normal states. Similarly the bivalent states to the stabilizing effect of the partial ionic cha- of carbon, : C •, the nitrogen ion, : N • and racter of the double bond, is observed when the atoms differ by 0,5 in electronegativity: NO X, Silicon,:Si •, are 0,44 v.e., 0,64 v.e., and 0,28 v.e., 7,69; CN A, 7,62 v.e. -
Four-Dimensional Regular Polytopes
faculty of science mathematics and applied and engineering mathematics Four-dimensional regular polytopes Bachelor’s Project Mathematics November 2020 Student: S.H.E. Kamps First supervisor: Prof.dr. J. Top Second assessor: P. Kiliçer, PhD Abstract Since Ancient times, Mathematicians have been interested in the study of convex, regular poly- hedra and their beautiful symmetries. These five polyhedra are also known as the Platonic Solids. In the 19th century, the four-dimensional analogues of the Platonic solids were described mathe- matically, adding one regular polytope to the collection with no analogue regular polyhedron. This thesis describes the six convex, regular polytopes in four-dimensional Euclidean space. The focus lies on deriving information about their cells, faces, edges and vertices. Besides that, the symmetry groups of the polytopes are touched upon. To better understand the notions of regularity and sym- metry in four dimensions, our journey begins in three-dimensional space. In this light, the thesis also works out the details of a proof of prof. dr. J. Top, showing there exist exactly five convex, regular polyhedra in three-dimensional space. Keywords: Regular convex 4-polytopes, Platonic solids, symmetry groups Acknowledgements I would like to thank prof. dr. J. Top for supervising this thesis online and adapting to the cir- cumstances of Covid-19. I also want to thank him for his patience, and all his useful comments in and outside my LATEX-file. Also many thanks to my second supervisor, dr. P. Kılıçer. Furthermore, I would like to thank Jeanne for all her hospitality and kindness to welcome me in her home during the process of writing this thesis. -
Regular Polyhedra Through Time
Fields Institute I. Hubard Polytopes, Maps and their Symmetries September 2011 Regular polyhedra through time The greeks were the first to study the symmetries of polyhedra. Euclid, in his Elements showed that there are only five regular solids (that can be seen in Figure 1). In this context, a polyhe- dron is regular if all its polygons are regular and equal, and you can find the same number of them at each vertex. Figure 1: Platonic Solids. It is until 1619 that Kepler finds other two regular polyhedra: the great dodecahedron and the great icosahedron (on Figure 2. To do so, he allows \false" vertices and intersection of the (convex) faces of the polyhedra at points that are not vertices of the polyhedron, just as the I. Hubard Polytopes, Maps and their Symmetries Page 1 Figure 2: Kepler polyhedra. 1619. pentagram allows intersection of edges at points that are not vertices of the polygon. In this way, the vertex-figure of these two polyhedra are pentagrams (see Figure 3). Figure 3: A regular convex pentagon and a pentagram, also regular! In 1809 Poinsot re-discover Kepler's polyhedra, and discovers its duals: the small stellated dodecahedron and the great stellated dodecahedron (that are shown in Figure 4). The faces of such duals are pentagrams, and are organized on a \convex" way around each vertex. Figure 4: The other two Kepler-Poinsot polyhedra. 1809. A couple of years later Cauchy showed that these are the only four regular \star" polyhedra. We note that the convex hull of the great dodecahedron, great icosahedron and small stellated dodecahedron is the icosahedron, while the convex hull of the great stellated dodecahedron is the dodecahedron. -
What Is a Polyhedron?
3. POLYHEDRA, GRAPHS AND SURFACES 3.1. From Polyhedra to Graphs What is a Polyhedron? Now that we’ve covered lots of geometry in two dimensions, let’s make things just a little more difficult. We’re going to consider geometric objects in three dimensions which can be made from two-dimensional pieces. For example, you can take six squares all the same size and glue them together to produce the shape which we call a cube. More generally, if you take a bunch of polygons and glue them together so that no side gets left unglued, then the resulting object is usually called a polyhedron.1 The corners of the polygons are called vertices, the sides of the polygons are called edges and the polygons themselves are called faces. So, for example, the cube has 8 vertices, 12 edges and 6 faces. Different people seem to define polyhedra in very slightly different ways. For our purposes, we will need to add one little extra condition — that the volume bound by a polyhedron “has no holes”. For example, consider the shape obtained by drilling a square hole straight through the centre of a cube. Even though the surface of such a shape can be constructed by gluing together polygons, we don’t consider this shape to be a polyhedron, because of the hole. We say that a polyhedron is convex if, for each plane which lies along a face, the polyhedron lies on one side of that plane. So, for example, the cube is a convex polyhedron while the more complicated spec- imen of a polyhedron pictured on the right is certainly not convex. -
3. Linear Programming and Polyhedral Combinatorics Summary of What Was Seen in the Introductory Lectures on Linear Programming and Polyhedral Combinatorics
Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the introductory lectures on linear programming and polyhedral combinatorics. Definition 3.1 A halfspace in Rn is a set of the form fx 2 Rn : aT x ≤ bg for some vector a 2 Rn and b 2 R. Definition 3.2 A polyhedron is the intersection of finitely many halfspaces: P = fx 2 Rn : Ax ≤ bg. Definition 3.3 A polytope is a bounded polyhedron. n n−1 Definition 3.4 If P is a polyhedron in R , the projection Pk ⊆ R of P is defined as fy = (x1; x2; ··· ; xk−1; xk+1; ··· ; xn): x 2 P for some xkg. This is a special case of a projection onto a linear space (here, we consider only coordinate projection). By repeatedly projecting, we can eliminate any subset of coordinates. We claim that Pk is also a polyhedron and this can be proved by giving an explicit description of Pk in terms of linear inequalities. For this purpose, one uses Fourier-Motzkin elimination. Let P = fx : Ax ≤ bg and let • S+ = fi : aik > 0g, • S− = fi : aik < 0g, • S0 = fi : aik = 0g. T Clearly, any element in Pk must satisfy the inequality ai x ≤ bi for all i 2 S0 (these inequal- ities do not involve xk). Similarly, we can take a linear combination of an inequality in S+ and one in S− to eliminate the coefficient of xk. This shows that the inequalities: ! ! X X aik aljxj − alk aijxj ≤ aikbl − alkbi (1) j j for i 2 S+ and l 2 S− are satisfied by all elements of Pk. -
Are Your Polyhedra the Same As My Polyhedra?
Are Your Polyhedra the Same as My Polyhedra? Branko Gr¨unbaum 1 Introduction “Polyhedron” means different things to different people. There is very little in common between the meaning of the word in topology and in geometry. But even if we confine attention to geometry of the 3-dimensional Euclidean space – as we shall do from now on – “polyhedron” can mean either a solid (as in “Platonic solids”, convex polyhedron, and other contexts), or a surface (such as the polyhedral models constructed from cardboard using “nets”, which were introduced by Albrecht D¨urer [17] in 1525, or, in a more mod- ern version, by Aleksandrov [1]), or the 1-dimensional complex consisting of points (“vertices”) and line-segments (“edges”) organized in a suitable way into polygons (“faces”) subject to certain restrictions (“skeletal polyhedra”, diagrams of which have been presented first by Luca Pacioli [44] in 1498 and attributed to Leonardo da Vinci). The last alternative is the least usual one – but it is close to what seems to be the most useful approach to the theory of general polyhedra. Indeed, it does not restrict faces to be planar, and it makes possible to retrieve the other characterizations in circumstances in which they reasonably apply: If the faces of a “surface” polyhedron are sim- ple polygons, in most cases the polyhedron is unambiguously determined by the boundary circuits of the faces. And if the polyhedron itself is without selfintersections, then the “solid” can be found from the faces. These reasons, as well as some others, seem to warrant the choice of our approach.