DOCSLIB.ORG

A Capstone Paper from Spring 2007

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Capstone Paper from Spring 2007 Adam Siege1 Senior Capstone Dr. Papantonopoulou Spring 2007 and their Rotational Groups I. Introduction The symmetry group for a physical object is the set of ways that object can be repositioned so that it maintains its symmetry, or looks the same. Any object of any dimension has such a group of symmetries. Some objects, like the letter J, have only one symmetry called the identity because the only way they maintain their symmetry is to remain in the same place. Other objects, like an equilateral triangle, have several symmetries because they can be repositioned by rotations or reflections and still maintain symmetry. The symmetries of any object form a mathematical group under symmetry. The set of three-dimensional objects called regular solids possess many beautiful symmetries and accordingly form interesting symmetry groups under their rotations. A Regular Solid, or Platonic solid, is a three-dimensional polyhedron such that each face is a regular polygon, all faces are isometric to each other, and all vertices are isometric to each other. For geometrical objects, isometric means congruent. Therefore every face of a regular solid has the same number of edges and every vertex connects to the same number of faces. It turns out that among all the three-dimensional objects that can possibly be created, only five regular solids can be formed. Each is named for its number of faces. The five regular solids are: 1. The Tetrahedron, a 4-sided solid with equilateral triangles as faces 2. The Hexahedron or Cube, a 6-sided solid with squares as faces 3. The Octahedron, an 8-sided solid with equilateral triangles as faces 4. The Dodecahedron, a 12-sided solid with regular pentagons as faces 5. The Icosahedron, a 20-sided solid with equilateral triangles as faces Figure 1.1 2. 4. 5. Anyv regular or non-regular polyhedron, which is any completely enclosed solid with regular or non-regular polygonal faces, has v vertices, e edges, and f faces. Euler 's Observation tells us that any polyhedron maintains the equality f + v - e = 2, which we note holds true for the regular solids in Table 1.2. For the regular solids, we know that each vertex is isometric to every other vertex. Therefore the same number of faces k meet at each vertex. We also know each face must be the same regular polygon, so each face as n sides. See Table 1.2 to see the properties of each regular solid. Table 1.2 Because of the symmetry created by a regular solid, we can find some equalities among the values of v, e,J; k, and n. Note that a regular n-gon has n vertices which implies each face of a regular solid touches n vertices. If we multiply the vertices of each face n by the number of facesf, we get a multiple of the number of vertices of the entire regular solid. We do not get the exact number v because each vertex is touching more than one face - each vertex is touching k faces. Therefore multiplying n-f will give us k times the number of vertices v. Hence, for all regular solids n .f = k .v . We can also count the edges in a similar manner. Each edge has one vertex on each end, so each edge touches two vertices. Therefore multiplying 2 by e will give us a multiple of the amount of vertices of the regular solid. Since each vertex touches k faces, it is also touching k edges. Hence the multiple of v will be k -v once again, and we conclude 2.e = k-v= n-f . 11. Proving How Many Regular Solids Exist It seems amazing that among all the 3-dimensional objects, only five regular solids exist. However we can prove that there are only five. Theorem: Only five regular solids exist. Proof: We found that any regular solid has the equality 2.e = k - v = n. f . Euler's Observation states f + v - e = 2 . Combining the equalities we get: 2.e 2.e 2en 2ek ekn -+--e=2 =-+---= 2 32en+2ek-ekn=2kn k n knknkn 1 1 1111 1 3-(2en+2ek-ekn)=-(2kn)z-+---=-. Note ->0 sinceno 2enk 2enk kn2e e 1111 111 polyhedron can have 0 or fewer edges. Thus - + - - - = - > 0 3 -+ - - - > 0 kn2e kn2 111 111 z- + - > - . Thus, any k and n combination where - + - > - will form a regular solid. kn2 kn2 Before we find such combinations, we must note the following restrictions: i.) First we note that n 2 3 since polygons must have at least 3 sides. ii.) Also note that k 2 3 in all cases. Clearly if k = 1, then the entire figure would be one sided, and hence two-dimensional. If all vertices only touched one side, there would only be one side. If k = 2 ,then two adjacent faces would be forced to lie in the same plane in order to keep the solid completely enclosed. Likewise the object would reduce to two dimensions. See Figure 2.1 for an illustration of this contradiction for 3,4,5, and 6 sided polygons, confirming k > 2 in all cases. Notice how if any of these sets of polygons were folded so that edges meet and form an enclosure, it would force adjacent faces to overlap. Figure 2.1 , ... iii.) Note that for any n, once we find some k" such that the combination of n and k" do not form a regular solid, then the combination of n and k for all k > k will also fail 111 to form a regular solid, by the following: Given k > k" where7 + - I-, it follows that kn2 iv.) Similarly, once we find some ii where we cannot form a regular solid for any k, then we cannot form a regular solid for any n > ii ,by the following: Given n > ii 1 1 111 where -+_I- forallk. Ifit'strueforallk, thenit'struefor k=3. Thus -+-5-. kn2 3ii2 11 11111 ii<n~-<~--+-<-+~I-. Thusforall n>ii,nand k=3 donotforma nn 3n3n2 regular solid. In note iii. above, we showed if n and k = 3 do not form a regular solid, then n and all k > 3 do not form a regular solid. Since k 2 3 in all cases then all n > ii cannot form a regular solid with any possible k. With these restrictions, we must only inspect a finite amount of combinations of n and k to find which combinations form regular solids. As stated above, for any n once we find a k that doesn't form a regular solid we can stop. Likewise once we find an n that cannot create a regular solid we can stop. See Table 2.2 where we check all possible combinations and show only five such combinations form regular solids. Table 2.2 Hence we have proven there are exactly five regular solids that can be formed. Finding such possible combinations can also be illustrated quite nicely. Since each vertex is isometric to each other, we may inspect one vertex and decide whether it can be a vertex on regular solid with a given n and k. We can represent a vertex in two dimensions by "flattening" the vertex so that we see all of its neighboring faces in one plane. To visualize the regular solid, imagine the polygons being folded into the paper until the edges of all polygons meet and begin to enclose the solid. The vertex representations that can form a regular solid are the ones that can be folded into the paper in such a way. If the edges already meet or overlap, then the object cannot be folded into the paper and will be stuck in two dimensions, thus failing to form a regular solid. This implies the condition that the interior angles of the faces meeting at a vertex must not be greater than 2n. Since an interior angle of an n-gon is 7r - 27r / n ,the sum of the interior angles of k faces is k[7r -2z / n] . Thus in order to form a regular solid, we must have a k and n such that k[n- 27r / n] < 27r (Pap. 466). In Figures 2.3-2.5 we see the vertex representations for the regular solids, as well as vertex representations of n and k combinations that do not form regular solids by having the sum of the interior angles greater than or equal to 271.. Compare the n and k combinations in the following Figures to Table 2.2 to see that both methods find the same - combinations of n and k to form regular solids. Figure 2.3, n = 3 : k = 3 Angle sum < 2n k=4 Angle sum < 2n (Tetrahedron) (Octahedron) Angle sum < 2x Angle sum = 2x (Icosahedron) (NOT a regular solid) Figure 2.4, n = 4 : "'-pAngle sum < 2a Angle sum = 2a (Cube) (NOT a regular solid) Figure 2.5, n = 5 : Angle sum < 2a Angle sum > 2a (Dodecahedron) (NOT a regular solid) - .. - .. The above condition for the sum of interior angles also tells us that no regular solid can be formed for polygons of six or more sides. See Figure 2.6 which shows a vertex joining three regular hexagons. Since the edges already meet, the figure is trapped in two-dimensions. This would form an interesting tessellation or floor tile, but not a regular solid. Also since any n-gon where n > 6 would have larger interior angles than the hexagon, this confms that n I 5 in order to form a regular solid. Figure 2.6, n = 6 \ / Angle swn=2a (NOT a regular solid) Recall for the objects in Figures 2.3-2.5, the description instructed the reader to "imagine the polygons being folded into the paper until the edges all polygons meet and begin to enclose the solid".
Load more

Recommended publications
  • A New Calculable Orbital-Model of the Atomic Nucleus Based on a Platonic-Solid Framework
    A new calculable Orbital-Model of the Atomic Nucleus based on a Platonic-Solid framework and based on constant Phi by Dipl. Ing. (FH) Harry Harry K. K.Hahn Hahn / Germany ------------------------------ 12. December 2020 Abstract : Crystallography indicates that the structure of the atomic nucleus must follow a crystal -like order. Quasicrystals and Atomic Clusters with a precise Icosahedral - and Dodecahedral structure indi cate that the five Platonic Solids are the theoretical framework behind the design of the atomic nucleus. With my study I advance the hypothesis that the reference for the shell -structure of the atomic nucleus are the Platonic Solids. In my new model of the atomic nucleus I consider the central space diagonals of the Platonic Solids as the long axes of Proton - or Neutron Orbitals, which are similar to electron orbitals. Ten such Proton- or Neutron Orbitals form a complete dodecahedral orbital-stru cture (shell), which is the shell -type with the maximum number of protons or neutrons. An atomic nucleus therefore mainly consists of dodecahedral shaped shells. But stable Icosahedral- and Hexagonal-(cubic) shells also appear in certain elements. Consta nt PhI which directly appears in the geometry of the Dodecahedron and Icosahedron seems to be the fundamental constant that defines the structure of the atomic nucleus and the structure of the wave systems (orbitals) which form the atomic nucelus. Albert Einstein wrote in a letter that the true constants of an Universal Theory must be mathematical constants like Pi (π) or e. My mathematical discovery described in chapter 5 shows that all irrational square roots of the natural numbers and even constant Pi (π) can be expressed with algebraic terms that only contain constant Phi (ϕ) and 1 Therefore it is logical to assume that constant Phi, which also defines the structure of the Platonic Solids must be the fundamental constant that defines the structure of the atomic nucleus.
    [Show full text]
  • The Roundest Polyhedra with Symmetry Constraints
    Article The Roundest Polyhedra with Symmetry Constraints András Lengyel *,†, Zsolt Gáspár † and Tibor Tarnai † Department of Structural Mechanics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary; [email protected] (Z.G.); [email protected] (T.T.) * Correspondence: [email protected]; Tel.: +36-1-463-4044 † These authors contributed equally to this work. Academic Editor: Egon Schulte Received: 5 December 2016; Accepted: 8 March 2017; Published: 15 March 2017 Abstract: Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the sphere, i.e., the roundest polyhedra. A new numerical optimization method developed previously by the authors has been applied to optimize polyhedra to best approximate a sphere if tetrahedral, octahedral, or icosahedral symmetry constraints are applied. In addition to evidence provided for various cases of face numbers, potentially optimal polyhedra are also shown for n up to 132. Keywords: polyhedra; isoperimetric problem; point group symmetry 1. Introduction The so-called isoperimetric problem in mathematics is concerned with the determination of the shape of spatial (or planar) objects which have the largest possible volume (or area) enclosed with given surface area (or circumference). The isoperimetric problem for polyhedra can be reflected in a question as follows: What polyhedron maximizes the volume if the surface area and the number of faces n are given? The problem can be quantified by the so-called Steinitz number [1] S = A3/V2, a dimensionless quantity in terms of the surface area A and volume V of the polyhedron, such that solutions of the isoperimetric problem minimize S.
    [Show full text]
  • On the Archimedean Or Semiregular Polyhedra
    ON THE ARCHIMEDEAN OR SEMIREGULAR POLYHEDRA Mark B. Villarino Depto. de Matem´atica, Universidad de Costa Rica, 2060 San Jos´e, Costa Rica May 11, 2005 Abstract We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler’s polyhedral formula. Contents 1 Introduction 2 1.1 RegularPolyhedra .............................. 2 1.2 Archimedean/semiregular polyhedra . ..... 2 2 Proof techniques 3 2.1 Euclid’s proof for regular polyhedra . ..... 3 2.2 Euler’s polyhedral formula for regular polyhedra . ......... 4 2.3 ProofsofArchimedes’theorem. .. 4 3 Three lemmas 5 3.1 Lemma1.................................... 5 3.2 Lemma2.................................... 6 3.3 Lemma3.................................... 7 4 Topological Proof of Archimedes’ theorem 8 arXiv:math/0505488v1 [math.GT] 24 May 2005 4.1 Case1: fivefacesmeetatavertex: r=5. .. 8 4.1.1 At least one face is a triangle: p1 =3................ 8 4.1.2 All faces have at least four sides: p1 > 4 .............. 9 4.2 Case2: fourfacesmeetatavertex: r=4 . .. 10 4.2.1 At least one face is a triangle: p1 =3................ 10 4.2.2 All faces have at least four sides: p1 > 4 .............. 11 4.3 Case3: threefacesmeetatavertes: r=3 . ... 11 4.3.1 At least one face is a triangle: p1 =3................ 11 4.3.2 All faces have at least four sides and one exactly four sides: p1 =4 6 p2 6 p3. 12 4.3.3 All faces have at least five sides and one exactly five sides: p1 =5 6 p2 6 p3 13 1 5 Summary of our results 13 6 Final remarks 14 1 Introduction 1.1 Regular Polyhedra A polyhedron may be intuitively conceived as a “solid figure” bounded by plane faces and straight line edges so arranged that every edge joins exactly two (no more, no less) vertices and is a common side of two faces.
    [Show full text]
  • Of Great Rhombicuboctahedron/Archimedean Solid
    Mathematical Analysis of Great Rhombicuboctahedron/Archimedean Solid Mr Harish Chandra Rajpoot M.M.M. University of Technology, Gorakhpur-273010 (UP), India March, 2015 Introduction: A great rhombicuboctahedron is an Archimedean solid which has 12 congruent square faces, 8 congruent regular hexagonal faces & 6 congruent regular octagonal faces each having equal edge length. It has 72 edges & 48 vertices lying on a spherical surface with a certain radius. It is created/generated by expanding a truncated cube having 8 equilateral triangular faces & 6 regular octagonal faces. Thus by the expansion, each of 12 originally truncated edges changes into a square face, each of 8 triangular faces of the original solid changes into a regular hexagonal face & 6 regular octagonal faces of original solid remain unchanged i.e. octagonal faces are shifted radially. Thus a solid with 12 squares, 8 hexagonal & 6 octagonal faces, is obtained which is called great rhombicuboctahedron which is an Archimedean solid. (See figure 1), thus we have Figure 1: A great rhombicuboctahedron having 12 ( ) ( ) congruent square faces, 8 congruent regular ( ) ( ) hexagonal faces & 6 congruent regular octagonal faces each of equal edge length 풂 ( ) ( ) We would apply HCR’s Theory of Polygon to derive a mathematical relationship between radius of the spherical surface passing through all 48 vertices & the edge length of a great rhombicuboctahedron for calculating its important parameters such as normal distance of each face, surface area, volume etc. Derivation of outer (circumscribed) radius ( ) of great rhombicuboctahedron: Let be the radius of the spherical surface passing through all 48 vertices of a great rhombicuboctahedron with edge length & the centre O.
    [Show full text]
  • A Technology to Synthesize 360-Degree Video Based on Regular Dodecahedron in Virtual Environment Systems*
    A Technology to Synthesize 360-Degree Video Based on Regular Dodecahedron in Virtual Environment Systems* Petr Timokhin 1[0000-0002-0718-1436], Mikhail Mikhaylyuk2[0000-0002-7793-080X], and Klim Panteley3[0000-0001-9281-2396] Federal State Institution «Scientific Research Institute for System Analysis of the Russian Academy of Sciences», Moscow, Russia 1 [email protected], 2 [email protected], 3 [email protected] Abstract. The paper proposes a new technology of creating panoramic video with a 360-degree view based on virtual environment projection on regular do- decahedron. The key idea consists in constructing of inner dodecahedron surface (observed by the viewer) composed of virtual environment snapshots obtained by twelve identical virtual cameras. A method to calculate such cameras’ projec- tion and orientation parameters based on “golden rectangles” geometry as well as a method to calculate snapshots position around the observer ensuring synthe- sis of continuous 360-panorama are developed. The technology and the methods were implemented in software complex and tested on the task of virtual observing the Earth from space. The research findings can be applied in virtual environment systems, video simulators, scientific visualization, virtual laboratories, etc. Keywords: Visualization, Virtual Environment, Regular Dodecahedron, Video 360, Panoramic Mapping, GPU. 1 Introduction In modern human activities the importance of the visualization of virtual prototypes of real objects and phenomena is increasingly grown. In particular, it’s especially required for training specialists in space and aviation industries, medicine, nuclear industry and other areas, where the mistake cost is extremely high [1-3]. The effectiveness of work- ing with virtual prototypes is significantly increased if an observer experiences a feeling of being present in virtual environment.
    [Show full text]
  • Steinitz's Theorem Project Report §1 Introduction §2 Basic Definitions
    Steinitz's Theorem Project Report Jon Hillery May 17, 2019 §1 Introduction Looking at the vertices and edges of polyhedra gives a family of graphs that we might expect has nice properties. It turns out that there is actually a very nice characterization of these graphs! We can use this characterization to find useful representations of certain graphs. §2 Basic Definitions We define a space to be convex if the line segment connecting any two points in the space remains entirely inside the space. This works for two-dimensional sets: and three-dimensional sets: Given a polyhedron, we define its 1-skeleton to be the graph formed from the vertices and edges of the polyhedron. For example, the 1-skeleton of a tetrahedron is K4: 1 Jon Hillery (May 17, 2019) Steinitz's Theorem Project Report Here are some further examples of the 1-skeleton of an icosahedron and a dodecahedron: §3 Properties of 1-Skeletons What properties do we know the 1-skeleton of a convex polyhedron must have? First, it must be planar. To see this, imagine moving your eye towards one of the faces until you are close enough that all of the other faces appear \inside" the face you are looking through, as shown here: This is always possible because the polyedron is convex, meaning intuitively it doesn't have any parts that \jut out". The graph formed from viewing in this way will have no intersections because the polyhedron is convex, so the straight-line rays our eyes see are not allowed to leave via an edge on the boundary of the polyhedron and then go back inside.
    [Show full text]
  • Bridges Conference Proceedings Guidelines Word
    Bridges 2019 Conference Proceedings Helixation Rinus Roelofs Lansinkweg 28, Hengelo, the Netherlands; [email protected] Abstract In the list of names of the regular and semi-regular polyhedra we find many names that refer to a process. Examples are ‘truncated’ cube and ‘stellated’ dodecahedron. And Luca Pacioli named some of his polyhedral objects ‘elevations’. This kind of name-giving makes it easier to understand these polyhedra. We can imagine the model as a result of a transformation. And nowadays we are able to visualize this process by using the technique of animation. Here I will to introduce ‘helixation’ as a process to get a better understanding of the Poinsot polyhedra. In this paper I will limit myself to uniform polyhedra. Introduction Most of the Archimedean solids can be derived by cutting away parts of a Platonic solid. This operation , with which we can generate most of the semi-regular solids, is called truncation. Figure 1: Truncation of the cube. We can truncate the vertices of a polyhedron until the original faces of the polyhedron become regular again. In Figure 1 this process is shown starting with a cube. In the third object in the row, the vertices are truncated in such a way that the square faces of the cube are transformed to regular octagons. The resulting object is the Archimedean solid, the truncated cube. We can continue the process until we reach the fifth object in the row, which again is an Archimedean solid, the cuboctahedron. The whole process or can be represented as an animation. Also elevation and stellation can be described as processes.
    [Show full text]
  • Section 1000.00 Omnitopology
    Table of Contents ● 1000.00 OMNITOPOLOGY ● 1001.00 Inherent Rationality of Omnidirectional Epistemology ■ 1001.10 Spherical Reference ❍ 1001.20 Field of Geodesic Event Relationships ❍ 1002.10 Omnidirectional Nucleus ❍ 1003.10 Isotropic-Vector-Matrix Reference ❍ 1004.10 An Omnisynergetic Coordinate System ❍ 1005.10 Inventory of Omnidirectional Intersystem Precessional Effects ❍ 1005.15 Volume and Area Progressions ■ 1005.20 Biospherical Patterns ■ 1005.30 Poisson Effect ■ 1005.40 Genetic Intercomplexity ■ 1005.50 Truth and Love: Linear and Embracing ■ 1005.60 Generalization and Polarization ■ 1005.611 Metabolic Generalizations ❍ 1006.10 Omnitopology Defined ■ 1006.20 Omnitopological Domains ■ 1006.30 Vector Equilibrium Involvement Domain ■ 1006.40 Cosmic System Eight-dimensionality ❍ 1007.10 Omnitopology Compared with Euler's Topology ❍ 1007.20 Invalidity of Plane Geometry ❍ 1008.10 Geodesic Spheres in Closest Packing ❍ 1009.00 Critical Proximity ■ 1009.10 Interference: You Really Can t Get There from Here ■ 1009.20 Magnitude of Independent Orbiting ■ 1009.30 Symmetrical Conformation of Flying-Star Teams ■ 1009.40 Models and Divinity ■ 1009.50 Acceleration ■ 1009.60 Hammer-Thrower ■ 1009.69 Comet ■ 1009.70 Orbital Escape from Earth s Critical-Proximity Programmability ■ 1009.80 Pea-Shooter Sling-Thrower. and Gyroscope: Gravity and Mass- Attraction ● 1010.00 Prime Volumes ■ 1010.10 Domain and Quantum ■ 1010.20 Nonnuclear Prime Structural Systems ❍ 1011.00 Omnitopology of Prime Volumes ■ 1011.10 Prime Enclosure ■ 1011.20 Hierarchy of Nuclear
    [Show full text]
  • Dodecahedron
    Dodecahedron Figure 1 Regular Dodecahedron vertex labeling using “120 Polyhedron” vertex labels. COPYRIGHT 2007, Robert W. Gray Page 1 of 9 Encyclopedia Polyhedra: Last Revision: July 8, 2007 DODECAHEDRON Topology: Vertices = 20 Edges = 30 Faces = 12 pentagons Lengths: 15+ ϕ = 1.618 033 989 2 EL ≡ Edge length of regular Dodecahedron. 43ϕ + FA = EL 1.538 841 769 EL ≡ Face altitude. 2 ϕ DFV = EL ≅ 0.850 650 808 EL 5 15+ 20 ϕ DFE = EL ≅ 0.688 190 960 EL 10 3 DVV = ϕ EL ≅ 1.401 258 538 EL 2 1 DVE = ϕ 2 EL ≅ 1.309 016 994 EL 2 711+ ϕ DVF = EL ≅ 1.113 516 364 EL 20 COPYRIGHT 2007, Robert W. Gray Page 2 of 9 Encyclopedia Polyhedra: Last Revision: July 8, 2007 DODECAHEDRON Areas: 15+ 20 ϕ 2 2 Area of one pentagonal face = EL ≅ 1.720 477 401 EL 4 2 2 Total face area = 31520+ ϕ EL ≅ 20.645 728 807 EL Volume: 47+ ϕ 3 3 Cubic measure volume equation = EL ≅ 7.663 118 961 EL 2 38( + 14ϕ) 3 3 Synergetics' Tetra-volume equation = EL ≅ 65.023 720 585 EL 2 Angles: Face Angles: All face angles are 108°. Sum of face angles = 6480° Central Angles: ⎛⎞35− All central angles are = 2arcsin ⎜⎟ ≅ 41.810 314 896° ⎜⎟6 ⎝⎠ Dihedral Angles: ⎛⎞2 ϕ 2 All dihedral angles are = 2arcsin ⎜⎟ ≅ 116.565 051 177° ⎝⎠43ϕ + COPYRIGHT 2007, Robert W. Gray Page 3 of 9 Encyclopedia Polyhedra: Last Revision: July 8, 2007 DODECAHEDRON Additional Angle Information: Note that Central Angle(Dodecahedron) + Dihedral Angle(Icosahedron) = 180° Central Angle(Icosahedron) + Dihedral Angle(Dodecahedron) = 180° which is the case for dual polyhedra.
    [Show full text]
  • Geometry and Arithmetic of Crystallographic Sphere Packings
    Geometry and arithmetic of crystallographic sphere packings Alex Kontorovicha,b,1 and Kei Nakamuraa aDepartment of Mathematics, Rutgers University, New Brunswick, NJ 08854; and bSchool of Mathematics, Institute for Advanced Study, Princeton, NJ 08540 Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved November 21, 2018 (received for review December 12, 2017) We introduce the notion of a “crystallographic sphere packing,” argument leading to Theorem 3 comes from constructing circle defined to be one whose limit set is that of a geometrically packings “modeled on” combinatorial types of convex polyhedra, finite hyperbolic reflection group in one higher dimension. We as follows. exhibit an infinite family of conformally inequivalent crystallo- graphic packings with all radii being reciprocals of integers. We (~): Polyhedral Packings then prove a result in the opposite direction: the “superintegral” Let Π be the combinatorial type of a convex polyhedron. Equiv- ones exist only in finitely many “commensurability classes,” all in, alently, Π is a 3-connectedz planar graph. A version of the at most, 20 dimensions. Koebe–Andreev–Thurston Theorem§ says that there exists a 3 geometrization of Π (that is, a realization of its vertices in R with sphere packings j crystallographic j arithmetic j polyhedra j straight lines as edges and faces contained in Euclidean planes) Coxeter diagrams having a midsphere (meaning, a sphere tangent to all edges). This midsphere is then also simultaneously a midsphere for the he goal of this program is to understand the basic “nature” of dual polyhedron Πb. Fig. 2A shows the case of a cuboctahedron Tthe classical Apollonian gasket.
    [Show full text]
  • 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.
    [Show full text]
  • DIMACS REU 2018 Project: Sphere Packings and Number Theory
    DIMACS REU 2018 Project: Sphere Packings and Number Theory Alisa Cui, Devora Chait, Zachary Stier Mentor: Prof. Alex Kontorovich July 13, 2018 Apollonian Circle Packing This is an Apollonian circle packing: I Draw two more circles, each of which is tangent to the original three Apollonian Circle Packing Here's how we construct it: I Start with three mutually tangent circles I Draw two more circles, each of which is tangent to the original three Apollonian Circle Packing Here's how we construct it: I Start with three mutually tangent circles Apollonian Circle Packing Here's how we construct it: I Start with three mutually tangent circles I Draw two more circles, each of which is tangent to the original three Apollonian Circle Packing I Start with three mutually tangent circles I Draw two more circles, each of which is tangent to the original three Apollonian Circle Packing I Start with three mutually tangent circles I Draw two more circles, each of which is tangent to the original three I Continue drawing tangent circles, densely filling space These two images actually represent the same circle packing! We can go from one realization to the other using circle inversions. Apollonian Circle Packing Apollonian Circle Packing These two images actually represent the same circle packing! We can go from one realization to the other using circle inversions. Circle Inversions Circle inversion sends points at a distance of rd from the center of the mirror circle to a distance of r=d from the center of the mirror circle. Circle Inversions Circle inversion sends points at a distance of rd from the center of the mirror circle to a distance of r=d from the center of the mirror circle.
    [Show full text]