1.
Symmetry group
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In abstract algebra, the symmetry group of an object is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, the objects may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be more precise by specifying what is meant by image or pattern. For symmetry of objects, one may also want to take their physical composition into account. The group of isometries of space induces an action on objects in it. The symmetry group is also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries that leave the figure invariant is called its symmetry group. The proper symmetry group of an object is equal to its symmetry group if. The proper symmetry group is then a subgroup of the orthogonal group SO. A discrete symmetry group is a group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances, the group of all symmetries of a sphere O is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups, for example, two 3D figures have mirror symmetry, but with respect to different mirror planes. Two 3D figures have 3-fold rotational symmetry, but with respect to different axes, two 2D patterns have translational symmetry, each in one direction, the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also involved in continuous symmetries. A figure with this group is non-drawable and up to arbitrarily fine detail homogeneous. The group generated by all translations, this group cannot be the group of a pattern, it would be homogeneous
2.
Icosahedral symmetry
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A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the set of symmetries, since it is the dual of the icosahedron. The set of orientation-preserving symmetries forms a group referred to as A5, the latter group is also known as the Coxeter group H3, and is also represented by Coxeter notation, and Coxeter diagram. Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups. Presentations corresponding to the above are, I, ⟨ s, t ∣ s 2, t 3,5 ⟩ I h, ⟨ s, t ∣ s 3 −2, t 5 −2 ⟩ and these correspond to the icosahedral groups being the triangle groups. The first presentation was given by William Rowan Hamilton in 1856, note that other presentations are possible, for instance as an alternating group. The icosahedral rotation group I is of order 60, the group I is isomorphic to A5, the alternating group of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the compound of five cubes, the group contains 5 versions of Th with 20 versions of D3, and 6 versions of D5. The full icosahedral group Ih has order 120 and it has I as normal subgroup of index 2. The group Ih is isomorphic to I × Z2, or A5 × Z2, with the inversion in the corresponding to element. Ih acts on the compound of five cubes and the compound of five octahedra and it acts on the compound of ten tetrahedra, I acts on the two chiral halves, and −1 interchanges the two halves. Notably, it does not act as S5, and these groups are not isomorphic, the group contains 10 versions of D3d and 6 versions of D5d. I is also isomorphic to PSL2, but Ih is not isomorphic to SL2, all of these classes of subgroups are conjugate, and admit geometric interpretations. Note that the stabilizer of a vertex/edge/face/polyhedron and its opposite are equal, stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate. Stabilizers of a pair of edges in Ih give Z2 × Z2 × Z2, there are 5 of these, stabilizers of an opposite pair of faces can be interpreted as stabilizers of the anti-prism they generate. g. Flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, in aluminum, the icosahedral structure was discovered experimentally three years after this by Dan Shechtman, which earned him the Nobel Prize in 2011. Icosahedral symmetry is equivalently the projective linear group PSL, and is the symmetry group of the modular curve X. The modular curve X is geometrically a dodecahedron with a cusp at the center of each polygonal face, similar geometries occur for PSL and more general groups for other modular curves
3.
The Fifty-Nine Icosahedra
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The Fifty-Nine Icosahedra is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie and it enumerates certain stellations of the regular convex or Platonic icosahedron, according to a set of rules put forward by J. C. P. Miller. First published by the University of Toronto in 1938, a Second Edition by Springer-Verlag followed in 1982, K. and D. Crennell completely reset the text and redrew the plates and diagrams for Tarquins 1999 Third Edition, also adding new reference material and photographs. Although Miller did not contribute to the book directly, he was a colleague of Coxeter. All parts composing the faces must be the same in each plane, the parts included in any one plane must have trigonal symmetry, without or with reflection. This secures icosahedral symmetry for the whole solid, the parts included in any plane must all be accessible in the completed solid. We exclude from consideration cases where the parts can be divided into two sets, each giving a solid with as much symmetry as the whole figure, but we allow the combination of an enantiomorphous pair having no common part. Rules to are symmetry requirements for the face planes, rule excludes buried holes, to ensure that no two stellations look outwardly identical. Rule prevents any disconnected compound of simpler stellations, Coxeter was the main driving force behind the work. He carried out the analysis based on Millers rules, adopting a number of techniques such as combinatorics. He observed that the stellation diagram comprised many line segments and he then developed procedures for manipulating combinations of the adjacent plane regions, to formally enumerate the combinations allowed under Millers rules. His graph, reproduced here, shows the connectivity of the various faces identified in the stellation diagram, based on this he tested all possible combinations against Millers rules, confirming the result of Coxeters more analytical approach. Flathers contribution was indirect, he made models of all 59. When he first met Coxeter he had made many stellations. He went on to complete the series of fifty-nine, which are preserved in the library of Cambridge University. The library also holds some non-Miller models, but it is not known whether these were made by Flather or by Millers later students, john Flinders Petrie was a lifelong friend of Coxeter and had a remarkable ability to visualise four-dimensional geometry. He and Coxeter had worked together on many mathematical problems and his direct contribution to the fifty nine icosahedra was the exquisite set of three-dimensional drawings which provide much of the fascination of the published work. For the Third Edition, Kate and David Crennell completely reset the text and redrew the illustrations and they also added a reference section containing tables, diagrams, and photographs of some of the Cambridge models
4.
Wenninger
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Father Magnus J. Wenninger OSB was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Born to German immigrants in Park Falls, Wisconsin, Joseph Wenninger always knew he was going to be a priest. From an early age, it was understood that his brother Heinie would take after their father and become a baker, the ad was for a preparatory school in Collegeville, Minnesota, associated with the Benedictine St. John’s University. While admitting to feeling homesick at first, Wenninger quickly made friends and, after a year, knew that this was where he needed to be. He was a student in a section of the school that functioned as a “minor seminary” – later moving on into St. John’s where he studied philosophy and theology. Wenninger became a Benedictine monk, he took on his monastic name Magnus, at the start of his career, Wenninger did not set out on a path one might expect would lead to his becoming the great polyhedronist that he is known as today. Rather, a few chance happenings and seemingly minor decisions shaped a course for Wenninger that led to his groundbreaking studies, shortly after becoming a priest, Wenninger’s Abbot informed him that their order was starting up a school in the Bahamas. It was decided that Wenninger would be assigned to teach at that school, in order to do this, it was necessary that he get a masters degree. Wenninger was sent to the University of Ottawa, in Canada, there he studied symbolic logic under Thomas Greenwood of the philosophy department. His thesis title was “The Concept of Number According to Roger Bacon, after completing his degree, Wenninger went to the school in the Bahamas, where he was asked by the headmaster to choose between teaching English or math. Wenninger chose math, since it seemed to be more in line with the topic of his MA thesis, however, not having taken many math courses in college, Wenninger admits to being able to teach by staying a few pages ahead of the students. He taught Algebra, Euclidean Geometry, Trigonometry and Analytic Geometry, after ten years of teaching, Wenninger felt he was becoming a bit stale. At the suggestion of his headmaster, Wenninger attended the Columbia Teachers College in summer sessions over a period in the late fifties. It was here that his interest in the “New Math” was formed, Wenninger died at the age of 97, at St Johns Abbey on Friday, February 17,2017. Wenninger’s first publication on the topic of polyhedra was the entitled, “Polyhedron Models for the Classroom”. He wrote to H. S. M. Coxeter and received a copy of Uniform polyhedra which had a complete list of all 75 uniform polyhedra, after this, he spent a great deal of time building various polyhedra. He made 65 of them and had them on display in his classroom, at this point, Wenninger decided to contact a publisher to see if there was any interest in a book. He had the models photographed and wrote the text, which he sent off to Cambridge University Press in London
5.
Euler characteristic
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It is commonly denoted by χ. The Euler characteristic was originally defined for polyhedra and used to prove theorems about them. Leonhard Euler, for whom the concept is named, was responsible for much of early work. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, any convex polyhedrons surface has Euler characteristic V − E + F =2. This equation is known as Eulers polyhedron formula and it corresponds to the Euler characteristic of the sphere, and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below and this version holds both for convex polyhedra and the non-convex Kepler-Poinsot polyhedra. Projective polyhedra all have Euler characteristic 1, like the real plane, while the surfaces of toroidal polyhedra all have Euler characteristic 0. The Euler characteristic can be defined for connected plane graphs by the same V − E + F formula as for polyhedral surfaces, the Euler characteristic of any plane connected graph G is 2. This is easily proved by induction on the number of determined by G. For trees, E = V −1 and F =1, if G has C components, the same argument by induction on F shows that V − E + F − C =1. One of the few graph theory papers of Cauchy also proves this result, via stereographic projection the plane maps to the two-dimensional sphere, such that a connected graph maps to a polygonal decomposition of the sphere, which has Euler characteristic 2. This viewpoint is implicit in Cauchys proof of Eulers formula given below, there are many proofs of Eulers formula. One was given by Cauchy in 1811, as follows and it applies to any convex polyhedron, and more generally to any polyhedron whose boundary is topologically equivalent to a sphere and whose faces are topologically equivalent to disks. Remove one face of the polyhedral surface, after this deformation, the regular faces are generally not regular anymore. The number of vertices and edges has remained the same, therefore, proving Eulers formula for the polyhedron reduces to proving V − E + F =1 for this deformed, planar object. If there is a face more than three sides, draw a diagonal—that is, a curve through the face connecting two vertices that arent connected yet. This adds one edge and one face and does not change the number of vertices, continue adding edges in this manner until all of the faces are triangular. This decreases the number of edges and faces by one each and does not change the number of vertices, remove a triangle with two edges shared by the exterior of the network, as illustrated by the third graph
6.
Isogonal figure
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
7.
Isohedral figure
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In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, in other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex polyhedra are the shapes that will make fair dice. They can be described by their face configuration, a polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral and they are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, a polyhedron which is isohedral and isogonal is said to be noble. A polyhedron is if it contains k faces within its symmetry fundamental domain. Similarly a k-isohedral tiling has k separate symmetry orbits, a monohedral polyhedron or monohedral tiling has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces, a facet-transitive or isotopic figure is a n-dimensional polytopes or honeycomb, with its facets congruent and transitive. The dual of an isotope is an isogonal polytope, by definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, an isotopic 3-dimensional figure is isohedral. An isotopic 4-dimensional figure is isochoric, edge-transitive Anisohedral tiling Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.367 Transitivity Olshevsky, George. Archived from the original on 4 February 2007
8.
Stellation diagram
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In geometry, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron, showing lines where other face planes intersect with this one. The lines cause 2D space to be divided up into regions, regions not intersected by any further lines are called elementary regions. Usually infinite regions are excluded from the diagram, along with any infinite portions of the lines, each elementary region represents a top face of one cell, and a bottom face of another. A collection of diagrams, one for each face type, can be used to represent any stellation of the polyhedron. A stellation diagram exists for every face of a given polyhedron, in face transitive polyhedra, symmetry can be used to require all faces have the same diagram shading. Semiregular polyhedra like the Archimedean solids will have different stellation diagrams for different kinds of faces, list of Wenninger polyhedron models The fifty nine icosahedra M Wenninger, Polyhedron models, Cambridge University Press, 1st Edn, Ppbk. Coxeter, Harold Scott MacDonald, Du Val, P. Flather, H. T. Petrie, J. F
9.
Stellation
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In geometry, stellation is the process of extending a polygon, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. The new figure is a stellation of the original, the word stellation comes from the Latin stellātus, starred, which in turn comes from Latin stella, star. In 1619 Kepler defined stellation for polygons and polyhedra, as the process of extending edges or faces until they meet to form a new polygon or polyhedron and he stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and great stellated dodecahedron. He also stellated the regular octahedron to obtain the stella octangula, stellating a regular polygon symmetrically creates a regular star polygon or polygonal compound. These polygons are characterised by the number of times m that the polygonal boundary winds around the centre of the figure, like all regular polygons, their vertices lie on a circle. M also corresponds to the number of vertices around the circle to get one end of a given edge to the other. A regular star polygon is represented by its Schläfli symbol, where n is the number of vertices, m is the used in sequencing the edges around it. Making m =1 gives the convex, if n and m do have a common divisor, then the figure is a regular compound. For example is the compound of two triangles or hexagram, while is a compound of two pentagrams. Some authors use the Schläfli symbol for such regular compounds, others regard the symbol as indicating a single path which is wound m times around n/m vertex points, such that one edge is superimposed upon another and each vertex point is visited m times. In this case a modified symbol may be used for the compound, a regular n-gon has /2 stellations if n is even, and /2 stellations if n is odd. Like the heptagon, the octagon also has two octagrammic stellations, one, being a star polygon, and the other, being the compound of two squares. A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound, the interior of the new polyhedron is divided by the faces into a number of cells. The face planes of a polyhedron may divide space into many such cells, for a symmetrical polyhedron, these cells will fall into groups, or sets, of congruent cells - we say that the cells in such a congruent set are of the same type. A common method of finding stellations involves selecting one or more cell types and this can lead to a huge number of possible forms, so further criteria are often imposed to reduce the set to those stellations that are significant and unique in some way. A set of cells forming a layer around its core is called a shell. For a symmetrical polyhedron, a shell may be made up of one or more cell types, based on such ideas, several restrictive categories of interest have been identified. Adding successive shells to the core leads to the set of main-line stellations