Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. It follows. Uniform polyhedra may be quasi-regular or semi-regular; the faces and vertices need not be convex, so many of the uniform polyhedra are star polyhedra. There are two infinite classes of uniform polyhedra together with 75 others. Infinite classes prisms antiprisms Convex exceptional 5 Platonic solids – regular convex polyhedra 13 Archimedean solids – 2 quasiregular and 11 semiregular convex polyhedra Star exceptional 4 Kepler–Poinsot polyhedra – regular nonconvex polyhedra 53 uniform star polyhedra – 5 quasiregular and 48 semiregularhence 5 + 13 + 4 + 53 = 75. There are many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John Skilling called the great disnub dirhombidodecahedron. Dual polyhedra to uniform polyhedra are face-transitive and have regular vertex figures, are classified in parallel with their dual polyhedron; the dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid.
The concept of uniform polyhedron is a special case of the concept of uniform polytope, which applies to shapes in higher-dimensional space. Coxeter, Longuet-Higgins & Miller define uniform polyhedra to be vertex-transitive polyhedra with regular faces, they define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional Euclidean space. There are some generalizations of the concept of a uniform polyhedron. If the connectedness assumption is dropped we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate we get the so-called degenerate uniform polyhedra; these require a more general definition of polyhedra. Grunbaum gave a rather complicated definition of a polyhedron, while McMullen & Schulte gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization.
Here an abstract polytope is a poset of its "faces" satisfying various condition, a realization is a function from its vertices to some space, the realization is called non-degenerate if any two distinct faces of the abstract polytope have distinct realizations. Some of the ways they can be degenerate are as follows: Hidden faces; some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are not counted as uniform polyhedra. Degenerate compounds; some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. Double covers. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron. There double covers have doubled faces and vertices, they are not counted as uniform polyhedra. Double faces. There are several polyhedra with doubled faces produced by Wythoff's construction.
Most authors do not remove them as part of the construction. Double edges. Skilling's figure has the property that it has double edges but its faces cannot be written as a union of two uniform polyhedra; the Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Theaetetus, Timaeus of Locri and Euclid. The Etruscans discovered the regular dodecahedron before 500 BC; the cuboctahedron was known by Plato. Archimedes discovered all of the 13 Archimedean solids, his original book on the subject was lost, but Pappus of Alexandria mentioned Archimedes listed 13 polyhedra. Piero della Francesca rediscovered the five truncation of the Platonic solids: truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, truncated icosahedron. Luca Pacioli republished Francesca's work in De divina proportione in 1509, adding the rhombicuboctahedron, calling it a icosihexahedron for its 26 faces, drawn by Leonardo da Vinci. Johannes Kepler was the first to publish the complete list of Archimedean solids, in 1619, as well as identified the infinite families of uniform prisms and antiprisms.
Kepler discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot discovered the other two. The set of four were named by Arthur Cayley. Of the remaining 53, Edmund Hess discovered two, Albert Badoureau discovered 36 more, Pitsch independently discovered 18, of which 3 had not been discovered. Together these gave 41 polyhedra; the geometer H. S. M. Coxeter did not publish. M. S. Longuet-Higgins and H. C. Longuet-Higgins independently discovered eleven of these. Lesavre and Mercier rediscovered five of them in 1947. Coxeter, Longuet-Higgins & Miller published the list of uniform polyhedra. Sopov (19
Harmonices Mundi
Harmonices Mundi is a book by Johannes Kepler. In the work, written in Latin, Kepler discusses harmony and congruence in geometrical forms and physical phenomena; the final section of the work relates his discovery of the so-called "third law of planetary motion". It is estimated that Kepler had begun working on Harmonices Mundi sometime near 1599, the year Kepler sent a letter to Michael Maestlin detailing the mathematical data and proofs that he intended to use for his upcoming text, which he planned to name De harmonia mundi. Kepler was aware that the content of Harmonices Mundi resembled that of the subject matter for Ptolemy’s Harmonica, but was not concerned, because the new astronomy Kepler would use, most notably the adoption of elliptic orbits in the Copernican system, allowed him to explore new theorems. Another important development that allowed Kepler to establish his celestial-harmonic relationships, was the abandonment of the Pythagorean tuning as the basis for musical consonance and the adoption of geometrically supported musical ratios.
Thus Kepler, could reason that his relationships gave evidence for God acting as a grand geometer, rather than a Pythagorean numerologist. The concept of musical harmonies intrinsically existing within the spacing of the planets existed in medieval philosophy prior to Kepler. Musica universalis was a traditional philosophical metaphor, taught in the quadrivium, was called the "music of the spheres". Kepler was intrigued by this idea while he sought explanation for a rational arrangement of the heavenly bodies, it should be noted that when Kepler uses the term “harmony” it is not referring to the musical definition, but rather, a broader definition encompassing congruence in Nature and the workings of both the celestial and terrestrial bodies. He notes musical harmony as being a product of man, derived from angles, in contrast to a harmony that he refers to as being a phenomenon that interacts with the human soul. In turn, this allowed Kepler to claim the Earth has a soul because it is subjected to astrological harmony.
Kepler divides The Harmony of the World into five long chapters: the first is on regular polygons. Chapters 1 and 2 of The Harmony of the World contain most of Kepler's contributions concerning polyhedra, he is interested with how polygons, which he defines as either regular or semiregular, can come to be fixed together around a central point on a plane to form congruence. His primary objective was to be able to rank polygons based on a measure of sociability, or rather, their ability to form partial congruence when combined with other polyhedra, he returns to this concept in Harmonices Mundi with relation to astronomical explanations. In the second chapter is the earliest mathematical understanding of two types of regular star polyhedra, the small and great stellated dodecahedron, he describes polyhedra in terms of their faces, similar to the model used in Plato's Timaeus to describe the formation of Platonic solids in terms of basic triangles. The book features illustrations of solids and tiling patterns, some of which are related to the golden ratio.
While medieval philosophers spoke metaphorically of the "music of the spheres", Kepler discovered physical harmonies in planetary motion. He found that the difference between the maximum and minimum angular speeds of a planet in its orbit approximates a harmonic proportion. For instance, the maximum angular speed of the Earth as measured from the Sun varies by a semitone, from mi to fa, between aphelion and perihelion. Venus only varies by a tiny 25:24 interval. Kepler explains the reason for the Earth's small harmonic range: The Earth sings Mi, Fa, Mi: you may infer from the syllables that in this our home misery and famine hold sway; the celestial choir Kepler formed was made up of a tenor, two bass, a soprano, two altos. Mercury, with its large elliptical orbit, was determined to be able to produce the greatest number of notes, while Venus was found to be capable of only a single note because its orbit is nearly a circle. At rare intervals all of the planets would sing together in "perfect concord": Kepler proposed that this may have happened only once in history at the time of creation.
Kepler reminds us that harmonic order is only mimicked by man, but has origin in the alignment of the heavenly bodies: Accordingly you won’t wonder any more that a excellent order of sounds or pitches in a musical system or scale has been set up by men, since you see that they are doing nothing else in this business except to play the apes of God the Creator and to act out, as it were, a certain drama of the ordination of the celestial movements.. Kepler discovers that all but one of the ratios of the maximum and minimum speeds of planets on neighboring orbits approximate musical harmonies within a margin of error of less than a diesis; the orbits of Mars and Jupiter produce the one exception to this rule, creating the inharmonic ratio of 18:19. The cause of this dissonance might be explained by the fact that the asteroid belt separates those two planetary orbits, as discovered in 18
Wythoff symbol
In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. A Wythoff symbol consists of a vertical bar, it represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 | 4 2 with Oh symmetry, 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D 2 h symmetry. With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space. In three dimensions, Wythoff's construction begins by choosing a generator point on the triangle. If the distance of this point from each of the sides is non-zero, the point must be chosen to be an equal distance from each edge.
A perpendicular line is dropped between the generator point and every face that it does not lie on. The three numbers in Wythoff's symbol, p, q and r, represent the corners of the Schwarz triangle used in the construction, which are π / p, π / q and π / r radians respectively; the triangle is represented with the same numbers, written. The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following: p | q r indicates that the generator lies on the corner p, p q | r indicates that the generator lies on the edge between p and q, p q r | indicates that the generator lies in the interior of the triangle. In this notation the mirrors are labeled by the reflection-order of the opposite vertex; the p, q, r values are listed before the bar. The one impossible symbol | p q r implies the generator point is on all mirrors, only possible if the triangle is degenerate, reduced to a point; this unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored.
The resulting figure has rotational symmetry only. The generator point can either be off each mirror, activated or not; this distinction creates 8 possible forms, neglecting one where the generator point is on all the mirrors. The Wythoff symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. A node is circled. There are seven generator points with each set of p, q, r: There are three special cases: p q | – This is a mixture of p q r | and p q s |, containing only the faces shared by both. | p q r – Snub forms are given by this otherwise unused symbol. | p q r s – A unique snub form for U75 that isn't Wythoff-constructible. There are 4 symmetry classes of reflection on the sphere, three in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are listed. Point groups: dihedral symmetry, p = 2, 3, 4 … tetrahedral symmetry octahedral symmetry icosahedral symmetry Euclidean groups: *442 symmetry: 45°-45°-90° triangle *632 symmetry: 30°-60°-90° triangle *333 symmetry: 60°-60°-60° triangleHyperbolic groups: *732 symmetry *832 symmetry *433 symmetry *443 symmetry *444 symmetry *542 symmetry *642 symmetry...
The above symmetry groups only include the integer solutions on the sphere. The list of Schwarz triangles includes rational numbers, determine the full set of solutions of nonconvex uniform polyhedra. In the tilings above, each triangle is a fundamental domain, colored by and odd reflections. Selected tilings created by the Wythoff con
Cantellation (geometry)
In geometry, a cantellation is an operation in any dimension that bevels a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex. The operation applies to regular tilings and honeycombs; this is rectifying its rectification. This operation is called expansion by Alicia Boole Stott, as imagined by taking the faces of the regular form moving them away from the center and filling in new faces in the gaps for each opened vertex and edge, it is represented by r or rr. For polyhedra, a cantellation operation offers a direct sequence from a regular polyhedron and its dual. Example cantellation sequence between a cube and octahedron For higher-dimensional polytopes, a cantellation offers a direct sequence from a regular polytope and its birectified form. A cuboctahedron would be a cantellated tetrahedron, as another example. Uniform polyhedron Uniform 4-polytope Coxeter, H. S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 Norman Johnson Uniform Polytopes, Manuscript N.
W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto, 1966 Weisstein, Eric W. "Expansion". MathWorld
Icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries; the set of orientation-preserving symmetries forms a group referred to as A5, the full symmetry group is the product A5 × Z2. The latter group is known as the Coxeter group H3, is represented by Coxeter notation, Coxeter diagram. Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries with the largest symmetry groups. 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 ⟩; these correspond to the icosahedral groups being the triangle groups.
The first presentation was given by William Rowan Hamilton in 1856, in his paper on icosian calculus. 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 permutations of five objects; this isomorphism can be realized by I acting on various compounds, notably the compound of five cubes, the compound of five octahedra, or either of the two compounds of five tetrahedra. The group contains 5 versions of Th with 20 versions of D3, 6 versions of D5; the full icosahedral group Ih has order 120. 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 center corresponding to element, where Z2 is written multiplicatively. Ih acts on the compound of five cubes and the compound of five octahedra, but −1 acts as the identity, it acts on the compound of ten tetrahedra: I acts on the two chiral halves, −1 interchanges the two halves.
Notably, it does not act as S5, these groups are not isomorphic. The group contains 6 versions of D5d. I is isomorphic to PSL2, but Ih is not isomorphic to SL2; the following groups all have order 120, but are not isomorphic: S5, the symmetric group on 5 elements Ih, the full icosahedral group 2I, the binary icosahedral groupThey correspond to the following short exact sequences and product 1 → A 5 → S 5 → Z 2 → 1 I h = A 5 × Z 2 1 → Z 2 → 2 I → A 5 → 1 In words, A 5 is a normal subgroup of S 5 A 5 is a factor of I h, a direct product A 5 is a quotient group of 2 I Note that A 5 has an exceptional irreducible 3-dimensional representation, but S 5 does not have an irreducible 3-dimensional representation, corresponding to the full icosahedral group not being the symmetric group. These can be related to linear groups over the finite field with five elements, which exhibit the subgroups and covering groups directly. In computational contexts, the rotation icosahedral group I above can be explicitly represented by the following 60 rotatio
Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, new facets are formed at separated elements. Equivalently this operation can be imagined by keeping facets in the same position but reducing their size; the expansion of a regular polytope creates a uniform polytope, but the operation can be applied to any convex polytope, as demonstrated for polyhedra in Conway polyhedron notation. For polyhedra, an expanded polyhedron has all the faces of the original polyhedron, all the faces of the dual polyhedron, new square faces in place of the original edges. According to Coxeter, this multidimensional term was defined by Alicia Boole Stott for creating new polytopes starting from regular polytopes to construct new uniform polytopes; the expansion operation is symmetric with respect to its dual. The resulting figure contains the facets of both the regular and its dual, along with various prismatic facets filling the gaps created between intermediate dimensional elements.
It has somewhat different meanings by dimension. In a Wythoff construction, an expansion is generated by reflections from the last mirrors. In higher dimensions, lower dimensional expansions can be written with a subscript, so e2 is the same as t0,2 in any dimension. By dimension: A regular polygon expands into a regular 2n-gon; the operation is identical to truncation for polygons, e = e1 = t0,1 = t and has Coxeter-Dynkin diagram. A regular polyhedron expands into a polyhedron with vertex figure p.4.q.4. This operation for polyhedra is called cantellation, e = e2 = t0,2 = rr, has Coxeter diagram. For example, a rhombicuboctahedron can be called an expanded cube, expanded octahedron, as well as a cantellated cube or cantellated octahedron. A regular 4-polytope expands into a new 4-polytope with the original cells, new cells in place of the old vertices, p-gonal prisms in place of the old faces, r-gonal prisms in place of the old edges; this operation for 4-polytopes is called runcination, e = e3 = t0,3, has Coxeter diagram.
A regular 5-polytope expands into a new 5-polytope with facets, × prisms, × prisms, × duoprisms. This operation has Coxeter diagram; the general operator for expansion of a regular n-polytope is t0,n-1. New regular facets are added at each vertex, new prismatic polytopes are added at each divided edge, face... ridge, etc. Conway polyhedron notation Weisstein, Eric W. "Expansion". MathWorld. Coxeter, H. S. M. Regular Polytopes. 3rd edition, Dover, ISBN 0-486-61480-8. Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto, 1966
Vertex (geometry)
In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices; the vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect, or any appropriate combination of rays and lines that result in two straight "sides" meeting at one place. A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called "convex" if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians. More a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, concave otherwise. Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope, in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices.
However, in graph theory, vertices may have fewer than two incident edges, not allowed for geometric vertices. There is a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, if a polygon is approximated by a smooth curve there will be a point of extreme curvature near each polygon vertex. However, a smooth curve approximation to a polygon will have additional vertices, at the points where its curvature is minimal. A vertex of a plane tiling or tessellation is a point. More a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x. There are two types of principal vertices: mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies in P. According to the two ears theorem, every simple polygon has at least two ears.
A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedron's surface has Euler characteristic V − E + F = 2, where V is the number of vertices, E is the number of edges, F is the number of faces; this equation is known as Euler's polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, a cube has 12 edges and 6 faces, hence 8 vertices. In computer graphics, objects are represented as triangulated polyhedra in which the object vertices are associated not only with three spatial coordinates but with other graphical information necessary to render the object such as colors, reflectance properties and surface normal. Weisstein, Eric W. "Polygon Vertex". MathWorld. Weisstein, Eric W. "Polyhedron Vertex". MathWorld. Weisstein, Eric W. "Principal Vertex". MathWorld