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
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century, he was born in London, received his BA and PhD from Cambridge, but lived in Canada from age 29. He was always called Donald, from his third name MacDonald, he was most noted for his work on higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10, he felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the Canadian Music Journal. Coxeter went up to Cambridge in 1926 to read mathematics. There he earned his BA in 1928, his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, Solomon Lefschetz.
Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics. In 1934 he spent a further year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto. In 1938 he and P. Du Val, H. T. Flather, John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays published by W. W. Rouse Ball in 1892, he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and a Fellow of the Royal Society in 1950, he met M. C. Escher in 1954 and the two became lifelong friends, he inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, he published twelve books. Since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor, he was made a Fellow of the Royal Society in 1950 and in 1997 he was awarded their Sylvester Medal.
In 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he received the Jeffery–Williams Prize. 1940: Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46: 380-407, MR 2,10 doi:10.1007/BF01181449 1942: Non-Euclidean Geometry, University of Toronto Press, MAA. 1954: "Uniform Polyhedra", Philosophical Transactions of the Royal Society A 246: 401–50 doi:10.1098/rsta.1954.0003 1949: The Real Projective Plane 1957: Generators and Relations for Discrete Groups 1980: Second edition, Springer-Verlag ISBN 0-387-09212-9 1961: Introduction to Geometry 1963: Regular Polytopes, Macmillan Company 1967: Geometry Revisited 1970: Twisted honeycombs 1973: Regular Polytopes, Dover edition, ISBN 0-486-61480-8 1974: Projective Geometry 1974: Regular Complex Polytopes, Cambridge University Press 1981:, Zero-Symmetric Graphs, Academic Press. 1985: Regular and Semi-Regular Polytopes II, Mathematische Zeitschrift 188: 559–591 1987 Projective Geometry ISBN 978-0-387-40623-7 1988: Regular and Semi-Regular Polytopes III, Mathematische Zeitschrift 200: 3–45 1995: F. Arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors: Kaleidoscopes — Selected Writings of H.
S. M. Coxeter. John Wiley and Sons ISBN 0-471-01003-0 1999: The Beauty of Geometry: Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler; the Coxeter Legacy: Reflections and Projections. Providence, R. I.: American Mathematical Society. ISBN 978-0-8218-3722-1. OCLC 62282754. Roberts, Siobhan. King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. New York: Walker & Company. ISBN 978-0-8027-1499-2. OCLC 71436884. Archival papers held at University of Toronto Archives and Records Management Services Harold Scott MacDonald Coxeter at the Mathematics Genealogy Project H. S. M. Coxeter, Erich W. Ellers, Branko Grünbaum, Peter McMullen, Asia Ivic Weiss Notices of the AMS: Volume 50, Number 10. Www.donaldcoxeter.com www.math.yorku.ca/dcoxeter webpages dedicated to him Jaron's World: Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C. Escher video of a lecture by H. S. M. Coxeter, April 28, 2000
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, 14 vertices of two types, it is a Catalan solid, the dual polyhedron of the cuboctahedron. The rhombic dodecahedron is a zonohedron, its polyhedral dual is the cuboctahedron. The long diagonal of each face is √2 times the length of the short diagonal, so that the acute angles on each face measure arccos, or 70.53°. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B; the rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron and the rhombic triacontahedron. The rhombic dodecahedron can be used to tessellate three-dimensional space.
It can be stacked to fill a space. This polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice, it is the Brillouin zone of body centered cubic crystals. Some minerals such as garnet form a rhombic dodecahedral crystal habit. Honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of, a hexagonal prism capped with half a rhombic dodecahedron; the rhombic dodecahedron appears in the unit cells of diamond and diamondoids. In these cases, four vertices are absent; the graph of the rhombic dodecahedron is nonhamiltonian. A rhombic dodecahedron can be dissected with its center into 4 trigonal trapezohedra; these rhombohedra are the cells of a trigonal trapezohedral honeycomb. This is analogous to the dissection of a regular hexagon dissected into rhombi, tiled in the plane as a rhombille. If the edge length of a rhombic dodecahedron is a, the radius of an inscribed sphere is r i = 6 3 a ≈ 0.816 496 5809 a, OEIS: A157697the radius of the midsphere is r m = 2 2 3 a ≈ 0.942 809 041 58 a, OEIS: A179587.and the radius of the circumscribed sphere is r o = 2 3 3 a ≈ 1.154 700 538 a, OEIS: A020832.
The area A and the volume V of the rhombic dodecahedron of edge length a are: A = 8 2 a 2 ≈ 11.313 7085 a 2 V = 16 3 9 a 3 ≈ 3.079 201 44 a 3 The rhombic dodecahedron has four special orthogonal projections along its axes of symmetry, centered on a face, an edge, the two types of vertex and fourfold. The last two correspond to the B2 and A2 Coxeter planes; the eight vertices where three faces meet at their obtuse angles have Cartesian coordinates: The coordinates of the six vertices where four faces meet at their acute angles are:, The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron, with permutation of coordinates and with parameter h = 1. The rhombic dodecahedron is a parallelohedron, a space-filling polyhedron, being the dual to the tetroctahedrille or half cubic honeycomb, described by two Coxeter diagrams: and. With D3d symmetry, it can be seen as an elongated trigonal trapezohedron. Other symmetry constructions of the rhombic dodecahedron are space-filling, as parallelotopes they are similar to variations of space-filling truncated octahedra.
For example, with 4 square faces, 60-degree rhombic faces, D4h dihedral symmetry, order 16. It be seen as a cuboctahedron with square pyramids augmented on the bottom. In 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the Bilinski dodecahedron, it has the same different geometry. The rhombic faces in this form have the golden ratio. Another topologically equivalent variation, sometimes called a deltoidal dodecahedron or trapezoidal dodecahedron, is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites, it has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by. Is the rhombic solution; as approaches 1/2, approaches infinity. (
Rubik's Cube is a 3-D combination puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. Called the Magic Cube, the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 via businessman Tibor Laczi and Seven Towns founder Tom Kremer, won the German Game of the Year special award for Best Puzzle that year. As of January 2009, 350 million cubes had been sold worldwide making it the world's top-selling puzzle game, it is considered to be the world's best-selling toy. On the original classic Rubik's Cube, each of the six faces was covered by nine stickers, each of one of six solid colours: white, blue, orange and yellow; the current version of the cube has been updated to coloured plastic panels instead, which prevents peeling and fading. In sold models, white is opposite yellow, blue is opposite green, orange is opposite red, the red and blue are arranged in that order in a clockwise arrangement. On early cubes, the position of the colours varied from cube to cube.
An internal pivot mechanism enables each face to turn thus mixing up the colours. For the puzzle to be solved, each face must be returned to have only one colour. Similar puzzles have now been produced with various numbers of sides and stickers, not all of them by Rubik. Although the Rubik's Cube reached its height of mainstream popularity in the 1980s, it is still known and used. Many speedcubers continue to practice similar puzzles. Since 2003, the World Cube Association, the Rubik's Cube's international governing body, has organised competitions worldwide and recognises world records. In March 1970, Larry D. Nichols invented a 2×2×2 "Puzzle with Pieces Rotatable in Groups" and filed a Canadian patent application for it. Nichols's cube was held together by magnets. Nichols was granted U. S. Patent 3,655,201 on 11 April 1972, two years before Rubik invented his Cube. On 9 April 1970, Frank Fox applied to patent an "amusement device", a type of sliding puzzle on a spherical surface with "at least two 3×3 arrays" intended to be used for the game of noughts and crosses.
He received his UK patent on 16 January 1974. In the mid-1970s, Ernő Rubik worked at the Department of Interior Design at the Academy of Applied Arts and Crafts in Budapest. Although it is reported that the Cube was built as a teaching tool to help his students understand 3D objects, his actual purpose was solving the structural problem of moving the parts independently without the entire mechanism falling apart, he did not realise that he had created a puzzle until the first time he scrambled his new Cube and tried to restore it. Rubik applied for a patent in Hungary for his "Magic Cube" on 30 January 1975, HU170062 was granted that year; the first test batches of the Magic Cube were produced in late 1977 and released in Budapest toy shops. Magic Cube was held together with interlocking plastic pieces that prevented the puzzle being pulled apart, unlike the magnets in Nichols's design. With Ernő Rubik's permission, businessman Tibor Laczi took a Cube to Germany's Nuremberg Toy Fair in February 1979 in an attempt to popularise it.
It was noticed by Seven Towns founder Tom Kremer and they signed a deal with Ideal Toys in September 1979 to release the Magic Cube worldwide. Ideal wanted at least a recognisable name to trademark; the puzzle made its international debut at the toy fairs of London, Paris and New York in January and February 1980. After its international debut, the progress of the Cube towards the toy shop shelves of the West was halted so that it could be manufactured to Western safety and packaging specifications. A lighter Cube was produced, Ideal decided to rename it. "The Gordian Knot" and "Inca Gold" were considered, but the company decided on "Rubik's Cube", the first batch was exported from Hungary in May 1980. After the first batches of Rubik's Cubes were released in May 1980, initial sales were modest, but Ideal began a television advertising campaign in the middle of the year which it supplemented with newspaper adverts. At the end of 1980 Rubik's Cube won a German Game of the Year special award, won similar awards for best toy in the UK, the US.
By 1981 Rubik's Cube had become a craze, it is estimated that in the period from 1980 to 1983 around 200 million Rubik's Cubes were sold worldwide. In March 1981 a speedcubing championship organised by the Guinness Book of World Records was held in Munich, a Rubik's Cube was depicted on the front cover of Scientific American that same month. In June 1981 The Washington Post reported that the Rubik's Cube is "a puzzle that's moving like fast food right now... this year's Hoola Hoop or Bongo Board", by September 1981 New Scientist noted that the cube had "captivated the attention of children of ages from 7 to 70 all over the world this summer."As most people could only solve one or two sides, numerous books were published including David Singmaster's Notes on Rubik's "Magic Cube" and Patrick Bossert's You Can Do the Cube. At one stage in 1981 three of the top ten best selling books in the US were books on solving the Rubik's Cube, the best-selling book of 1981 was James G. Nourse's The Simple Solution to Rubik's Cube which sold over 6 million copies.
In 1981 the Museum of Modern Art in New York exhibited a Rubik's Cube, at the 1982 World's Fair in Knoxville, Tennessee a six-foot Cube was put on display. ABC Television developed a cartoon show called Rubik, the Amazing Cube. In June 1982 the First Rubik's Cube World Championship
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
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions, its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space. Honeycombs are constructed in ordinary Euclidean space, they may be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. There are infinitely many honeycombs, which have only been classified; the more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane. In particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary space.
Another interesting family is the Hill tetrahedra and their generalizations, which can tile the space. A 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of uniform polyhedral cells, having all vertices the same. There are 28 convex examples in Euclidean 3-space called the Archimedean honeycombs. A honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell; every regular honeycomb is automatically uniform. However, there is just the cubic honeycomb. Two are quasiregular: The tetrahedral-octahedral honeycomb and gyrated tetrahedral-octahedral honeycombs are generated by 3 or 2 positions of slab layer of cells, each alternating tetrahedra and octahedra. An infinite number of unique honeycombs can be created by higher order of patterns of repeating these slab layers. A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric.
In the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a space-filling polyhedron. A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, ruling out any of the Platonic solids other than the cube. Five space-filling polyhedra can tessellate 3-dimensional euclidean space using translations only, they are called parallelohedra: Cubic honeycomb Hexagonal prismatic honeycomb Rhombic dodecahedral honeycomb Elongated dodecahedral honeycomb. Bitruncated cubic honeycomb or truncated octahedraOther known examples of space-filling polyhedra include: The Triangular prismatic honeycomb; the gyrated triangular prismatic honeycomb. The Voronoi cells of the carbon atoms in diamond are this shape; the trapezo-rhombic dodecahedral honeycomb Isohedral tilings. Sometimes, two or more different polyhedra may be combined to fill space. Besides many of the uniform honeycombs, another well known example is the Weaire–Phelan structure, adopted from the structure of clathrate hydrate crystals Weaire–Phelan structure Documented examples are rare.
Two classes can be distinguished: Non-convex cells which pack without overlapping, analogous to tilings of concave polygons. These include a packing of the small stellated rhombic dodecahedron, as in the Yoshimoto Cube. Overlapping of cells whose positive and negative densities'cancel out' to form a uniformly dense continuum, analogous to overlapping tilings of the plane. In 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size; the regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge. Apart from this effect, the hyperbolic honeycombs obey the same topological constraints as Euclidean honeycombs and polychora; the 4 compact and 11 paracompact regular hyperbolic honeycombs and many compact and paracompact uniform hyperbolic honeycombs have been enumerated. For every honeycomb there is a dual honeycomb, which may be obtained by exchanging: cells for vertices. Faces for edges; these are just the rules for dualising four-dimensional 4-polytopes, except that the usual finite method of reciprocation about a concentric hypersphere can run into problems.
The more regular honeycombs dualise neatly: The cubic honeycomb is self-dual. That of octahedra and tetrahedra is dual to that of rhombic dodecahedra; the slab honeycombs derived from uniform plane tilings are dual to each other in the same way that the tilings are. The duals of the remaining Archimedean honeycombs are all cell-transitive and have been described by Inchbald. Honeycombs can be self-dual. All n-dimensional hypercubic honeycombs with Schläfli symbols, are self-dual. List of uniform tilings Regular honeycombs Infinite skew polyhedron Plesiohedron Coxeter, H. S. M.: Regular Polytopes. Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. pp. 164–199. ISBN 0-486-23729-X. Chapter 5: Polyhedra packing and space filling Critchlow, K.: Order in space. Pearce, P.: Structure in nature is a strategy for design. Goldberg, Michael Three Infinite Families of Tetrahedral Space-Fillers Journal of Combinatorial Theory A, 16, pp. 348–354, 1974.
Goldberg, Michael The space-filling pentahedra, Journal of Combinatorial Theory, Series A Volume 13, Issue 3, November 1972, Pages 437-443 [
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.
R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.
The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.
When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form