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A Bill picture is a sequence of nested regular polygons in which subsequent polygons are each rotated so that they begin one vertex further. The term was coined by Trott (2004, pp. 88-89) and commemorates Swiss artist Max Bill, who in 1938 created a picture showing a similar arrangement of the equilateral triangle through octagon (Huttingerr 1978, Bill 1987).The figure above shows the Bill picture including regular polygons up through theregular dodecagon.

The gyroid, illustrated above, is an infinitely connected periodic minimal surface containing no straight lines (Osserman 1986) that was discovered by Schoen (1970). Große-Brauckmann and Wohlgemuth (1996) proved that the gyroid is embedded.The gyroid is the only known embedded triply periodic minimal surface with triple junctions. In addition, unlike the five triply periodic minimal surfaces studied by Anderson et al. (1990), the gyroid does not have any reflectional symmetries (Große-Brauckmann 1997).The image above shows a metal print of the gyroid created by digital sculptor BathshebaGrossman (https://www.bathsheba.com/).

Consider the plane quartic curve defined bywhere homogeneous coordinates have been used here so that can be considered a parameter (the plot above shows the curve for a number of values of between and 2), over a field of characteristic 3. Hartshorne (1977, p. 305) terms this "a funny curve" since it is nonsingular, every point is an inflection point, and the dual curve is isomorphic to but the natural map is purely inseparable.The surface in complex projective coordinates (Levy 1999, p. ix; left figure), and with the ideal surface determined by the equation(Thurston 1999, p. 3; right figure) is more properly known as the Klein quarticor Klein curve. It has constant zero Gaussian curvature.Klein (1879; translation reprinted in 1999) discovered that this surface has a number of remarkable properties, including an incredible 336-fold symmetry when mirror reflections are allowed (Levy 1999, p. ix; Thurston..

An -gonal trapezohedron, also called an antidipyramid, antibipyramid, or deltohedron (not to be confused with a deltahedron), is a dual polyhedra of an -antiprism. Unfortunately, the name for these solids is not particularly well chosen since their faces are not trapezoids but rather kites. The trapezohedra are isohedra.The 3-trapezohedron (trigonal trapezohedron) is a rhombohedron having all six faces congruent. A special case is the cube (oriented along a space diagonal), corresponding to the dual of the equilateral 3-antiprism (i.e., the octahedron).A 4-trapezohedron (tetragonal trapezohedron) appears in the upper left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).The trapezohedra generated by taking the duals of the equilateral antiprisms have side length , half-heights (half the peak-to-peak distance) , surface areas , and volumes..

"Escher's solid" is the solid illustrated on the right pedestal in M. C. Escher's Waterfall woodcut (Bool et al. 1982, p. 323). It is obtained by augmenting a rhombic dodecahedron until incident edges become parallel, corresponding to augmentation height of for a rhombic dodecahedron with unit edge lengths.It is the first rhombic dodecahedron stellation and is a space-filling polyhedron. Its convex hull is a cuboctahedron.It is implemented in the Wolfram Languageas PolyhedronData["EschersSolid"].It has edge lengths(1)(2)surface area and volume(3)(4)and moment of inertia tensor(5)The skeleton of Escher's solid is the graph of the disdyakis dodecahedron.Escher's solid can also be viewed as a polyhedron compound of three dipyramids (nonregular octahedra) with edges of length 2 and ...

The pentagonal dipyramid is one of the convex deltahedra, and Johnson solid . It is also the dual polyhedron of the pentagonal prism and is an isohedron.It is implemented in the Wolfram Language as PolyhedronData["Dipyramid", 5].A pentagonal dipyramid appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).For a pentagonal dipyramid having a base with unit edge lengths, the circumradiusof the base pentagon is(1)In order for the top and bottom edges to also be of unit length, the polyhedron must be of height(2)The ratio of is therefore given by(3)where is the golden ratio.The surface area and volume of a unit pentagonal dipyramid are(4)(5)

The elongated square dipyramid with unit edge lengths is Johnson Solid .An elongated square dipyramid (having a central ribbon composed of rectangles instead of squares) appears in the top center as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).

Dürer's solid, also known as the truncated triangular trapezohedron, is the 8-faced solid depicted in an engraving entitled Melancholia I by Albrecht Dürer (The British Museum, Burton 1989, Gellert et al. 1989), the same engraving in which Dürer's magic square appears, which depicts a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Although Dürer does not specify how his solid is constructed, Schreiber (1999) has noted that it appears to consist of a distorted cube which is first stretched to give rhombic faces with angles of , and then truncated on top and bottom to yield bounding triangular faces whose vertices lie on the circumsphere of the azimuthal cube vertices.It is implemented in the Wolfram Languageas PolyhedronData["DuererSolid"].The skeleton of Dürer's solid is the Dürer graph (i.e., generalized Petersen graph ).Starting..

The stella octangula is a polyhedron compound composed of a tetrahedron and its dual (a second tetrahedron rotated with respect to the first). The stella octangula is also (incorrectly) called the stellated tetrahedron, and is the only stellation of the octahedron. A wireframe version of the stella octangula is sometimes known as the merkaba and imbued with mystic properties.The name "stella octangula" is due to Kepler (1611), but the solid was known earlier to many others, including Pacioli (1509), who called it the "octaedron elevatum," and Jamnitzer (1568); see Cromwell (1997, pp. 124 and 152).It is implemented in the Wolfram Languageas PolyhedronData["StellaOctangula"].A stella octangula can be inscribed in a cube, deltoidal icositetrahedron, pentagonal icositetrahedron, rhombic dodecahedron, small triakis octahedron, and tetrakis hexahedron, (E. Weisstein, Dec. 24-25,..

In general, a triakis octahedron is a non-regular icositetrahedron that can be constructed as a positive augmentation of regular octahedron. Such a solid is also known as a trisoctahedron, especially to mineralogists (Correns 1949, p. 41; Berry and Mason 1959, p. 127). While the resulting icositetrahedron is not regular, its faces are all identical. The small triakis octahedron, called simply the triakis octahedron by Holden (1971, p. 55), is the 24-faced dual polyhedron of the truncated cube and is Wenninger dual . The addition of the word "small" is necessary to distinguish it from the great triakis octahedron, which is the dual of the stellated truncated hexahedron. The small triakis octahedron It can be constructed by augmentation of a unit edge-length octahedron by a pyramid with height .A small triakis octahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's..

The small rhombicuboctahedron is the 26-faced Archimedean solid consisting of faces . Although this solid is sometimes also called the truncated icosidodecahedron, this name is inappropriate since true truncation would yield rectangular instead of square faces. It is uniform polyhedron and Wenninger model . It has Schläfli symbol r and Wythoff symbol .The solid may also be called an expanded (or cantellated) cube or octahedron sinceit may be constructed from either of these solids by the process of expansion.A small rhombicuboctahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).It is implemented in the Wolfram Languageas PolyhedronData["SmallRhombicuboctahedron"].Its dual polyhedron is the deltoidal icositetrahedron, also called the trapezoidal icositetrahedron. The inradius of the..

The (first) rhombic dodecahedron is the dual polyhedron of the cuboctahedron (Holden 1971, p. 55) and Wenninger dual . Its sometimes also called the rhomboidal dodecahedron (Cotton 1990), and the "first" may be included when needed to distinguish it from the Bilinski dodecahedron (Bilinski 1960, Chilton and Coxeter 1963).A rhombic dodecahedron appears in the upper right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).The rhombic dodecahedron is implemented in the WolframLanguage as PolyhedronData["RhombicDodecahedron"].The 14 vertices of the rhombic dodecahedron are joined by 12 rhombiof the dimensions shown in the figure below, where(1)(2)(3)(4)(5)The rhombic dodecahedron can be built up by a placing six cubes on the faces of a seventh, in the configuration of a metal "jack" (left figure). Joining..

A cuboctahedron, also called the heptaparallelohedron or dymaxion (the latter according to Buckminster Fuller; Rawles 1997), is Archimedean solid with faces . It is one of the two convex quasiregular polyhedra. It is uniform polyhedron and Wenninger model . It has Schläfli symbol and Wythoff symbol .A cuboctahedron appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43), as well is in the mezzotint "Crystal" (Bool et al. 1982, p. 293).It is implemented in the Wolfram Languageas PolyhedronData["Cuboctahedron"].It is shown above in a number of symmetric projections.The dual polyhedron is the rhombic dodecahedron. The cuboctahedron has the octahedral group of symmetries. According to Heron, Archimedes ascribed the cuboctahedron to Plato (Heath 1981; Coxeter 1973, p. 30). The polyhedron..

Dürer's magic square is a magic square with magic constant 34 used in an engraving entitled Melancholia I by Albrecht Dürer (The British Museum, Burton 1989, Gellert et al. 1989). The engraving shows a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Dürer's magic square is located in the upper right-hand corner of the engraving. The numbers 15 and 14 appear in the middle of the bottom row, indicating the date of the engraving, 1514.Dürer's magic square has the additional property that the sums in any of the four quadrants, as well as the sum of the middle four numbers, are all 34 (Hunter and Madachy 1975, p. 24). It is thus a gnomon magic square. In addition, any pair of numbers symmetrically placed about the center of the square sums to 17, a property making the square even more magical...

A maze, also known as a labyrinth, as is a set of passages (with impermeable walls). The goal of the maze is to start at one given point and find a path through the passages that reaches a second given point.The back of a clay accounting tablet from Pylos, Greece is illustrated above (Wolfram 2002, p. 43). Legend has it that it was the plan for the labyrinth housing the minotaur in the palace at Knossos, Crete, and that it was designed by Daedalus. It is also said that it was a logo for the city of Troy-or perhaps the plan of some of its walls (Wolfram 2002, p. 873).The above pattern (in either its square or rounded form) has appeared with remarkably little variation in a large variety of places all over the world-from Cretan coins, to graffiti at Pompeii, to the floor of the cathedral at Chartres, to carvings in Peru, to logos for aboriginal tribes. For probably three thousand years, it has been the single most common design used for mazes (Wolfram..

"Spikey" is the logo of Wolfram Research, makers of Mathematica and the Wolfram Language. In its original (Version 1) form, it is an augmented icosahedron with an augmentation height of , not to be confused with the great stellated dodecahedron, which is a distinct solid. This gives it 60 equilateral triangular faces, making it a deltahedron. More elaborate forms of Spikey have been used for subsequent versions of Mathematica. In particular, Spikeys for Version 2 and up are based on a hyperbolic dodecahedron with embellishments rather than an augmented icosahedron (Trott 2007, Weisstein 2009).The "classic" (Version 1) Spikey illustrated above is implemented in theWolfram Language as PolyhedronData["MathematicaPolyhedron"].The skeleton of the classic Spikey is the graph of thetriakis icosahedron.A glyph corresponding to the classic Spikey, illustrated above, is available as the character \[MathematicaIcon]..

The small stellated dodecahedron is the Kepler-Poinsot solids whose dual polyhedron is the great dodecahedron. It is also uniform polyhedron , Wenninger model , and is the first stellation of the dodecahedron (Wenninger 1989). The small stellated dodecahedron has Schläfli symbol and Wythoff symbol . It is concave, and is composed of 12 pentagrammic faces ().The small stellated dodecahedron appeared ca. 1430 as a mosaic by Paolo Uccello on the floor of San Marco cathedral, Venice (Muraro 1955). It was rediscovered by Kepler (who used th term "urchin") in his work Harmonice Mundi in 1619, and again by Poinsot in 1809.The skeleton of the small stellated dodecahedron is isomorphic to the icosahedralgraph.Schläfli (1901, p. 134) did not recognize the small stellated dodecahedron as a regular solid because it violates the polyhedral formula, instead satisfying(1)where is the number of vertices, the number of edges,..

The regular icosahedron (often simply called "the" icosahedron) is the regular polyhedron and Platonic solid illustrated above having 12 polyhedron vertices, 30 polyhedron edges, and 20 equivalent equilateral triangle faces, .The regular icosahedron is also uniform polyhedron and Wenninger model . It is described by the Schläfli symbol and Wythoff symbol . Coxeter et al. (1999) have shown that there are 58 icosahedron stellations (giving a total of 59 solids when the icosahedron itself is included).The regular icosahedron is implemented in the Wolfram Language as Icosahedron[], and precomputed properties are available as PolyhedronData["Icosahedron"].Two icosahedra constructed in origami are illustrated above (Gurkewitz and Arnstein 1995, p. 53). This construction uses 30 triangle edge modules, each made from a single sheet of origami paper.Two icosahedra appears as polyhedral "stars"..

The regular dodecahedron, often simply called "the" dodecahedron, is the Platonic solid composed of 20 polyhedron vertices, 30 polyhedron edges, and 12 pentagonal faces, . It is also uniform polyhedron and Wenninger model . It is given by the Schläfli symbol and the Wythoff symbol .The regular dodecahedron is implemented in the Wolfram Language as Dodecahedron[], and precomputed properties are available as PolyhedronData["Dodecahedron"].There are 43380 distinct nets for the regular dodecahedron, the same number as for the icosahedron (Bouzette and Vandamme, Hippenmeyer 1979, Buekenhout and Parker 1998). Questions of polyhedron coloring of the regular dodecahedron can be addressed using the Pólya enumeration theorem.The image above shows an origami regular dodecahedron constructed using six dodecahedron units, each consisting of a single sheet of paper (Kasahara and Takahama 1987, pp. 86-87).A..

The trefoil knot , also called the threefoil knot or overhand knot, is the unique prime knot with three crossings. It is a (3, 2)-torus knot and has braid word . The trefoil and its mirror image are not equivalent, as first proved by Dehn (1914). In other words, the trefoil knot is not amphichiral. It is, however, invertible, and has Arf invariant 1.Its laevo form is implemented in the WolframLanguage, as illustrated above, as KnotData["Trefoil"].M. C. Escher's woodcut "Knots" (Bool et al. 1982, pp. 128 and 325; Forty 2003, Plate 71) depicts three trefoil knots composed of differing types of strands. A preliminary study (Bool et al. 1982, p. 123) depicts another trefoil.The animation above shows a series of gears arranged along a Möbiusstrip trefoil knot (M. Trott).The bracket polynomial can be computed as follows.(1)(2)Plugging in(3)(4)gives(5)The corresponding Kauffman polynomial..

The spherical curve taken by a ship which travels from the south pole to the north pole of a sphere while keeping a fixed (but not right) angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles.It is given by the parametric equations(1)(2)(3)where(4)and is a constant. Plugging in therefore gives(5)(6)(7)It is a special case of a loxodrome.The arc length, curvature,and torsion are all slightly complicated expressions.A series of spherical spirals are illustrated in Escher's woodcuts "Sphere Surface with Fish" (Bool et al. 1982, pp. 96 and 318) and "Sphere Spirals" (Bool et al. 1982, p. 319; Forty 2003, Plate 67).

The Poincaré hyperbolic disk is a two-dimensional space having hyperbolic geometry defined as the disk , with hyperbolic metric(1)The Poincaré disk is a model for hyperbolic geometry in which a line is represented as an arc of a circle whose ends are perpendicular to the disk's boundary (and diameters are also permitted). Two arcs which do not meet correspond to parallel rays, arcs which meet orthogonally correspond to perpendicular lines, and arcs which meet on the boundary are a pair of limits rays. The illustration above shows a hyperbolic tessellation similar to M. C. Escher's Circle Limit IV (Heaven and Hell) (Trott 1999, pp. 10 and 83).The endpoints of any arc can be specified by two angles around the disk and . Define(2)(3)Then trigonometry shows that in the above diagram,(4)(5)so the radius of the circle forming the arc is and its center is located at , where(6)The half-angle subtended by the arc is then(7)so(8)The..

The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322-323). The strip bearing his name was invented by Möbius in 1858, although it was independently discovered by Listing, who published it, while Möbius did not (Derbyshire 2004, p. 381). Like the cylinder, it is not a true surface, but rather a surface with boundary (Henle 1994, p. 110).The Möbius strip has Euler characteristic (Dodson and Parker 1997, p. 125).According to Madachy (1979), the B. F. Goodrich Company patented a conveyor belt in the form of a Möbius strip which lasts twice as long as conventional belts. M. C. Escher was fond of portraying Möbius strips, and..

The necker cube is an illusion in which a two-dimensional drawing of an array of cubes appears to simultaneously protrude from and intrude into the page.A Necker cube appears on the banner shown in Escher's lithographs "Metamorphosis I" (Bool et al. 1982, p. 271; Forty 2003, p. 39), "Cycle" (Bool et al. 1982, p. 274), and "Convex and Concave". It is also the basis for the arcade game Q*bert.Depending on the view point chosen for projection, the cubes may be composed of one,two, or three types of rhombi.The Necker cube is also a tiling that was used in ancient times, including as a mosaic on the floor of one of the houses in Pompeii, as illustrated in the photograph above (courtesy of S. Jaskulowski).The image above shows a Necker cube pattern emblazoned on a quilt created by Janice Ewing using a pattern by Karen Combs. ..

The wallpaper groups are the 17 possible plane symmetry groups. They are commonly represented using Hermann-Mauguin-like symbols or in orbifold notation (Zwillinger 1995, p. 260).orbifold notationHermann-Mauguin symbolop12222p2**pmxxpg*2222pmm22*pmg22xpggx*cm2*22cmm442p4*442p4m4*2p4g333p3*333p3ml3*3p3lm632p6*632p6mPatterns created with Artlandia SymmetryWorks for each of these groups are illustrated above.Beautiful patterns can be created by repeating geometric and artistic motifs according to the symmetry of the wallpaper groups, as exemplified in works by M. C. Escher and in the patterns created by I. Bakshee in the Wolfram Language using Artlandia, illustrated above.For a description of the symmetry elements present in each space group, see Coxeter (1969, p. 413)...

A strange loop is a phenomenon in which, whenever movement is made upwards or downwards through the levels of some hierarchical system, the system unexpectedly arrives back where it started. Hofstadter (1989) uses the strange loop as a paradigm in which to interpret paradoxes in logic (such as Grelling's paradox, the liar's paradox, and Russell's antinomy) and calls a system in which a strange loop appears a tangled hierarchy.Canon 5 from Bach's Musical Offering (sometimes known as Bach's endlessly rising canon) is a musical piece that continues to rise in key, modulating through the entire chromatic scale until it ends in the same key in which it began. This is the first example cited by Hofstadter (1989) as a strange loop.Other examples include the endlessly rising stairs in M. C. Escher 1960 lithograph Ascending and Descending, the endlessly falling waterfall in his 1961 lithograph Waterfall, and the pair of hands drawing each..

An impossible figure in which a stairway in the shape of a square appears to circulate indefinitely while still possessing normal steps (Penrose and Penrose 1958). The Dutch artist M. C. Escher included a Penrose stairway in his mind-bending illustration "Ascending and Descending" (Bool et al. 1982, p. 321; Forty 2003, Plate 68). Distorted variations of the stairway are also depicted in Escher's "House of Stairs" (Bool et al. 1982, p. 301; Forty 2003, Plate 40).In the 1998 film The Avengers, Uma Thurman is shown walking down a Penrose stairway and ending up back where she began.

Consider an arbitrary one-dimensional map(1)(with implicit parameter ) at the onset of chaos. After a suitable rescaling, the Feigenbaum function(2)is obtained. This function satisfies(3)with .Proofs for the existence of an even analytic solution to this equation, sometimes called the Feigenbaum-Cvitanović functional equation, have been given by Campanino and Epstein (1981), Campanino et al. (1982), and Lanford (1982, 1984).The picture above illustrate the Feigenbaum function for the logistic map with ,(4)along the real axis (M. Trott, pers. comm., Sept. 9, 2003).The images above show two views of a sculpture presented by Stephen Wolfram to Mitchell Feigenbaum on the occasion of his 60th birthday that depicts the Feigenbaum function in the complex plane. The sculpture (photos courtesy of A. Young) was designed by M. Trott and laser-etched into a block of glass by Bathsheba Grossman (https://www.bathsheba.com/)...

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