geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

or nonplanar

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...

s, allowing the figure to extend indefinitely without folding round to form a

closed surface In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as g ...

. Skew apeirohedra have also been called polyhedral sponges. Many are directly related to a

convex uniform honeycomb In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. Twenty-eight such honeycombs are known: * the familiar cubic honeycomb and 7 tr ...

, being the

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

al surface of a

honeycomb A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey ...

with some of the cells removed. Characteristically, an infinite skew polyhedron divides 3-dimensional space into two halves. If one half is thought of as ''solid'' the figure is sometimes called a partial honeycomb.

Regular skew apeirohedra

According to

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

, in 1926

John Flinders Petrie In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...

generalized the concept of

regular skew polygon The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrume ...

s (nonplanar polygons) to ''regular skew polyhedra'' (apeirohedra). Coxeter and Petrie found three of these that filled 3-space: There also exist

chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...

skew apeirohedra of types , , and . These skew apeirohedra are

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

edge-transitive In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given t ...

, and

face-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...

, but not mirror symmetric . Beyond Euclidean 3-space, in 1967 C. W. L. Garner published a set of 31 regular skew polyhedra in hyperbolic 3-space.

Gott's regular pseudopolyhedrons

J. Richard Gott John Richard Gott III (born February 8, 1947) is a professor of astrophysics, astrophysical sciences at Princeton University. He is known for his work on time travel and the Doomsday argument. Exotic matter time travel theories Paul Davies's ...

in 1967 published a larger set of seven infinite skew polyhedra which he called ''regular pseudopolyhedrons'', including the three from Coxeter as , , and and four new ones: , , , .The Symmetries of things, Pseudo-platonic polyhedra, p.340-344 Gott relaxed the definition of regularity to allow his new figures. Where Coxeter and Petrie had required that the vertices be symmetrical, Gott required only that they be congruent. Thus, Gott's new examples are not regular by Coxeter and Petrie's definition. Gott called the full set of ''regular polyhedra'', ''regular tilings'', and ''regular pseudopolyhedra'' as regular generalized polyhedra, representable by a Schläfli symbol, with by p-gonal faces, ''q'' around each vertex. However neither the term "pseudopolyhedron" nor Gott's definition of regularity have achieved wide usage. Crystallographer A.F. Wells in 1960's also published a list of skew apeirohedra.

Melinda Green The Buddhabrot is the probability distribution over the trajectories of points that escape the Mandelbrot fractal. Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddha, seated in a meditation pose with a forehead ...

publishe
many more
in 1998.

Prismatic forms

There are two ''prismatic'' forms: # : 5 squares on a vertex (Two parallel

square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of th ...

s connected by cubic holes.) # : 8 triangles on a vertex (Two parallel

triangle tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular Tessellation, tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of ...

s connected by

octahedral In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...

holes.)

Other forms

is also formed from parallel planes of ''triangular tilings'', with alternating octahedral holes going both ways. is composed of 3 coplanar pentagons around a vertex and two perpendicular pentagons filling the gap. Gott also acknowledged that there are other periodic forms of the regular planar tessellations. Both the ''square tiling'' and ''triangular tiling'' can be curved into approximating infinite cylinders in 3-space.

Theorems

He wrote some theorems: # For every regular polyhedron : (p-2)*(q-2)<4. For Every regular tessellation: (p-2)*(q-2)=4. For every regular pseudopolyhedron: (p-2)*(q-2)>4. # The number of faces surrounding a given face is p*(q-2) in any regular generalized polyhedron. # Every regular pseudopolyhedron approximates a negatively curved surface. # The seven regular pseudopolyhedron are repeating structures.

Uniform skew apeirohedra

There are many other

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...

(

) skew apeirohedra. Wachmann, Burt and Kleinmann (1974) discovered many examples but it is not known whether their list is complete. A few are illustrated here. They can be named by their

vertex configuration In geometry, a vertex configurationCrystallography ...

, although it is not a unique designation for skew forms. Others can be constructed as augmented chains of polyhedra:

References

, ''Regular Polytopes'', Third edition, (1973), Dover edition, * ''Kaleidoscopes: Selected Writings of H.S.M. Coxeter'', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

** (Paper 2) H.S.M. Coxeter, "The Regular Sponges, or Skew Polyhedra", ''

Scripta Mathematica ''Scripta Mathematica'' was a quarterly journal published by Yeshiva University devoted to the philosophy, history, and expository treatment of mathematics. It was said to be, at its time, "the only mathematical magazine in the world edited by spe ...

'' 6 (1939) 240-244. *

John H. Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...

, Heidi Burgiel,

Chaim Goodman-Strauss Chaim Goodman-Strauss (born June 22, 1967 in Austin TX) is an American mathematician who works in convex geometry, especially aperiodic tiling. He is on the faculty of the University of Arkansas and is a co-author with John H. Conway of ''The Sym ...

, (2008) ''The Symmetries of Things'', (Chapter 23, Objects with prime symmetry, pseudo-platonic polyhedra, p340-344) *

* A. F. Wells, ''Three-Dimensional Nets and Polyhedra'', Wiley, 1977

*A. Wachmann, M. Burt and M. Kleinmann, ''Infinite polyhedra'', Technion, 1974. 2nd Edn. 2005. * E. Schulte, J.M. Will
On Coxeter's regular skew polyhedra
Discrete Mathematics, Volume 60, June–July 1986, Pages 253–262

External links

* * *

18 SYMMETRY OF POLYTOPES AND POLYHEDRA, Egon Schulte: 18.3 REGULAR SKEW POLYHEDRA

{{Polyhedra Polyhedra