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 c ...

, the heptagrammic-order heptagonal tiling is a

regular 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 instrum ...

star-tiling of the

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

. It has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...

of . The vertex figure

heptagram A heptagram, septagram, septegram or septogram is a seven-point star drawn with seven straight strokes. The name ''heptagram'' combines a numeral prefix, '' hepta-'', with the Greek suffix ''-gram''. The ''-gram'' suffix derives from ''γρ� ...

s are ,

. The heptagonal faces overlap with

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

Related tilings

It has the same

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...

as the regular

order-7 triangular tiling In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of . Hurwitz surfaces The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is t ...

, . The full set of edges coincide with the edges of a

heptakis heptagonal tiling In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional ...

. :

It is related to a Kepler-Poinsot polyhedron, the

great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel penta ...

, , which is polyhedron and a

-3 regular star-tiling on the sphere (resembling a regular icosahedron in this state, similarly to this tessellation resembling the order-7 triangular tiling): : Great dodecahedron tiling

References

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

, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) *

External links

* * Heptagonal tilings Hyperbolic tilings Isogonal tilings Isohedral tilings Regular tilings Heptagrammic-order tilings {{geometry-stub