Order-7 Heptagrammic Tiling on:
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. The overlapping heptagrams subdivide the hyperbolic plane into isosceles triangles, 14 of which form each heptagram.
Each point of the hyperbolic plane that does not lie on a heptagram edge belongs to the central heptagon of one heptagram, and is in one of the points of exactly one other heptagram. The
It is related to a Kepler-Poinsot polyhedron, the 
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 order-7 heptagrammic tiling is a 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'' ...
by overlapping 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.
Description
This tiling is aregular
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, and 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 heptagrams forming the tiling are of type , . The overlapping heptagrams subdivide the hyperbolic plane into isosceles triangles, 14 of which form each heptagram.
Each point of the hyperbolic plane that does not lie on a heptagram edge belongs to the central heptagon of one heptagram, and is in one of the points of exactly one other heptagram. The winding number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tu ...
of each heptagram around its points is one, and the winding number around the central heptagon is two, so adding these two numbers together, each point of the plane is surrounded three times; that is, the 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 ...
of the tiling is 3.
In the Euclidean plane, a heptagram of type would have angles of 3/7 at its vertices, but in the hyperbolic plane heptagrams can have the sharper vertex angle 2/7 that is needed to make exactly seven other heptagrams meet up at the center of each heptagram of the tiling.
Related tilings
It has the samevertex 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. The valance 6 vertices in this tiling are false-vertices in the 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� ...
mic one caused by crossed edges.
:
It is related to a Kepler-Poinsot polyhedron, the small stellated dodecahedron
In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeti ...
, , which is polyhedron and a 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 ...
-3 regular star-tiling on the sphere:
:
References
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) *See also
External links
* * Hyperbolic tilings Isogonal tilings Isohedral tilings Regular tilings Heptagrammic tilings Order-7 tilings {{hyperbolic-geometry-stub