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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 ...
, a decagram is a 10-point star polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
. There is one regular decagram, containing the vertices of a regular decagon
In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°.
A self-intersecting ''regular decagon'' i ...
, but connected by every third point. Its 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 more ...
is .
The name ''decagram'' combines a numeral prefix
Numeral or number prefixes are prefixes derived from Numeral (linguistics), numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example:
* unicycle, bicycle, tricycle (1 ...
, ''deca-
''Deca'' (American and British English spelling differences#-re, -er, International spelling as used by the International Bureau of Weights and Measures;
'', with the Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
suffix '' -gram''. The ''-gram'' suffix derives from ''γραμμῆς'' (''grammēs'') meaning a line.
Regular decagram
For a regular decagram with unit edge lengths, the proportions of the crossing points on each edge are as shown below.
Applications
Decagrams have been used as one of the decorative motifs ingirih tiles
''Girih'' tiles are a set of five tiles that were used in the creation of Islamic geometric patterns using strapwork ('' girih'') for decoration of buildings in Islamic architecture. They have been used since about the year 1200 and their arrang ...
.
:
Isotoxal variations
An isotoxal polygon has two vertices and one edge. There are isotoxal decagram forms, which alternates vertices at two radii. Each form has a freedom of one angle. The first is a variation of a double-wound of a pentagon , and last is a variation of a double-wound of a pentagram . The middle is a variation of a regular decagram, .Related figures
A regular decagram is a 10-sidedpolygram
PolyGram N.V. was a multinational entertainment company and major music record label formerly based in the Netherlands. It was founded in 1962 as the Grammophon-Philips Group by Dutch corporation Philips and German corporation Siemens, to be a ...
, represented by symbol , containing the same vertices as regular decagon
In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°.
A self-intersecting ''regular decagon'' i ...
. Only one of these polygrams, (connecting every third point), forms a regular star polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
, but there are also three ten-vertex polygrams which can be interpreted as regular compounds:
* is a compound of five degenerate digon
In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visua ...
s 5
* is a compound of two pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aroun ...
s 2
* is a compound of two pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
s 2.
can be seen as the 2D equivalent of the 3D compound of dodecahedron and icosahedron
In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound.
As a compound
It can be seen as the compound of an icosahedron and dodecahedron. It is one of four compounds constructed from a Platonic solid or Ke ...
and 4D compound of 120-cell and 600-cell; that is, the compound of two pentagonal polytope
In geometry, a pentagonal polytope is a regular polytope in ''n'' dimensions constructed from the H''n'' Coxeter group. The family was named by H. S. M. Coxeter, because the two-dimensional pentagonal polytope is a pentagon. It can be named by i ...
s in their respective dual positions.
can be seen as the two-dimensional equivalent of the three-dimensional compound of small stellated dodecahedron and great dodecahedron
The compound of small stellated dodecahedron and great dodecahedron is a polyhedron compound where the great dodecahedron is internal to its dual, the small stellated dodecahedron.
This can be seen as one of the two three-dimensional equivalen ...
or compound of great icosahedron and great stellated dodecahedron
There are two different compounds of great icosahedron and great stellated dodecahedron: one is a dual compound and a stellation of the great icosidodecahedron, the other is a stellation of the icosidodecahedron.
Dual compound
It can be seen as ...
through similar reasons. It has six four-dimensional analogues, with two of these being compounds of two self-dual star polytopes, like the pentagram itself; the compound of two great 120-cells
Compound may refer to:
Architecture and built environments
* Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall
** Compound (fortification), a version of the above fortified with defensive struct ...
and the compound of two grand stellated 120-cells. A full list can be seen at Polytope compound#Compounds with duals.
Deeper truncations of the regular pentagon and pentagram can produce intermediate star polygon forms with ten equally spaced vertices and two edge lengths that remain 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 ...
(any two vertices can be transformed into each other by a symmetry of the figure).Coxeter, The Densities of the Regular polytopes I, p.43 If d is odd, the truncation of the polygon is naturally . But if not, it consists of two coincident 's; two, because each side arises from an original side and once from an original vertex. Thus the density of a polygon is unaltered by truncation.
See also
* List of regular polytopes and compounds#StarsReferences
{{Polygons 10 (number) 10