# Decagram (geometry)

Regular decagram
A regular decagram
TypeRegular star polygon
Edges and vertices10
Schläfli symbol{10/3}
t{5/3}
Coxeter diagram
Symmetry groupDihedral (D10)
Internal angle (degrees)72°
Dual polygonself
Propertiesstar, cyclic, equilateral, isogonal, isotoxal

In geometry, a decagram is a 10-point star polygon. There is one regular decagram, containing the vertices of a regular decagon, but connected by every third point. Its Schläfli symbol is {10/3}.[1]

The name decagram combines a numeral prefix, deca-, with the Greek suffix -gram. The -gram suffix derives from γραμμῆς (grammēs) meaning a line.[2]

## 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 in girih tiles.[3]

## 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 covering of a pentagon {5}, and last is a variation of a double covering of a pentagram {5/2}. The middle is a variation of a regular decagram, {10/3}.

 {(5/2)α} {(5/3)α} {(5/4)α}

## Related figures

A regular decagram is a 10-sided polygram, represented by symbol {10/n}, containing the same vertices as regular decagon. Only one of these polygrams, {10/3} (connecting every third point), forms a regular star polygon, but there are also three ten-vertex polygrams which can be interpreted as regular compounds:

Form Convex Compound Star polygon Compounds
Image
Symbol {10/1} = {10} {10/2} = 2{5} {10/3} {10/4} = 2{5/2} {10/5} = 5{2}

{10/2} can be seen as the 2D equivalent of the 3D compound of dodecahedron and icosahedron and 4D compound of 120-cell and 600-cell; that is, the compound of two pentagonal polytopes in their respective dual positions.

{10/4} can be seen as the two-dimensional equivalent of the three-dimensional compound of small stellated dodecahedron and great dodecahedron or compound of great icosahedron and great stellated dodecahedron 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 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 (any two vertices can be transformed into each other by a symmetry of the figure).[6][7][8]

Isogonal truncations of pentagon and pentagram
Quasiregular Isogonal Quasiregular
Double covering

t{5} = {10}

t{5/4} = {10/4} = 2{5/2}

t{5/3} = {10/3}

t{5/2} = {10/2} = 2{5}

## References

1. ^ Barnes, John (2012), Gems of Geometry, Springer, pp. 28–29, ISBN 9783642309649.
2. ^ γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
3. ^ Sarhangi, Reza (2012), "Polyhedral Modularity in a Special Class of Decagram Based Interlocking Star Polygons", Bridges 2012: Mathematics, Music, Art, Architecture, Culture (PDF), pp. 165–174.
4. ^ Regular polytopes, p 93-95, regular star polygons, regular star compounds
5. ^ Coxeter, Introduction to Geometry, second edition, 2.8 Star polygons p.36-38
6. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum.
7. ^ *Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954). "Uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. The Royal Society. 246 (916): 411. Bibcode:1954RSPTA.246..401C. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446.CS1 maint: ref=harv (link)
8. ^ Coxeter, The Densities of the Regular polytopes I, p.43 If d is odd, the truncation of the polygon {p/q} is naturally {2n/d}. But if not, it consists of two coincident {n/(d/2)}'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.