# Decagram (geometry)

Regular decagram
A regular decagram
Type Regular star polygon
Edges and vertices 10
Schläfli symbol {10/3}
t{5/3}
Coxeter diagram
Symmetry group Dihedral (D10)
Internal angle (degrees) 72°
Dual polygon self
Properties star, 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 combine 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]

## 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.

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.
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.