Decagram (geometry)

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Regular decagram
Regular star polygon 10-3.svg
A regular decagram
Type Regular star polygon
Edges and vertices 10
Schläfli symbol {10/3}
t{5/3}
Coxeter diagram CDel node 1.pngCDel 10.pngCDel rat.pngCDel d3.pngCDel node.png
CDel node 1.pngCDel 5-3.pngCDel node 1.png
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[edit]

For a regular decagram with unit edge lengths, the proportions of the crossing points on each edge are as shown below.

Decagram lengths.svg

Applications[edit]

Decagrams have been used as one of the decorative motifs in girih tiles.[3]

Girih tiles.svg

Related figures[edit]

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 Regular polygon 10.svg Regular star figure 2(5,1).svg Regular star polygon 10-3.svg Regular star figure 2(5,2).svg Regular star figure 5(2,1).svg
Symbol {10/1} = {10} {10/2} = 2{5} {10/3} {10/4} = 2{5/2} {10/5} = 5{2}

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
Regular polygon truncation 5 1.svg
t{5} = {10}
Regular polygon truncation 5 2.svg Regular polygon truncation 5 3.svg Regular star polygon 5-2.svg
t{5/4} = {10/4} = 2{5/2}
Regular star truncation 5-3 1.svg
t{5/3} = {10/3}
Regular star truncation 5-3 2.svg Regular star truncation 5-3 3.svg Regular polygon 5.svg
t{5/2} = {10/2} = 2{5}

See also[edit]

References[edit]

  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. 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.