Polygram (geometry)#Regular compound polygons
{{short description|Mathematical term in geometry}}
{{for|Dutch record label|PolyGram}}
File:Regular Star Polygons-en.svg
In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides. All polygons are polygrams, but they can also include disconnected sets of edges, called a compound polygon. For example, a regular pentagram, {5/2}, has 5 sides, and the regular hexagram, {6/2} or 2{3}, has 6 sides divided into two triangles.
A regular polygram {p/q} can either be in a set of regular star polygons (for gcd(p,q) = 1, q > 1) or in a set of regular polygon compounds (if gcd(p,q) > 1).{{Mathworld |urlname=Polygram |title=Polygram}}
Etymology
The polygram names combine a numeral prefix, such as penta-, with the Greek suffix -gram (in this case generating the word pentagram). The prefix is normally a Greek cardinal, but synonyms using other prefixes exist. The -gram suffix derives from γραμμῆς (grammos) meaning a line.[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dgrammh%2F γραμμή], Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
Generalized regular polygons
{{Further information|Regular polygon#Regular star polygons}}
A regular polygram, as a general regular polygon, is denoted by its Schläfli symbol {p/q}, where p and q are relatively prime (they share no factors) and q ≥ 2. For integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement.{{cite book |last=Coxeter |first=Harold Scott Macdonald |title=Regular polytopes |publisher=Courier Dover Publications |page=[https://archive.org/details/regularpolytopes0000coxe/page/93 93] |year=1973 |isbn=978-0-486-61480-9 }}
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Regular compound polygons
{{Further information|List of regular polytopes and compounds#Two dimensions}}
In other cases where n and m have a common factor, a polygram is interpreted as a lower polygon, {n/k, m/k}, with k = gcd(n,m), and rotated copies are combined as a compound polygon. These figures are called regular compound polygons.
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|+ Some regular polygon compounds !colspan=3|Triangles... !colspan=2|Squares... !Pentagons... !colspan=2|Pentagrams... |
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See also
- {{section link|List of regular polytopes and compounds|Stars}}
References
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- Cromwell, P.; Polyhedra, CUP, Hbk. 1997, {{ISBN|0-521-66432-2}}. Pbk. (1999), {{ISBN|0-521-66405-5}}. p. 175
- Grünbaum, B. and G.C. Shephard; Tilings and patterns, New York: W. H. Freeman & Co., (1987), {{ISBN|0-7167-1193-1}}.
- Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993), ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43–70.
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)
- Robert Lachlan, An Elementary Treatise on Modern Pure Geometry. London: Macmillan, 1893, p. 83 polygrams. [https://archive.org/details/elementarytreati00lachuoft]
- Branko Grünbaum, Metamorphoses of polygons, published in The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994)
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