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Difference between revisions of "Regular polygons"

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Polygons all angles of which are equal and all sides of which are equal. Cf. [[Polygon|Polygon]] for more details.
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====Comments====
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A regular polygon is a [[Polygon|polygon]], all angles of which are equal and all sides of which are equal.
In other words, a regular polygon is a polygon whose vertices all lie on one circle while its sides all touch a concentric circle. If the polygon is convex and has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080780/r0807801.png" /> vertices, it is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080780/r0807803.png" />-gon, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080780/r0807804.png" />. If it is non-convex and its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080780/r0807805.png" /> sides surround its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080780/r0807806.png" /> times, it is called a star polygon, or a star <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080780/r0807808.png" />-gon of density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080780/r0807809.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080780/r08078010.png" />. For instance, the pentagram is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080780/r08078011.png" />.
 
  
====References====
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In other words, a regular polygon is a polygon whose vertices all lie on one circle while its sides all touch a concentric circle. If the polygon is convex and has $n$ vertices, it is called an $n$-gon, and is denoted by $\{n\}$. If it is non-convex and its $n$ sides surround its centre $d$ times, it is called a star polygon, or a star $n$-gon of density $d$, and is denoted by $\{n/d\}$. For instance, the pentagram is $\{5/2\}$.
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"H.S.M. Coxeter,   "Regular complex polytopes" , Cambridge Univ. Press (1990) pp. Chapt. 1</TD></TR></table>
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====References====  
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|valign="top"|{{Ref|Co}}||valign="top"| H.S.M. Coxeter, "Regular complex polytopes", Cambridge Univ. Press (1990) pp. Chapt. 1
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Latest revision as of 17:23, 26 July 2012

2010 Mathematics Subject Classification: Primary: 51M05 [MSN][ZBL]

A regular polygon is a polygon, all angles of which are equal and all sides of which are equal.

In other words, a regular polygon is a polygon whose vertices all lie on one circle while its sides all touch a concentric circle. If the polygon is convex and has $n$ vertices, it is called an $n$-gon, and is denoted by $\{n\}$. If it is non-convex and its $n$ sides surround its centre $d$ times, it is called a star polygon, or a star $n$-gon of density $d$, and is denoted by $\{n/d\}$. For instance, the pentagram is $\{5/2\}$.

References

[Co] H.S.M. Coxeter, "Regular complex polytopes", Cambridge Univ. Press (1990) pp. Chapt. 1
How to Cite This Entry:
Regular polygons. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Regular_polygons&oldid=27206