Triacontatetragon

Regular triacontatetragon
A regular triacontatetragon
Type Regular polygon
Edges and vertices 34
Schläfli symbol {34}, t{17}
Coxeter diagram
Symmetry group Dihedral (D34), order 2×34
Internal angle (degrees) 160+7/17°
Dual polygon Self
Properties Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a triacontatetragon or triacontakaitetragon is a thirty-four-sided polygon or 34-gon.[1] The sum of any triacontatetragon's interior angles is 5760 degrees.

Regular triacontatetragon

A regular triacontatetragon is represented by Schläfli symbol {34} and can also be constructed as a truncated 17-gon, t{17}, which alternates two types of edges.

One interior angle in a regular triacontatetragon is (2880/17)°, meaning that one exterior angle would be (180/17)°.

The area of a regular triacontatetragon is (with t = edge length)

and its inradius is

The factor is a root of the equation .

The circumradius of a regular triacontatetragon is

As 34 = 2 × 17 and 17 is a Fermat prime, a regular triacontatetragon is constructible using a compass and straightedge.[2][3][4] As a truncated 17-gon, it can be constructed by an edge-bisection of a regular 17-gon. This means that the values of and may be expressed in terms of nested radicals.

Dissection

34-gon with 544 rhombs

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular triacontatetragon, m=17, it can be divided into 136: 8 sets of 17 rhombs. This decomposition is based on a Petrie polygon projection of a 17-cube.

Examples

Triacontatetragram

A triacontatetragram is a 34-sided star polygon. There are seven regular forms given by Schläfli symbols {34/3}, {34/5}, {34/7}, {34/9}, {34/11}, {34/13}, and {34/15}, and nine compound star figures with the same vertex configuration.


{34/3}
{34/5}
{34/7}
{34/9}
{34/11}
{34/13}
{34/15}

Many isogonal triacontatetragrams can also be constructed as deeper truncations of the regular heptadecagon {17} and heptadecagrams {17/2}, {17/3}, {17/4}, {17/5}, {17/6}, {17/7}, and {17/8}. These also create eight quasitruncations: t{17/9} = {34/9}, t{17/10} = {34/10}, t{17/11} = {34/11}, t{17/12} = {34/12}, t{17/13} = {34/13}, t{17/14} = {34/14}, t{17/15} = {34/15}, and t{17/16} = {34/16}. Some of the isogonal triacontatetragrams are depicted below, as a truncation sequence with endpoints t{17}={34} and t{17/16}={34/16}.[6]


t{17}={34}
t{17/16}={34/16}

References

  1. ↑ "Ask Dr. Math: Naming Polygons and Polyhedra". mathforum.org. Retrieved 2017-09-05.
  2. ↑ W., Weisstein, Eric. "Constructible Polygon". mathworld.wolfram.com. Retrieved 2017-09-01.
  3. ↑ Chepmell, C. H. (1913-03-01). "A construction of the regular polygon of 34 sides". Mathematische Annalen. 74 (1): 150–151. doi:10.1007/bf01455349. ISSN 0025-5831.
  4. ↑ White, Charles Edgar (1913). Theory of Irreducible Cases of Equations and Its Applications in Algebra, Geometry, and Trigonometry. p. 79.
  5. ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  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
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