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Triacontatetragon

Regular triacontatetragon
Regular polygon 34.svg
A regular triacontatetragon
Type Regular polygon
Edges and vertices 34
Schläfli symbol {34}, t{17}
Coxeter diagram CDel node 1.pngCDel 3x.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 17.pngCDel node 1.png
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
34-gon rhombic dissection.svg 34-gon-dissection-star.svg 34-gon rhombic dissection2.svg 34-gon rhombic dissectionx.svg 34-gon-dissection-random.svg

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.

Regular star polygon 34-3.svg
{34/3}
Regular star polygon 34-5.svg
{34/5}
Regular star polygon 34-7.svg
{34/7}
Regular star polygon 34-9.svg
{34/9}
Regular star polygon 34-11.svg
{34/11}
Regular star polygon 34-13.svg
{34/13}
Regular star polygon 34-15.svg
{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]

Regular polygon truncation 17 1.svg
t{17}={34}
CDel node 1.pngCDel 17.pngCDel node 1.png
Regular polygon truncation 17 2.svg Regular polygon truncation 17 3.svg Regular polygon truncation 17 4.svg Regular polygon truncation 17 5.svg Regular polygon truncation 17 6.svg Regular polygon truncation 17 7.svg Regular polygon truncation 17 8.svg Regular polygon truncation 17 9.svg
t{17/16}={34/16}
CDel node 1.pngCDel 17.pngCDel rat.pngCDel 16.pngCDel node 1.png

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