# Tetracontagon

Regular tetracontagon | |
---|---|

A regular tetracontagon | |

Type | Regular polygon |

Edges and vertices | 40 |

Schläfli symbol | {40}, t{20}, tt{10}, ttt{5} |

Coxeter diagram | |

Symmetry group | Dihedral (D_{40}), order 2×40 |

Internal angle (degrees) | 171° |

Dual polygon | Self |

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

In geometry, a **tetracontagon** or **tessaracontagon** is a forty-sided polygon or 40-gon.^{[1]}^{[2]} The sum of any tetracontagon's interior angles is 6840 degrees.

## Contents

## Regular tetracontagon

A *regular tetracontagon* is represented by Schläfli symbol {40} and can also be constructed as a truncated icosagon, t{20}, which alternates two types of edges. Furthermore, it can also be constructed as a twice-truncated decagon, tt{10}, or a thrice-truncated pentagon, ttt{5}.

One interior angle in a regular tetracontagon is 171°, meaning that one exterior angle would be 9°.

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

and its inradius is

The factor is a root of the octic equation .

The circumradius of a regular tetracontagon is

As 40 = 2^{3} × 5, a regular tetracontagon is constructible using a compass and straightedge.^{[3]} As a truncated icosagon, it can be constructed by an edge-bisection of a regular icosagon. This means that the values of and may be expressed in radicals as follows:

## Construction of a regular tetracontagon

### Circumcircle is given

- Construct first the side length JE
_{1}of a pentagon. - Transfer this on the circumcircle, there arises the intersection E
_{39}. - Connect the point E
_{39}with the central point M, there arises the angle E_{39}ME_{1}with 72°. - Halve the angle E
_{39}ME_{1}, there arise the intersection E_{40}and the angle E_{40}ME_{1}with 9°. - Connect the point E
_{1}with the point E_{40}, there arises the first side length*a*of the tetracontagon. - Finally you transfer the segment E
_{1}E_{40}(side length*a*) repeatedly counterclockwise on the circumcircle until arises a regular tetracontagon.

**The golden ratio**

### Side length is given

- Draw a segment E
_{40}E_{1}whose length is the given*side length a*of the tetracontagon. - Extend the segment E
_{40}E_{1}by more than two times. - Draw each a circular arc about the points E
_{1}and E_{40}, there arise the intersections A and B. - Draw a vertical straight line from point B through point A.
- Draw a parallel line too the segment AB from the point E
_{1}to the circular arc, there arises the intersection D. - Draw a circle arc about the point C with the radius CD till to the extension of the side length, there arises the intersection F.
- Draw a circle arc about the point E
_{40}with the radius E_{40}F till to the vertical straight line, there arises the intersection G and the angle E_{40}GE_{1}with 36°. - Draw a circle arc about the point G with radius E
_{40}G till to the vertical straight line, there arises the intersection H and the angle E_{40}HE_{1}with 18°. - Draw a circle arc about the point H with radius E
_{40}H till to the vertical straight line, there arises the central point M of the circumcircle and the angle E_{40}ME_{1}with 9°. - Draw around the central point M with radius E
_{40}M the circumcircle of the tetracontagon. - Finally transfer the segment E
_{40}E_{1}(side length*a*) repeatedly counterclockwise on the circumcircle until to arises a regular tetracontagon.

**The golden ratio**

## Symmetry

The *regular tetracontagon* has Dih_{40} dihedral symmetry, order 80, represented by 40 lines of reflection. Dih_{40} has 7 dihedral subgroups: (Dih_{20}, Dih_{10}, Dih_{5}), and (Dih_{8}, Dih_{4}, Dih_{2}, Dih_{1}). It also has eight more cyclic symmetries as subgroups: (Z_{40}, Z_{20}, Z_{10}, Z_{5}), and (Z_{8}, Z_{4}, Z_{2}, Z_{1}), with Z_{n} representing π/*n* radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^{[4]} He gives **d** (diagonal) with mirror lines through vertices, **p** with mirror lines through edges (perpendicular), **i** with mirror lines through both vertices and edges, and **g** for rotational symmetry. **a1** labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular tetracontagons. Only the **g40** subgroup has no degrees of freedom but can seen as directed edges.

## Dissection

regular |
Isotoxal |

Coxeter states that every zonogon (a 2*m*-gon whose opposite sides are parallel and of equal length) can be dissected into *m*(*m*-1)/2 parallelograms.
These tilings are contained as subsets of vertices, edges and faces in orthogonal projections *m*-cubes^{[5]}
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the *regular tetracontagon*, *m*=20, and it can be divided into 190: 10 squares and 9 sets of 20 rhombs. This decomposition is based on a Petrie polygon projection of a 20-cube.

## Tetracontagram

A tetracontagram is a 40-sided star polygon. There are seven regular forms given by Schläfli symbols {40/3}, {40/7}, {40/9}, {40/11}, {40/13}, {40/17}, and {40/19}, and 12 compound star figures with the same vertex configuration.

Picture | {40/3} |
{40/7} |
{40/9} |
{40/11} |
{40/13} |
{40/17} |
{40/19} |
---|---|---|---|---|---|---|---|

Interior angle | 153° | 117° | 99° | 81° | 63° | 27° | 9° |

Many isogonal tetracontagrams can also be constructed as deeper truncations of the regular icosagon {20} and icosagrams {20/3}, {20/7}, and {20/9}. These also create four quasitruncations: t{20/11}={40/11}, t{20/13}={40/13}, t{20/17}={40/17}, and t{20/19}={40/19}. Some of the isogonal tetracontagrams are depicted below, as a truncation sequence with endpoints t{20}={40} and t{20/19}={40/19}.^{[6]}

t{20}={40} |
|||||

t{20/19}={40/19} |

## References

**^**Gorini, Catherine A. (2009),*The Facts on File Geometry Handbook*, Infobase Publishing, p. 165, ISBN 9781438109572.**^***The New Elements of Mathematics: Algebra and Geometry**by Charles Sanders Peirce (1976), p.298***^**Constructible Polygon**^****The Symmetries of Things**, Chapter 20**^**Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141**^**The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994),*Metamorphoses of polygons*, Branko Grünbaum