A regular tetracontagon is represented by Schläfli symbol and can also be constructed as a truncatedicosagon, t, which alternates two types of edges. Furthermore, it can also be constructed as a twice-truncated decagon, tt, or a thrice-truncated pentagon, ttt. One interior angle in a regular tetracontagon is 171°, meaning that one exterior angle would be 9°. The area of a regular tetracontagon is and its inradius is The factor is a root of the octic equation. The circumradius of a regular tetracontagon is As 40 = 23 × 5, a regular tetracontagon is constructible using a compass and straightedge. 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:
Draw a segment whose length is the given side length a of the tetracontagon.
Extend the segment by more than two times.
Draw each a circular arc about the points E1 and E40, 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 from the point E1 to the circular arc, there arises the intersection D.
Draw a circle arc about the point C with the radius till to the extension of the side length, there arises the intersection F.
Draw a circle arc about the point E40 with the radius till to the vertical straight line, there arises the intersection G and the angle E40GE1 with 36°.
Draw a circle arc about the point G with radius till to the vertical straight line, there arises the intersection H and the angle E40HE1 with 18°.
Draw a circle arc about the point H with radius till to the vertical straight line, there arises the central point M of the circumcircle and the angle E40ME1 with 9°.
Draw around the central point M with radius the circumcircle of the tetracontagon.
Finally transfer the segment repeatedly counterclockwise on the circumcircle until to arises a regular tetracontagon.
The golden ratio
Symmetry
The regular tetracontagon has Dih40dihedral symmetry, order 80, represented by 40 lines of reflection. Dih40 has 7 dihedral subgroups:, and. It also has eight more cyclic symmetries as subgroups:, and, with Zn representing π/n radian rotational symmetry. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d with mirror lines through vertices, p with mirror lines through edges, 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
states that every zonogon can be dissected into m/2 parallelograms. These tilings are contained as subsets of vertices, edges and faces in orthogonal projectionsm-cubes 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,,,,,, and, and 12 compound star figures with the same vertex configuration.
Picture
Interior angle
153°
117°
99°
81°
63°
27°
9°
Picture
=2
=4
=5
=2
=8
=10
Interior angle
162°
144°
135°
126°
108°
90°
Picture
=4
=2
=5
=8
=2
=20
Interior angle
72°
54°
45°
36°
18°
0°
Many isogonal tetracontagrams can also be constructed as deeper truncations of the regular icosagon and icosagrams,, and. These also create four quasitruncations: t=, t=, t=, and t=. Some of the isogonal tetracontagrams are depicted below, as a truncation sequence with endpoints t= and t=.
