# Pentagonal duotegum

The pentagonal duotegum or pedit, also known as the pentagonal-pentagonal duotegum, the 5 duotegum, or the 5-5 duotegum, is a noble duotegum that consists of 25 tetragonal disphenoids and 10 vertices, with 10 cells joining at each vertex. It is also the 10-4 step prism and the square funk tegum. It is the first in an infinite family of isogonal pentagonal hosohedral swirlchora and also the first in an infinite family of isochoric pentagonal dihedral swirlchora.

Pentagonal duotegum Rank4
TypeNoble
SpaceSpherical
Notation
Bowers style acronymPedit
Coxeter diagramm5o2m5o
Elements
Cells25 tetragonal disphenoids
Faces50 isosceles triangles
Edges10+25
Vertices10
Vertex figurePentagonal tegum
Measures (based on pentagons of edge length 1)
Edge lengthsBase (10): 1
Lacing (25): $\sqrt{\frac{5+\sqrt5}{5}} ≈ 1.20300$ Circumradius$\sqrt{\frac{5+\sqrt5}{10}} ≈ 0.85065$ Inradius$\sqrt{\frac{5+2\sqrt5}{40}} ≈ 0.48662$ Central density1
Related polytopes
ArmyPedit
RegimentPedit
DualPentagonal duoprism
ConjugatePentagrammic duotegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH2≀S2, order 200
ConvexYes
NatureTame

The ratio between the longest and shortest edges is 1:$\frac{\sqrt{25+5\sqrt5}}{5}$ ≈ 1:1.20300.

## Vertex coordinates

The vertices of a pentagonal duotegum based on two pentagons of edge length 1, centered at the origin, are given by:

• $\left(±\frac{1}{2},\, -\sqrt{\frac{5+2\sqrt{5}}{20}},\,0,\,0\right),$
• $\left(±\frac{1+\sqrt{5}}{4},\, \sqrt{\frac{5-\sqrt{5}}{40}},\,0,\,0\right),$
• $\left(0,\, \sqrt{\frac{5+\sqrt{5}}{10}},\,0,\,0\right),$
• $\left(0,\,0,\,±\frac{1}{2},\, -\sqrt{\frac{5+2\sqrt{5}}{20}}\right),$
• $\left(0,\,0,\,±\frac{1+\sqrt{5}}{4},\, \sqrt{\frac{5-\sqrt{5}}{40}}\right),$
• $\left(0,\,0,\,0,\, \sqrt{\frac{5+\sqrt{5}}{10}}\right).$