# Rectified dodecagonal duoprism

Rectified dodecagonal duoprism
Rank4
TypeIsogonal
SpaceSpherical
Notation
Elements
Cells144 tetragonal disphenoids, 24 rectified dodecagonal prisms
Faces576 isosceles triangles, 144 squares, 24 dodecagons
Edges288+576
Vertices288
Vertex figureWedge
Measures (based on dodecagons of edge length 1)
Edge lengthsLacing edges (576): ${\displaystyle \sqrt3-1 ≈ 0.73205}$
Edges of dodecagons (288): 1
Circumradius${\displaystyle \sqrt{6+\sqrt3} ≈ 2.78066}$
Central density1
Related polytopes
DualJoined dodecagonal duotegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(12)≀S2, order 1152
ConvexYes
NatureTame

The rectified dodecagonal duoprism or retwadip is a convex isogonal polychoron that consists of 24 rectified dodecagonal prisms and 144 tetragonal disphenoids. 3 rectified dodecagonal prisms and 2 tetragonal disphenoids join at each vertex. It can be formed by rectifying the dodecagonal duoprism.

It can also be formed as the convex hull of 2 oppositely oriented semi-uniform dodecagonal duoprisms, where the edges of one dodecagon are ${\displaystyle \sqrt6-\sqrt2 ≈ 1.03528}$ times as long as the edges of the other.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle \frac{1+\sqrt3}{2}}$ ≈ 1:1.36603.

## Vertex coordinates

The vertices of a rectified dodecagonal duoprism based on dodecagons of edge length 1, centered at the origin, are given by:

• ${\displaystyle \left(±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2},\,±\sqrt2,\,±\sqrt2\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2},\,±\frac{\sqrt6-\sqrt2}{2},\,±\frac{\sqrt2+\sqrt6}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2},\,±\frac{\sqrt2+\sqrt6}{2},\,±\frac{\sqrt6-\sqrt2}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{2+\sqrt3}{2},\,±\sqrt2,\,±\sqrt2\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{2+\sqrt3}{2},\,±\frac{\sqrt6-\sqrt2}{2},\,±\frac{\sqrt2+\sqrt6}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{2+\sqrt3}{2},\,±\frac{\sqrt2+\sqrt6}{2},\,±\frac{\sqrt6-\sqrt2}{2}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt3}{2},\,±\frac12,\,±\sqrt2,\,±\sqrt2\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt3}{2},\,±\frac12,\,±\frac{\sqrt6-\sqrt2}{2},\,±\frac{\sqrt2+\sqrt6}{2}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt3}{2},\,±\frac12,\,±\frac{\sqrt2+\sqrt6}{2},\,±\frac{\sqrt6-\sqrt2}{2}\right),}$
• ${\displaystyle \left(±2,\,0,\,±\frac{\sqrt2+\sqrt6},\,0\right),}$
• ${\displaystyle \left(±2,\,0,\,0,\,±\frac{\sqrt2+\sqrt6}{2}\right),}$
• ${\displaystyle \left(0,\,±2,\,±\frac{\sqrt2+\sqrt6},\,0\right),}$
• ${\displaystyle \left(0,\,±2,\,0,\,±\frac{\sqrt2+\sqrt6}{2}\right),}$
• ${\displaystyle \left(±2,\,0,\,±\frac{\sqrt2+\sqrt6}{4},\,±\frac{3\sqrt2+\sqrt6}{4}\right),}$
• ${\displaystyle \left(0,\,±2,\,±\frac{\sqrt2+\sqrt6}{4},\,±\frac{3\sqrt2+\sqrt6}{4}\right),}$
• ${\displaystyle \left(±2,\,0,\,±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{\sqrt2+\sqrt6}{4}\right),}$
• ${\displaystyle \left(0,\,±2,\,±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{\sqrt2+\sqrt6}{4}\right),}$
• ${\displaystyle \left(±\sqrt3,\,±1,\,±\frac{\sqrt2+\sqrt6},\,0\right),}$
• ${\displaystyle \left(±\sqrt3,\,±1,\,0,\,±\frac{\sqrt2+\sqrt6}{2}\right),}$
• ${\displaystyle \left(±1,\,±\sqrt3,\,±\frac{\sqrt2+\sqrt6},\,0\right),}$
• ${\displaystyle \left(±1,\,±\sqrt3,\,0,\,±\frac{\sqrt2+\sqrt6}{2}\right),}$
• ${\displaystyle \left(±\sqrt3,\,±1,\,±\frac{\sqrt2+\sqrt6}{4},\,±\frac{3\sqrt2+\sqrt6}{4}\right),}$
• ${\displaystyle \left(±1,\,±\sqrt3,\,±\frac{\sqrt2+\sqrt6}{4},\,±\frac{3\sqrt2+\sqrt6}{4}\right),}$
• ${\displaystyle \left(±\sqrt3,\,±1,\,±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{\sqrt2+\sqrt6}{4}\right),}$
• ${\displaystyle \left(±1,\,±\sqrt3,\,±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{\sqrt2+\sqrt6}{4}\right),}$