# Pentagonal antiditetragoltriate

Pentagonal antiditetragoltriate
Rank4
TypeIsogonal
Notation
Elements
Cells25+25 tetragonal disphenoids, 50 rectangular pyramids, 10 pentagonal prisms
Faces100+100 isosceles triangles, 50 rectangles, 10 pentagons
Edges50+50+100
Vertices50
Vertex figureBiaugmented triangular prism
Measures (based on same duoprisms as optimized pentagonal ditetragoltriate)
Edge lengthsEdges of smaller pentagon (50): 1
Lacing edges (50): ${\displaystyle {\sqrt {\frac {10-{\sqrt {5}}+{\sqrt {50+20{\sqrt {5}}}}}{5}}}\approx 1.41855}$
Edges of larger pentagon (50): ${\displaystyle {\frac {2+{\sqrt {5-{\sqrt {5}}}}}{2}}\approx 1.83125}$
Circumradius${\displaystyle {\sqrt {\frac {15+2{\sqrt {5}}+2{\sqrt {25+5{\sqrt {5}}}}}{10}}}\approx 1.77488}$
Central density1
Related polytopes
DualPentagonal antitetrambitriate
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH2≀S2, order 200
ConvexYes
NatureTame

The pentagonal antiditetragoltriate or paddet is a convex isogonal polychoron and the third member of the antiditetragoltriate family. It consists of 10 pentagonal prisms, 50 rectangular pyramids, and 50 tetragonal disphenoids of two kinds. 2 pentagonal prisms, 4 tetragonal disphenoids, and 5 rectangular pyramids join at each vertex. However, it cannot be made scaliform.

It can be formed as the convex hull of 2 oppositely oriented semi-uniform pentagonal duoprisms where the larger pentagon is more than ${\displaystyle {\sqrt {5}}-1}$ times the edge length of the smaller one.

The grand antiprism can be thought of as the convex hull of two inversely oriented pentagonal antiditetragoltriates, with the pentagons having a ratio of 1:${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$ ≈ 1:1.61803.