Twisted Antiprism

n

3

height of solid

1.3

faces to include

bottom

top

lower lateral

upper lateral

twist

0

polygon/line

cylinder opacity

0.07

hide cylinder

vertex notation

show lines that determine

Blaschke point

solid

function graph

net

The vertices on top are also on a circle of radius 1.

A twisted antiprism is obtained from an antiprism by rotating its top face by

-π/2

or

π/2

; it has a nontrivial infinitesimal isometric deformation. The case of the twisted triangular antiprism is known as Wunderlich's (or Schoenhardt's) octahedron. According to the Blaschke–Liebmann theorem, four nonadjacent faces of an infinitesimally flexible octahedron meet at a point. It seems that the theorem can be extended to the twisted antiprisms.

Let

ABC

be any triangular face of the twisted antiprism, where

A

and

B

are on the bottom face and

C

is on the top face. The rotation about the axis

AB

preserves the lengths of the sides of the triangle. If this rotation is done for all the triangles, the top face remains an equilateral triangle. Its side length is a function of the rotation angle

t

, and the derivative of that function at

t=0

is 0.