5-3-3-3-3

The above polyhedron was discovered by Janella Large and Mick Ayrton in 2005.  It can be derived by taking a frequency-2 pyramid and truncating the apex to form a frustum.  Two such frustums can then be connected with a ring of triangles (or cingulum).  It could therefore be termed a pentagonal bi-frusto-cingulum.  The 5-3-3-3-3 acrons in the frustum normally contain coplanar triangles, however the cingulum introduces sufficient distortion to make these acrons convex.

It is part of an infinite family of bifrustocingulums for all coprime n/d with 2 < n/d < 6.  Examples are provided for n/d = 4, n/d=5/2, n/d=7/2 and n/d=7/3.  Retrograde isomorphs are also possible with a lower limit of 6/5 being postulated.  Examples with n/d=5/3 and n/d=7/4 are given.

The family described above is one of the simpler members of a group of families where a frustum can be taken from a frequency-m pyramid with n-cingulums inserted.  The above case is (m,n) = (2,1).  For (m,n) = (2,0) two cases are possible. A bi-frustum with coplanar faces, and a gyro-bi-frustum which is more commonly known as a snub anti-prism.  An example is provided of a pentagonal case for (m,n)=(3,1).  These families share much in common with Paul Gailiunas's "Twisted Domes" (pdf)