Wednesday, 13 June 2018

The Disphenocingulum

As a result of contacts on twitter I have become interested in polyhedra, in particular the "Johnson Solids" which are formed of regular polygons but are not the usual suspects.

In particular a drawing was posted of the "disphenocingulum".  As a result I decided to make a model in card and ended up with two different versions. The 12 orange triangles form the "cingulum". The "spheno" parts form the roof and keel:

Corresponding flat diagrams of the pieces and their connections are:

The version on the right appears to conform to the patterns shown in the Wikipedia and MathWorld entries for Johnson Solid 90, but the version on the left does not.

So is my new version an alternative ("isotope") of the disphenocingulum? Or is one of them not an authentic Johnson Solid because it has a pair of triangles that are coplanar? It is difficult to tell from the models if some pairs of triangles are at an angle to each other or are flat together. The angle may be very small. In fact the one on the right seems flatter to me than that on the left.