__Spheno-prisms__

A prismatic family exists where the
two prismatic bases are connected by sphenoids. This family is 3-uniform.
The base **n/d**-gon is this time surrounded by (type **A**) vertices
of the form **n/d.4.3.3.4 **Type (**B**) vertices of the form **3.3.3.3**
and type (**C**) vertices of the form **4.4.3.3** also occur. Both
the type (**B**) and the type (**C**) vertices are equatorial. The
type (**C**) vertices are also internal to the solid. The linking sphenoids
are best seen in this model of the 7/3 spheno-prism where
only a single loop of the sphenoids around the prismatic symmetry axis
is shown. The internal structure can also be seen in this model of one
hemisphere of a 7/3 spheno-prism. See also the models of the Elongated
Spheno-prism for further models showing the internal structure.

The family is convex for **2 <=n/d
< 3**, **n/d=3** is the limiting case and is planar (see below).
Setting **n/d=2** gives a 'disphenoid' or to give it its Johnson Solid
name an elongated square dipyramid, this
is the only **n**-gonal convex member of this family. Spheno-prisms
are a specific form of edge expanded bi-prisms, or EEB's. Click here
for further information about the general family of EEB's.

*Note: This family is closely related
to the "Accordion Polyhedra" discussed in Professor Bonnie Stewart's "Adventures
Among the Toroids" 2 ^{nd} Ed. pp 151. Using Professor Stewart's
terminology these would be "Everted Square Plated Accordions"*

The height (*H*) of a spheno-prism is given by:

Click here for proof.

For the n-gonal cases, this gives H=sqrt(2)
for **n/d=2** and proves the limiting zero height for **n/d=3**.
The height formula can also be applied to spheno-prisms with retrograde
bases (i.e. **n/d<2**) but in these cases the solid is not locally
convex. For example the 5/3 spheno-prism

Due to the internal type (**C**)
vertices, solid models of the solids are dominated by the type (**B**)
vertices and are generally rather uninteresting, as the following models are **HEDRON**
'switch' files, I recommend viewing them in
translucent mode.

A full list of locally convex spheno-prisms
with **d>1** and **n<=12 **is below. Their retrograde equivalents are
also given.

5/2 | 7/3 | 8/3 | 9/4 | 11/4 | 11/5 | 12/5 |

**Next: Elongated Spheno-Prisms**
**Back:
To Prisms**
**Back: To Index**