A further family is that of **Gyro-elongated
Spheno-Prisms**. The 7/3 Gyro-elongated Spheno-prism
is shown above. The square cingulum in the Elongated Spheno-Prism is replaced
here by a cingulum of **6n** triangles, which wraps around the prismatic
symmetry axis **d** times. The base **n/d**-gon is still surrounded
by (type **A**) vertices of the form **n/d.4.3.3.4 **Type (**B**)
vertices are now of the form **3.3.3.3.3** and type (**C**) vertices
of the form **4.3.3.3.3**. The cingulum introduces some distortion to
the Spheno-prism cap and the type (**B**) vertices are now pulled nearer
to the solid's equator than the Type (**C**) vertices. The convexity
of this family is clearly seen in this 7/3 Gyro-elongated
Spheno-prism with only
one loop of the cingulum around the prismatic symmetry axis.

As a result of the distortion the term 'Gyro-elongated Spheno-Prism' is not technically correct, alternatives would be either Pseudo-Gyro-elongated Spheno-Prism or following on from the families only truly convex member (see below) the term Spheno-cingulum. Having given that health warning, I will however use the term Gyro-elongated Spheno-Prism in this page and its linked models.

The lower limit for convexity is still
**n/d=2**,
this is a 'Gyro-elongated Disphenoid' or to give it its formal name as
a Johnson Solid, a disphenocingulum, this
is the only **n**-gonal convex member of this family. However, due to
the distortion mentioned above the upper limit is now somewhat less than
**n/d=3.
**This
is apparent from this model of the Triangular Gyro-elongated
Spheno-Prism (with
**n/d=3**) The
type (**B**) vertices clearly show an element of concavity as there
is a definite valley between the two triangles connected to the prismatic
face. The upper limit for convexity is between 2.8 and 3 as can be seen
from this 14/5 Gyro-elongated Spheno-prism or this
example with
only one loop of the cingulum around the prismatic symmetry axis.

The distortion of the Spheno-prismatic
caps means that in many examples the type (**C**) vertices are not internal.
From the models generated with **n<=12**, if **d=2,4 or 5** then
the type (**C**) vertices are visible in a solid model, but not if **d=3**.
I do not know if this remains the case for **n>12**.

A full list of locally convex Gyro-elongated
Spheno-prisms with **d>1** and **n<=12 **is below. The "Cutaway" models
have one sphenoidal
cap removed. These are possibly the most effective models at revealing
the structure of the cingulum.

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