Chapter 15 of Professor B.M.
Stewart's "*Adventures Among the Toroids*" (ISBN 0-686-119 36-3),
focuses on toroids with a truncated dodecahedral convex hull.

Each of the decagonal faces
of a truncated dodecahedron (**T5**)
can be aligned with the pentagonal faces of a dodecahedron
(**D5**). These faces can then be joined by a combination of a
pentagonal
cupola (**Q5**) and a pentagonal antiprism
(**S5**).

This gives rise to a family
of toroids of the form **T5 / a Q5S5(D5) **where **2>=a>=12.**

The "Gem" of this family
is the beautiful toroid **T5 / 12 Q5S5(D5)** with genus** ***p=11.*

Stewart's observations about this toroid include the following:

"*the intersection of this
toroid with its convex closure is not a network of faces ... but is a complex
of 20 triangular faces, isolated in appearance, but each connected from
every vertex at a sketeton edge*"

"*despite its dodecahedral
appearance, the toroid is made up entirely of square and triangular faces.*"

In a further attempt to illustrate
the interior of this toroid, the image below simply shows the antiprismatic
faces surrounding the dodecahedron, or
in Stewart's terminology, the **12 S5(D5)** section.

Cross sections of this toroid
are also of interest. Many of the vertices of the toroid lie
in distinct planes. These are shown on the diagram below. Two
pairs of planes are shown, labelled **L1- L1'** and **L2 - L2'**.

The images below link to
two models of the toroid cut at the **L1** and **L2** planes respectively
with the edges in the relevant planes highlighted in yellow. Note
the central pentagon on the left hand image, the edges shown here are those
joining two anti-prisms, both of which have been removed by the slicing
operation.

Note that the vertices on
the **L2** plane form regular pentagons. These can be added to
the toroid to form a non convex toroid of genus* p=6*.

A fascinating slice of the
original toroid can also be generated by taking the area between the **L2
**and
**L2'
**planes.
This is itself a regularly faced toroid of genus
*p=1**.* .

The original toroid can
be thought of as six of these 'rings' connected by the icosahedral (i.e.
triangular) faces of the truncated dodecahedron.

The pentagonal faces are
identical to those which would be formed had the truncated
dodecahedron and the central dodecahedron
been linked by pentagonal rotundas.
The size of the rotundas however means that they interfere. To link
corresponding faces with a rotunda prevent the use of a further linkage
(by either a rotunda or by a cupola/anti-prism) at any of the neighbouring
decagonal faces. An example of a "**T5**" toroid containing a
rotunda is this **T5 / 6Q5S5 R5 (D5)**.

As an aside: A 'great' version of this figure also exists.

The model linked (here) shows the frame of the original toroid, with the sixty pentagonal "faces" superimposed. The image below (and linked model) shows just these pentagonal "faces".