Chapter 10 of Professor B.M.
Stewart's "*Adventures Among the Toroids*" (ISBN 0-686-119 36-3),
focuses on toroids with a truncated cuboctahedral convex hull.

The truncated
cuboctahedron (**K4**) can be partitioned into a central rhombicuboctahedron
(**E4**) surrounded by six square cupolas
(**Q4**), eight triangular cupolas
(**Q3**) and twelve cubes (**P4**).

This suggests a family of
toroids of the form **K4 / aQ4bQ3cP4(E4) **where **0<=a<=6**,
**0<=b<=8** and **0<=c<=12**.

The special case of **a=0**,
**b=0**,
**c=12**
is illustrated above. This is the Stewart Toroid **K4 / 12P4(E4)**
with genus *p=11 *(combo)

With **a=6**, **b=0**,
**c=0**
another special case is illustrated above. This is the Stewart
Toroid **K4 / 6Q4(E4)** with genus *p=5 *(combo)

The third special case is
**a=0**,
**b=8**,
**c=0**
illustrated above. This is the Stewart Toroid **K4 / 8Q3(E4)**
with genus *p=7 *(combo)

The above are, as have been described, special cases. Toroids exist for a number of combinations of **a**,
**b**
and **c** for **a+b+c >=2**. Stewart counted a total of 518
toroids of this form.

__Two independent Tunnels__

If
we take a variation on the toroid **K4 / 12P4(E4)
**and drill only four of the external squares, we can form a toroid of the
form **K4 / 4P4(E4).** If we have been careful in our selection of
squares for drilling we can then remove the central octagonal
prism from the rhombicuboctahedron to leave
the Stewart Toroid **K4 / 2P4(Q4), 2P4(Q4)** (combo)
which has two independent tunnels. Stewart suggests that with 54 faces
this seems a record low number for such a toroid.

__Two storey Toroids__

The central rhombicuboctahedron
in the above toroids can itself be tunnelled. This gives rise to
the potential for toroids of the form **A / (B / C)**. A very
attractive example is this **K4 / 12P4 (E4 / Q4P4)** where a rod consisting
of a cuboctahedron and a cube
has been re-inserted into the rhombicuboctahedral hole. This model
has genus *p=12*.

combo1
(with the **Q4P4 **rod solid) - and remaining faces as frames

combo2
(with **E4 / Q4P4** solid)

combo3
(with all but the **K4** hull solid)