In his book "*Adventures
Among the Toroids*" (ISBN 0-686-119 36-3), Professor B.M. Stewart provides
a study of "quasi-convex, aplanar, tunnelled, orientable polyhedra having
regular faces with disjoint interiors". To classify as a "Stewart
Toroid" a polyhedron ("P") must fulfil the following requirements:

(**R**): Each face of
P must be regular

(**A**): Faces of P which
share an edge must not be coplanar

(**Q**): It must be "quasi-convex",
i.e. every edge of the convex closure of P is an edge of P.

(**T**): P must be tunnelled.
i.e. P contains an excavation which amends the genus of P

The faces of P must also not intersect. So polyhedra containing pentagrams and other star-polygons are excluded.

The notation used in Professor Stewart's
book to describe the toroids is fairly simple. The notation **A / B**
means the convex closure is polyhedron **A** and an excavation has been
made to remove polyhedron **B**. Where the notation reads
**A
/ nB(C)** then **nB** means that **n** excavations have been made
to remove polyhedra **B** and a central excavation has been made to
remove polyhedron **C**. The prefix **g** is used to mean that one
copy of a polyhedron has been gyrated with respect to another. Multi-tiered
toroids have an extended notation of the form **A / (B / C)**, in these
cases polyhedron **C** has been removed from polyhedron
**B**, the
resulting toroid **B / C** is then removed from polyhedron
**A**.
In effect a **C** shaped "rod" is inserted into the
**B** shaped
"hole" in **A**.

The individual polyhedra utilised below are described using shorthand notation (as used by Stewart) as follows:

**B4**: cuboctahedron
**E4**: rhombicuboctahedron
**K3**: truncated
octahedron
**K4**: truncated
cuboctahedron
**P3:** triangular
prism
**P4**: cube
**P6:** hexagonal
prism
**Q3**:
triangular cupola
**Q4**: square
cupola
**Q5**: pentagonal
cupola
**R5**: pentagonal
rotunda
**S3**: octahedron
**S5**: pentagonal
anti-prism
**T3**: truncated
tetrahedron
**T4**: truncated
cube

**J91**: bilunabirotunda

**Z4**: Stewart 'Z4'

For those (simpler) Stewart Toroids that have been modelled, two models are presented:

(i) a '**switchable**' model
with all faces following the solid -> translucent -> frame cycle

(ii) a '**combo**' model
where the convex closure is as above and the excavation is left solid.

The references to page numbers are to the relevant page in Stewart's book (2nd edition).

To start, two toroids that are
**NOT** Stewart Toroids. They are not quasi-convex and they also break the non-intersecting
face rule. They are included here as with just 12 faces, I believe them to be the least faced toroids that can be generated with regular faces.
The least faced Stewart
Toroid ( **Q3P6 / P3Q3** - see third image below) has 21 faces. The
first of the 12 faced toroids (left hand image) consists of two hexagonal prisms, one gyrated with
regard to the other.
Using Stewart notation it is a **P6 / gP6**. The second (right hand
image) (solid, frame)
is a faceting of the icosahedron, discovered by
Dr Richard Klitzing during a systematic search of such facetings. This
second toroid also appears to be the toroid with the least vertices, having just
12. More of Dr Klitzing's images are
hosted by Ulrich Mikloweit here

Turning now to **Stewart
Toroids**, amongst the simpler ones are:

**Q3P6 / P3Q3** (*page
24*) combo

(minimum number of faces:
21)

**Q3Q3 / S3S3** (*page
31*) combo

(minimum number of vertices:
15)

**T4 / gQ4P4Q4** (*page
29*) combo

**K3 / Q3T3** *(page
34)* combo

(See also Szilassi
polyhedron for a further example

of the above toroid.)

**Q4(T4)Q4 / B4(P4)B4 ***(page
39)* combo

Toroids with a trucated octahedral
hull are the simplest toroids with an apolar tunnel and with a genus p>1.
The ultimate toroid in this class is K3 / 4Q3(S3) "with genus p=3 and a
minimal number of faces" [30]

**K3 / 4Q3(S3)** *(page
45)* combo

One
of Stewart's more remarkable discoveries was this drilling of the Johnson Solid:
bilunabirotunda by a non-convex polyhedron he
names only '**Z4**'. The Z4
itself was obtained by sectioning a bilunabirotunda.
The J91/Z4 combination is then used frequently in the
generation of more convoluted Stewart Toroids.

Professor Stewart also explored toroids that could be formed isogonally, in the case of triangular faces, the minimal toroid proved to be this 48 faced ring of eight octahedra.

**8S3
***(page 51)*