Flying Further into the K5 Galaxy
An extension to Chapter XVI of 'Adventures Among The Toroids' by B.M.Stewart 

In his work ‘Adventures Among the Toroids: A study of Orientable Polyhedra with Regular Faces’, [1] Professor B.M. Stewart (1980) introduced a family of toroidal polyhedra which would become synonymous with his name.

The Stewart toroids as they would become known are polyhedra (‘P’) meeting the following four conditions, which Stewart sets out in Chapters II and VIII:

(R): each face of P is a regular polygon

(A): faces of P which share an edge are not coplanar

(Q): every edge of the convex closure of P is an edge of P. (The ‘Q’ represents the term ‘quasi-convex’)

(T): a subset of the faces of P forms at least one ‘tunnel’ through P amending its genus. This last condition is more formally defined by Stewart on page 75.

Since the second edition of Stewart’s work in 1980, the only extensions have been by W. Alex Doskey (2002) [2] and by Robert Webb (2003) [3].  Doskey and Webb introduced several toroids which, while they met the four conditions (R)(A)(Q)(T) above, had faces in the convex closure that were not themselves regular. Stewart had considered this possibility but deliberately restricted himself to polyhedra where the convex closure only has regular faces, a condition that Stewart termed (Q").

On these pages I present a number of new Stewart toroids that meet Stewart’s stricter (R)(A)(Q")(T) conditions.  All have the convex closure of a truncated icosidodecahedron, in Stewart's terminology K5.  
In sections 1 to 4, I present toroids with interesting properties including:
(i) up to six independent tunnels,
(ii) tunnels passing through others up to six times,
(iii) interlinked tunnels, and
(iv) a genus p=53 Stewart Toroid. 

In section 5, I provide a solution to the problem posed at the end of Ex 153, which Stewart was not able to solve, i.e. that of passing a Z4(Z4(P4)Z4)Z4 rod through the hole of the "Zulu Chair". I then explore variations on this solution.

Toroids based on K5 with Q5S5(m*)S5Q5 tunnels

Toroids based on K5 with W+(m)W+ tunnels

Threaded Examples Involving Dodecahedra

4 Toroids based on K5 with Z4Y5Z4 tunnels

5 A Solution to Stewart Exercise 153


Names are from Stewart except those denoted with # which are my own additions.

A5 - Stewart A5 - (Stewart page 154)
A5'' - Stewart A5'' - (Stewart page 156)
C - the 'container' polyhedron in T5 / Q5S5(D5) described by Stewart (page 178) but not named #.
D5 - dodecahedron.
E5 - rhombicosidodecahedron.
gQ5 - gyrated pentagonal cupola.
- Gyrated Stewart Z4 – (Stewart page 132). Specifically when excavated from E5 such that the symmetry axis of the Z4 is parallel to the square-decagon edges of E5.
G3 - Stewart G3 – (Stewart page 129).
I5 - icosahedron.
J63 - tridiminished icosahedron.
J91 - bilunabirotunda.
J92 - hebesphenorotunda.
K5 - truncated icosidodecahedron.
m - Stewart m – (Stewart page 156). Stewart also uses m for 'meta' orientations, but this usage does not appear in this paper.
m* - Stewart m* - (Stewart page 156)
P'' - a ring polyhedron forming the central part of Stewart's "Zulu footstool" – (Stewart page 179).
P4 - cube
Q5 - pentagonal cupola.
R5 - pentagonal rotunda.
S* - a rod made up of Q5S5(S5(gQ52)D5)S5Q5 or any variation on such - see section 1 #.
Sn - n-gonal antiprism.
T5 - truncated dodecahedron.
W - Stewart W – (Stewart page 169).
W" - Stewart W" – (Stewart page 168).
W+ - the complex WY3-6 - see section 2 #.
X - Stewart X – (Stewart page 134).
Yn - n-gonal pyramid.
Z4 -
Stewart Z4 – (Stewart page 132).

A note on nomenclature

These pages follow the nomenclature of Stewart as far as is possible. However, as Stewart does not contain a nomenclature section, some aspects of the nomenclature must be collected from the text. The conventions used in this paper are as follows with X and Y below standing for any polyhedron or combination of polyhedra:

nX - n copies of polyhedron X forming separate entrances in the enclosing polyhedron (Stewart page 43). Multiple polyhedra can follow the n without brackets (Stewart page 43)
Xn - X repeated n times in a single tunnel. Stewart appears inconsistent in his use of this, e.g. Q32 / S3S3 (Stewart page 31) and only seems to use it in the case of Qn2
XnA - X augmented onto the previous polyhedron n times (1 is omitted) (Stewart page 74)
XA is also used for a retracted polyhedron (Stewart page 114)
X-nE or X-n - X excavated from the previous polyhedron n times (1 is sometimes omitted) (Stewart page 74). The E is sometimes omitted (Stewart page 132)
(X) - X is central to the toroid or is the inner polyhedron connecting multiple tunnels (Stewart page 16).
X,Y - Two independent tunnels (Stewart page 116)
n(X) - Where the entire tunnel is bracketed this means n independent tunnels (Stewart page 188)
pX•qY - The decomposition of a polyhedron into p copies of X and q copies of Y. This is used in place of the character ⊕ (Unicode U+2295) used by Stewart as the latter is not always available.

Extensions to Stewart's nomenclature

[XY]n - The combination of polyhedra XY in the square brackets is repeated in sequence n times;
XnC or [XY]nC - X or the combination of polyhedra XY in the square brackets is repeated in sequence n times and creates a closed ring.

Further Resources

3D models of all the figures in this paper are available here. This file contains VRML (*.wrl), *.OFF,  Stella (*.stel) and HEDRON input (*.txt) files.


This paper was made possible using Robert Webb's excellent Great Stella program ( and by multiple references to Alex Doskey's Stewart Toroids website (, where he has provided a most valuable resource by modelling most of the toroids in Stewart's book. My thanks also to Dr Richard Klitzing ( for inspiration by pointing out to me that I had somewhat neglected K5 based toroids this website, to Roger Kaufman for his VRML2OFF utility, and to Scott Vorthmann for his VRML Revival project which gave me the motivation to put these pages online.

All VRML files were generated using Great Stella and post-processed with VRML2OFF and HEDRON.


1. Stewart, B. M.(1980) Adventures among the toroids: a study of quasiconvex, aplanar, tunnelled, orientable polyhedra of positive genus having regular faces with disjoint interiors. Okemos, Mich:Stewart, Print. Page references are to the Second Edition.
2. Doskey, W.A. (2002) New Stewart toroids: Exploration of models with irregular-faced convex hulls, Symmetry: Culture and Science, 13(1-2), 33-46. Also at
3. Webb, R. (2003) A Genus-41 Stewart Toroid,