An extension to Chapter XVI of 'Adventures Among The Toroids' by B.M.Stewart

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)

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 Z

A

A

C - the 'container' polyhedron in T

D

E

gQ

gZ4 - Gyrated Stewart Z

G

I

J

J

J

K

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).

P

Q

R

S* - a rod made up of Q

S

T

W - Stewart W – (Stewart page 169).

W" - Stewart W" – (Stewart page 168).

W

X - Stewart X – (Stewart page 134).

Y

Z

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)

X

X

X

X

(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]

X

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.

Credits

This paper was made possible using Robert Webb's excellent Great Stella program (www.software3d.com) and by multiple references to Alex Doskey's Stewart Toroids website (polyhedra.doskey.com/Stewart00.html), 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 (bendwavy.org/klitzing/home.htm) for inspiration by pointing out to me that I had somewhat neglected K

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

References

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 polyhedra.doskey.com/PrismExpansions.html

3. Webb, R. (2003) A Genus-41 Stewart Toroid, www.software3d.com/WebbToroid.php