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Volume 34, Number 4, pages 425-451 (2023)
https://doi.org/10.26830/symmetry_2023_4_425
FLYING FURTHER INTO THE "K5-GALAXY"
AN EXTENSION TO CHAPTER XVI OF ADVENTURES AMONG THE TOROIDS BY B.M. STEWART
Jim McNeill
Email: jim@orchidpalms.com
Abstract: In chapter XVI of Adventures Among the Toroid, Stewart discusses toroids with a convex closure of a truncated icosidodecahedron, or in Stewart's terminology a "K5". He describes a toroid with up to four independent tunnels - the maximum he was able to achieve. He also posits the possibility of a toroid in which one tunnel passes through another but was unable to generate an example. In this paper, I present toroids with interesting properties including (i) up to six independent tunnels, (ii) tunnels passing through others up to six times, and (iii) interlinked tunnels. Then, 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".
Keywords: Polyhedron, Stewart Toroid, Truncated Icosidodecahedron
References:
Doskey, W.A. (2002) New Stewart toroids: Exploration of models with irregular-faced convex hulls, Symmetry: Culture and Science, 13(1-2), 33-46.
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.
Webb, R. (2003) A Genus-41 Stewart Toroid, https://www.software3d.com/WebbToroid.php. https://doi.org/10.1071/SR01077
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