''Adventures Among the Toroids: A study of orientable polyhedra with regular faces'' is a book on

toroidal polyhedra In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topological genus () of 1 or greater. Notable examples include the Császár and Szilassi polyhedra. Variations in definition Toroidal polyhedr ...

that have

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

s as their faces. It was written, hand-lettered, and illustrated by mathematician

Bonnie Stewart Bonnie Madison Stewart (July 10, 1914 – April 15, 1994) was a professor of mathematics at Michigan State University from 1940 to 1980. He earned his Ph.D. from the University of Wisconsin–Madison in 1941, under the supervision of Cyrus Colton M ...

, and self-published under the imprint "Number One Tall Search Book" in 1970. Stewart put out a second edition, again hand-lettered and self-published, in 1980. Although out of print, the Basic Library List Committee of the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

has recommended its inclusion in undergraduate mathematics libraries.

Topics

The

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

s, known to antiquity, have all faces regular polygons, all symmetric to each other (each face can be taken to each other face by a symmetry of the polyhedron). However, if less symmetry is required, a greater number of polyhedra can be formed while having all faces regular. The

convex polyhedra A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

with all faces regular were catalogued in 1966 by Norman Johnson (after earlier study e.g. by

Martyn Cundy Henry Martyn Cundy (23 December 1913 – 25 February 2005) was a mathematics teacher and professor in United Kingdom, Britain and Malawi as well as a singer, musician and poet. He was one of the founders of the School Mathematics Project to ref ...

and A. P. Rollett), and have come to be known as the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

s. ''Adventures Among the Toroids'' extends the investigation of polyhedra with regular faces to non-convex polyhedra, and in particular to polyhedra of higher

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

than the sphere. Many of these polyhedra can be formed by gluing together smaller polyhedral pieces, carving polyhedral tunnels through them, or piling them into elaborate towers. The toroidal polyhedra described in this book, formed from regular polygons with no self-intersections or flat angles, have come to be called

Stewart toroid In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topological genus () of 1 or greater. Notable examples include the Császár and Szilassi polyhedra. Variations in definition Toroidal polyhedr ...

The second edition is rewritten in a different page format, letter sized in landscape mode compared to the tall and narrow by page size of the first edition, with two columns per page. It includes new material on knotted polyhedra and on rings of regular octahedra and regular dodecahedra; as the ring of dodecahedra forms the outline of a

golden rhombus In geometry, a golden rhombus is a rhombus whose diagonals are in the golden ratio: : = \varphi = \approx 1.618~034 Equivalently, it is the Varignon parallelogram formed from the edge midpoints of a golden rectangle. Rhombi with this shape form ...

, it can be extended to make skeletal pentagon-faced versions of the convex polyhedra formed from the golden rhombus, including the

Bilinski dodecahedron In geometry, the Bilinski dodecahedron is a Convex set, convex polyhedron with twelve Congruence (geometry), congruent golden rhombus faces. It has the same topology but a different geometry than the face-transitive rhombic dodecahedron. It is a ...

rhombic icosahedron The rhombic icosahedron is a polyhedron shaped like an oblate sphere. Its 20 faces are congruent golden rhombi; 3, 4, or 5 faces meet at each vertex. It has 5 faces (green on top figure) meeting at each of its 2 poles; these 2 vertices lie on it ...

, and

rhombic triacontahedron In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...

. The second edition also includes the

Császár polyhedron In geometry, the Császár polyhedron () is a nonconvex toroidal polyhedron with 14 triangular faces. This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron ...

and

Szilassi polyhedron In geometry, the Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces. Coloring and symmetry The 14 vertices and 21 edges of the Szilassi polyhedron form an embedding of the Heawood graph onto the surf ...

, toroidal polyhedra with non-regular faces but with pairwise adjacent vertices and faces respectively, and constructions by Alaeglu and Giese of polyhedra with irregular but congruent faces and with the same numbers of edges at every vertex.

Audience and reception

The second edition describes its intended audience in an elaborate subtitle, a throwback to times when long subtitles were more common: "a study of Quasi-Convex, aplanar, tunneled orientable polyhedra of positive genus having regular faces with disjoint interiors, being an elaborate description and instructions for the construction of an enormous number or new and fascinating mathematical models of interest to students of euclidean geometry and topology, both secondary and collegiate, to designers, engineers and architects, to the scientific audience concerned with molecular and other structural problems, and to mathematicians, both professional and dilettante, with hundreds of exercises and search projects, many outlined for self-instruction". Reviewer

H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

summarizes the book as "a remarkable combination of sound mathematics, art, instruction and humor", while

Henry Crapo Henry Howland Crapo (pronounced ''Cray-poe''; May 24, 1804 – July 23, 1869) was a businessman and politician who was the 14th Governor of Michigan from 1865–1869, during the end of the American Civil War and the beginning of Reconstruction. ...

calls it "highly recommended" to others interested in polyhedra and their juxtapositions. Mathematician Joseph A. Troccolo calls a method of constructing physical models of polyhedra developed in the book, using cardboard and rubber bands, "of inestimable value in the classroom". One virtue of this technique is that it allows for the quick disassembly and reuse of its parts.

References

{{reflist, refs= {{citation, title=Review of ''Adventures Among the Toroids'' (1st ed.), first=H. S. M., last=Coxeter, authorlink=Harold Scott MacDonald Coxeter, journal=

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

, mr=0275266 {{citation, title=Review of ''Adventures Among the Toroids'' (2nd ed.), first=H. S. M., last=Coxeter, authorlink=Harold Scott MacDonald Coxeter, journal=

, mr=0588511, year=1982 {{citation, title=Review of ''Adventures Among the Toroids'' (2nd ed.), first=Henry, last=Crapo, authorlink=Henry Crapo (mathematician), journal=Structural Topology, volume=5, year=1980, url=http://www-iri.upc.es/people/ros/StructuralTopology/ST5/st5-09-a5-ocr.pdf, pages=45–48 {{citation, title=Adventures Among the Toroids (unreviewed listing), work=MAA Reviews, publisher=

, url=https://www.maa.org/press/maa-reviews/adventures-among-the-toroids, accessdate=2020-08-01 {{citation, title=Review of ''Adventures Among the Toroids'' (1st ed.), journal=

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure mathematics, pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Informa ...

, zbl=0214.47703, language=German {{citation, title=Review of ''Adventures Among the Toroids'' (2nd ed.), journal=

, zbl=0443.52005 {{citation, url=https://www.software3d.com/PolyNav/PolyNavigator.php, last=Webb, first=Robert, title=Stella: Polyhedron Navigator, journal=Symmetry: Culture and Science, volume=11, issue=1–4, pages=231–268, year=2000 {{citation , last = Troccolo , first = Joseph A. , date = March 1976 , issue = 3 , journal = The Mathematics Teacher , jstor = 27960432 , pages = 220–224 , title = The algebra and geometry of polyhedra , volume = 69 {{citation , last = Prichett , first = Gordon D. , date = January 1976 , issue = 1 , journal = The Mathematics Teacher , jstor = 27960351 , pages = 5–10 , title = Three-dimensional discovery , volume = 69

External links

Virtual reality models of Stewart's polyhedra
Alex Doskey
Bonnie Stewarts Hohlkörper
(in German), Christoph Pöppe, on the German-language site of ''Scientific American'' Polyhedra Mathematics books 1970 non-fiction books 1980 non-fiction books Self-published books

Topics

Audience and reception

See also

References

External links