Guy's Polyhedra Pages

Essays, current research
and random resources

One of the lost icosahedra

Cornoid with a hollow vertex figure

A spacefilling hendecahedron

News

Significant changes over the last couple of years or so.

23 Mar 2023. Filling Polytopes. A seductive fallacy added.
24 Jan 2023. It's a Long Way to the Stars. Theory expanded on, second half rearranged.
15 Jan 2023. Weaire-Phelan Bubbles. Weighted Voroni construction clarified, new refs, general cleanup.
31 Dec 2022. Dualising Polyhedra New, unfinished: A unified understanding of historical inconsistences and paradoxes.
10 Nov 2022. Introduction to Quasicrystals. Renamed from "A 3-D Quasicrystal Structure?" and expanded.
13 July 2022. A Critique of Abstract Polytopes. "Anaploid" or non-simple pieces named. Minor cleanup and corrections.
2 July 2022. Morphic Polytopes. Anaploid" or non-simple pieces named and further discussed. General cleanup and corrections.
6 May 2022. Polytopes, Duality and Precursors. Revised and completed: The final piece of general theory underlying stellation and facetting.
2 Jan 2022. Published 1 Jan 2022: Inchbald, Guy; "Morphic Polytopes and Symmetries". In Darvas, György (Ed.); Complex Symmetries, Birkhäuser, 2022. Pages 57-70. Hardcover ISBN 978-3-030-88058-3. Softcover ISBN 978-3-030-88061-3. eBook ISBN 978-3-030-88059-0. DOI https://doi.org/10.1007/978-3-030-88059-0.

Space-filling polyhedra

Illustrated essays, published and unpublished. Most include downloadable "nets" for printing off and making card models.

Five Space-Filling Polyhedra
Substantially as in The Mathematical Gazette 80, November 1996, p.p. 466-475.
My most popular polyhedron page. The text is reproduced by kind permission, with some revision and additions.
- 29 Sept 2018. Folding Space-Filling Bisymmetric Hendecahedron for a Large-Scale Art Installation, Wu, Jiangmei and Inchbald, Guy; Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture, Pages 483–486. (External link to PDF)
The Archimedean Honeycomb Duals
Substantially as in The Mathematical Gazette 81, July 1997, p.p. 213-219.
A remarkable family of 14 polyhedra. The text is reproduced by kind permission.
Introduction to Quasicrystals
Formerly "A 3-D Quasicrystal Structure?", now includes a wider discussion.
Weaire-Phelan Bubbles
The closest to ideal bubbles yet found.

Stellation and facetting

Star polyhedra include some of the most beautiful mathematical shapes around (see for example my icosahedron pages). Their mathematical theory remains surprisingly primitive and incomplete.

It's a Long Way to the Stars
or, why stellation theory is in such a mess.
Stellation and Facetting - A Brief History
Title says it all, really.
Facetting Diagrams
Substantially as in The Mathematical Gazette 86, July 2002, p.p. 208-215.
The facetting diagram may be used to find facettings of a polyhedron, in reciprocal manner to finding new stellations from the stellation diagram.
The regular star (Kepler-Poinsot) polyhedra:
- A.-L. Cauchy, Researches on Polyhedra, Part I
  Paper deriving Poinsot's regular star polyhedra by stellating the regular convex solids, and proving that the set is complete. English translation from: Recherches sur les polyèdres, Prèmiere partie, Journal de l' École Polytechnique, 16 (1813). Includes PDF file for downloading/printing.
- J. Bertrand, Note on the Theory of Regular Polyhedra
  Paper deriving Poinsot's regular star polyhedra by facetting the regular convex solids. English translation from: Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de l'Académie des Sciences 46 (1858). Includes PDF file for downloading/printing.
Stellating the icosahedron and facetting the dodecahedron - index page
Introduction and links to the rest. The main pages are listed below:
- Some Lost Stellations of the Icosahedron
  This page brings together many of the lost stellations of the icosahedron that I have come across so far, including two discovered, probably by H.T. Flather, before publication of the famous fifty-nine.
- In Search of the Lost Icosahedra
  Substantially as in The Mathematical Gazette 86, July 2002, p.p. 208-215.
  Flaws in the rationale for The Fifty-Nine Icosahedra lead to a re-evaluation.
- Towards Stellating the Icosahedron and Faceting the Dodecahedron
  Substantially as in Symmetry: Culture and Science, Vol. 11, 1-4, 2000, p.p. 269-291.
  Building a unified approach.
- Tidy Dodecahedra and Icosahedra
  Unpublished. Includes a discussion of Bridge's 1974 paper on Facetting the dodecahedron.

General theory of polytopes and polyhedra

The serious interest department.

It's a Long Way to the Stars
or, The sorry state of polyhedron theory today.
Ditela, Polytopes and Dyads
A new name for closed line segments completes the pantheon of names for polytopes in any number of dimensions.
A Critique of Abstract Polytopes
Abstract polytope theory as currently formulated is powerful and is changing our ideas, but it has flaws.
Polytopes - Abstract and Real
By Norman Johnson, edited by me. Real polytopes are a consistent mathematical formulation of the true geometric figures that we instinctively think of as polygons, polyhedra and so on. Jonhson's theory of "real" polytopes is a valid alternative to my "morphic" theory and this is the only place it has ever been published.
Morphic theory and its antecedents:
- Morphic Polytopes
  Towards a new definition of polyhedra and higher polytopes, seeking to reconcile abstract and more traditional topological approaches.
- Inchbald, Guy; "Morphic Polytopes and Symmetries". In Darvas, György (Ed.); Complex Symmetries, Birkhäuser, 2022. Pages 57-70. Hardcover ISBN 978-3-030-88058-3. Softcover ISBN 978-3-030-88061-3. eBook ISBN 978-3-030-88059-0. DOI https://doi.org/10.1007/978-3-030-88059-0.
- Filling polytopes
  In polytope theory, filling is shown to be of fundamental importance. Traditional theory ignores filling, and so is incomplete. Fillings are not unique and this must be recognised.
- Polytopes: Degeneracy and Tidiness
  Geometrical untidiness is distinguished from from topological degeneracy. Some types of untidiness and degeneracy are discussed. Features located at infinity can have two opposing images. Unfinished.
- Polytopes, Duality and Precursors
  Precursors underlie the structures of both a polyhedron and its dual.
Vertex Figures
Mathematicians have used different definitions of the vertex figure for different purposes. Different types are examined and classified leading, via the idea of the complete vertex, to a more general definition.
Dualising Polyhedra
We have many inconsistent notions of what a dual polyhedron is. This note seeks a unified understanding and resolves some longstanding paradoxes. Unfinished.
Trimethoric (and Trisynaptic) Polyhedra
Substantially as in Mathematics and Informatics Quarterly 2/2001, Vol. 11, p.p. 71-74.
Trimethoric and trisynaptic polyhedra represent two seldom recognised classes.
Is there a Self-Dual Hendecahedron?
Thanks to those who to told me yes, there is one called the canonical form. But are there also any "non-canonical" solutions?