News
Significant changes over the last couple of years or so.
- 15 May 2021. Filling Polytopes. Revised and completed: The problem of holes inside polytopes and how to resolve it.
- 25 Apr 2021. Morphic Polytopes. Update (New 30 Aug 2019): A new definition of polyhedra and higher polytopes, seeking to reconcile abstract and more traditional topological approaches.
- 19 Mar 2021. Weaire-Phelan Bubbles. Update: Added downloadable net of the Kelvin bubble, the problem clarified.
- 19 Sept 2020. A Critique of Abstract Polytopes. Update (New 17 July 2020): Abstract polytope theory as currently formulated is powerful and is changing our ideas, but it has flaws.
- 17 July 2020. Ditela, Polytopes and Dyads. Update: The nullon named, miscellaneous small revisions.
- 8 July 2020. Polytopes, Duality and Precursors. Update: Rewrite to adopt schemas from theoretical physics and hence to clarify abstract theory; unfinished.
Space-filling polyhedra
Illustrated essays, published and unpublished. Most have links to printable "nets" for 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.
- 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.
- A 3-D Quasicrystal Structure?
A possible candidate for a 3-dimensional, aperiodic crystal structure.
- 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.
- Icosahedral Precursors
Unfinished and subject to revision. New rules for stellation, founded in abstract and morphic theories.
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.
- Filling polytopes
In polytope theory, filling is shown to be of fundamental importance. Traditional theory ignores filling, and so is incomplete. This essay is incomplete too.
- 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 both the faces of a polyhedron and the vertex figures of its dual. Their theory and usefulness remain a work in hand.
- 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.
- 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?
Miscellaneous
Other polyhedron resources.