Annotated Bibliography

When asked for one outstanding book from which to begin learning about polyhedra, I recommend one of these:

• W.W.R. Ball / H.S.M. Coxeter, Mathematical Recreations and Essays. (one great polyhedra section)
• H.S.M. Coxeter, Regular Polytopes. (excellent, college level)
• Peter R. Cromwell, Polyhedra. (very comprehensive)
• George W. Hart and Henri Picciotto, Zome Geometry. (Hands-on models throughout) (more information)
• H. Martyn Cundy and A.P. Rollett, Mathematical Models, (one great polyhedra section)
• Alan Holden, Shapes, Spaces and Symmetry. (very visual
• Peter Pearce, Structure in Nature is a Strategy for Design. (outstanding) 
• Magnus Wenninger, Polyhedron Models. (most comprehensive instructions for paper models)

Hugh Apsimon, "Three facially regular polyhedra", Canadian Journal of Mathematics, pp. 326-330, 1950.

Shows three infinite polyhedra constructed from equilateral triangles, with 12, 9, or 8 at a vertex. The latter is a cubic lattice of alternately left- and right-handed snub cubes joined at their squares (with the squares then removed).

Brief survey of ancient dodecahedral artifacts.

Describes the one hollow bronze Roman icosahedron reported.

A short history. It includes references describing Platonic solids being carved in stone circa 2000 B.C.

Self-published booklet about the use of zonohedra in an architectural system and the Zometool plastic polyhedral construction toy. (See also Fivefold Symmetry by Hargittai, below.)

Describes compounds of 4 cubes and 4 octahedra.

An essential classic of recreational mathematics with a pithy chapter on polyhedra written by Coxeter. This plus many other interesting topics make this an excellent book.

Gentle introduction to polytopes and the geometry of four or more dimensions. Nicely illustrated.

Perspective manual with many drawings of polyhedra, including several unusual "symmetrohedra." (in Italian).

A miscellany of geometric drawings and tables, including polyhedral patterns.

Gives photographs and nets for constructing all of the Johnson solids.

Discusses the mathematical conditions of when it is possible to dissect a given polyhedron into a finite number of pieces and reassemble them into another given polyhedron.

Classic turn-of-the-century text (in German) summarizing everything known at the time about polyhedra. Contains drawings, plates, and discussion, including some polyhedral topics not mentioned in the English language literature as far as I know.

Good source for crystallographic polyhedra and the 230 space groups.

Describes an excellent program for generating stellations.

Original presentation (in French) of the Catalan solids (the duals to the Archimedean solids) plus some combinatorics.

Great book about wooden take-apart puzzles based on polyhedral shapes, written by a most ingenious puzzle designer.  It is available online.

Mathematical summary of results about the rigidity of polyhedra and tensegrity structures.

Analysis of polyhedral tensegrity structures.

Interesting version of Euclid, containing pop-up paper models of polyhedra, including rhombic dodecahedron of 2nd type.

Broad presentation of geometry with sections on platonic solids, the golden ratio, polyhedral symmetry, and four-dimensional polytopes.

Collection of mathematical essays; not elementary.

A collection of Coxeter's papers, mainly mathematical, on a range of topics, especially polyhedra and polytopes. Also contains a nice biography. (But too expensive!)

The first complete list of the uniform polyhedra. The essential mathematical paper on the nonconvex uniform polyhedra.

Classic enumeration of the 59 stellations of the icosahedron, with figures and historical notes. The 1999 edition is updated with new diagrams plus photos of some of Flather's original paper models.

Collection of papers on Escher's work, with analyses of his use of tessellations and polyhedra.

Polyhedra, space-fillers, tessellations, sphere packings, and their relationships, with lots of line drawings.

A must-see for anyone interested in polyhedra. Much art, history, and math, in a well illustrated book with lots of nice touches. At 450 pages, with many references, this is by far the most comprehensive book on polyhedra yet printed.

Describes the Csaszar polyhedron: fourteen triangular faces forming a torus.

An outstanding classic. (I think I had it out from my public library as a youth for two or three years straight.) It has instructions for making many models including Archimedeans, duals, some compounds, some stellations, and two non-convex quasi-regular polyhedra and their duals. Plus plenty of good stuff other than polyhedra.

Describes the compound of five dodecahedra.

Traces the effects of Piero's writings on renaissance polyhedral developments.

See Federico, below.

Two-part article analyzing and enumerating "hollow-faced" regular polyhedra.

The earliest use of nets to represent polyhedra.

Discusses the stellation of the rhombic triacontahedron.

Uses 3D line drawings (via a pair of red/blue "3D glasses") to illustrate the symmetries of the cube, its relations to other polyhedra, some dissections, and how to pass a cube through another cube.

An assortment of interesting papers by various authors on geometry and art, with some polyhedral topics, including one, "Art and Mathematics: The Platonic Solids" by Emmer.

Mathematica code for generating zonohedra.

Art of Maurits Cornelis Escher with his own commentary.

Art of Maurits Cornelis Escher with commentary by Ernst on Escher's life and art, including several pages on his use of polyhedra.

The urtext. Just do it.

Translation and analysis of Descartes' 1637 book which includes his famous angle deficit theorem.

Description of zonohedra and their properties.

Renaissance history of the rediscovery of the Archimedean polyhedra.

Stellates the chiral icositetrahedron.

Video describing the Platonic and Archimedean solids and their duals, showing many models, some computer animations, and a few mineral crystals, with a section of historical perspective.

Methods of dissecting  shapes and reassembling the pieces into other shapes, with three chapters on dissecting polyhedra.

Describes the convex equilateral deltahedra. (in Dutch)

Impressive assemblage of modular origami polyhedra, with photos and instructions.

Collection of sixteen articles by architects on applications of polyhedra in architecture. (See Hanaor, Tomlow)

Some polyhedral puzzles and miscellany.

Discusses the relations between polyhedra and art, stretching things a bit far in places.

Illustrates a number of infinite polyhedra constructed of regular polygons.

Curious, nicely illustrated, 38 page booklet with original constructions for the Archimedean duals. Watch for errors.

Brief, well illustrated, 18 page booklet.

Mathematical text focusing on combinatorial issues.

Short note pointing out that a consistent set of definitions allows for more regular polyhedra than are standardly counted.

Lists all uniform ways to pack uniform polyhedra in 3-space.  Includes several omitted in similar lists by Andreini, Critchlow, and Williams.

Detailed framework for a general notion of polyhedra in which the faces are basically a path of edges, and so may be nonplanar, or the edges may go around more than once, or may be infinite, e.g., a helix.

Mathematical paper discusses subtle inconsistencies in naive notions of duality.

An illustrated, step-by-step, how-to-fold-and-construct guide.

A follow-on guide with Archimedeans, Buckyballs, and dual models.

Not about polyhedra, but a beautiful book with 100 plates by Haeckel, starting with an icosahedral radiolarian.

Discusses a variety of tensegrity structures, including single and double layer polyhedral forms.

Mathematical description of an algorithm (used here) for computing the descriptions of the uniform polyhedra.

A broad collection of papers by assorted authors on various aspects of 5-fold symmetry. Several discuss icosahedral symmetry in nature, art, and architecture. Includes zonohedron papers by Stephen C. Baer ("The Discovery of Space Frames with Fivefold Symmetry") and David Booth ("The New Zome Primer").

A rich compendium of hundreds of photos and drawings, this introduction to symmetry illustrates many geometric patterns in nature, art, and architecture.

Introduction to symmetry in many forms, especially its role in chemistry, with many examples of polyhedral chemical structures.

Unpublished mathematical paper presents many interesting compounds not previously described.

An algorithm for finding the canonical form of any given polyhedron.

A class of polyhedra with zones, which generalizes zonohedra, and their use in sculpture.

History of icosahedral symmetry and its use in sculpture.

Zonohedra construction algorithm. (online version for subscribers)

Construction algorithm for icosahedral geodesic domes and related polyhedra.

Properties of the 'propello-Platonic' polyhedra, and their use in sculpture.

Use of the rhombic triacontahedron and other polyhedra in a large public sculpture.

Describes a sculpture, and a paper model of it, related to the compound of five tetrahedra.

A technique for making a polyhedron which envelops a given set of segments, plus its application to projections of polytopes.

Illustrates and briefly describes sculpture with polyhedral symmetries.

Illustrates and briefly describes 3D printings of various polyhedral models.

Discusses Leonardo's polyhedra models and shows 3D printings of them.

Does for the rhombic enneacontahedron what Kowalewski did for the rhombic triacontahedron.

Sculpture based on polyhedral stellations, make by rapid prototyping or laser-cut acrylic.

Learn about geometry and polyhedra by making beautiful 3D models.

Outstanding step-by-step construction manual for folding paper strips into polyhedra. Includes a nice introduction to the mathematics.

Well illustrated popular overview. Excellent introduction to polyhedra.

Interesting constructions with polyhedral symmetries.

One chapter explains how to make nested ivory Chinese balls with polyhedral symmeties, and how to turn a dodecahedron or icosahedron on a lathe.

Describes the general stellation process, and shows various unusual stellations.

Computational method for locating vertex coordinates, with exact formulas for angles.

Beautiful engravings of polyhedral imaginings, including the earliest presentations of the great dodecahedron, the great stellated dodecahedron, and the first stellation of the icosahedron, long before the mathematicians envisioned them.

A book which you cut up, score, fold, and glue to make polyhedra: the compound of ten tetrahedra and the compound of five cubes.

Another book which you cut up, score, fold, and glue. This one makes two stellations of the icosahedron: the final stellation and the one called Fg1 in Coxeter et al. (numbered sixth in Wenninger).

A series of three books which you cut up, score, fold, and glue, which cover a wide range of Platonic, Archimedean, Kepler-Poinsot, compound, and stellated polyhedra.

Another series of three books which you cut up, score, fold, and glue, which include some folding and flexing polyhedral models.

Definitive enumeration of the "Johnson solids".

Plans for various collapsible polyhedra which spring into 3D from flatness, using rubber bands, clever hinges, and rotating joints.

Presents the polyhedron which is related to the great rhombicuboctahedron in the same way that the pseudo rhombicuboctahedron is related to the rhombicuboctahedron.

Many images of attractive polyhedra formed by taking the convex hull of symmetrically placed polygons.

A wide-ranging smorgasbord of topics, with considerable discussion of polyhedra and many references.

Non-rigorous computations of strut lengths in tensegrity structures and geodesic domes.

Finally translated into English after 372 years, Kepler presents the Archimedean solids, the small and great stellated dodecahedra, and the rhombic dodecahedron and triacontahedron.

English translation of challenging 1884 German monograph, with accessible introductory chapters on symmetry groups of polyhedra.

Wonderful book shows how to dissect the five-colored rhombic triacontahedron into twenty three-colored rhombic parallelepipeds. Also discusses the thirty six-colored cubes.  (Watch for mathematical typos in the English translation.) (Sold by Zometool Inc.)

Fascinating analysis of Euler's theorem in its many variants. Essential reading for its vision, its detailed historical notes, and its lively use of the theorem to make larger points about the nature of mathematics itself.

Three self-published books in a series. The first illustrates the relation between a polyhedron and its dual by means of a continuous sequence of intermediate rectangle-connected "transpolyhedra." The latter two present graphic arrangements of various polyhedra to illustrate their relationships (akin to the Periodic Table of the Elements.)

One I read of, but haven't seen personally, on my list to look up.

Classic text with a good discussion of polyhedra.

An original of design science; emphasizes counting.

Shows a study for Escher's Stars featuring the first stellation of the rhombic dodecahedron instead of the compound of three octahedra.

Discusses some properties of the stellations of the rhombic dodecahedron.

Mathematical text which focuses on issues related to convexity.

Analysis of the famous painting and its mathematical roots.  Speculative in parts.

Describes a port of Zvi Har'El's algorithm to the Mathematica software environment (which I used here).

Describes Mathematica software (which I used here) to construct the 59 stellations.

Authoritative brief history with many references.

General technique of analysis and nomenclature for stellating polyhedra, applied to the triacontahedron. Describes the 226 fully supported stellations, with photos of some models.

Exact formulas for constructing all 77 kinds of uniform polyhedra---finally, fifty years after they were first enumerated!

Stellates the trapezoidal hexacontahedron.

Instructions for sixteen different modular origami polyhedra.

Beautifully crafted, but very expensive, book with photographs of thousands of models, emphasizing the author's artistic explorations. A bit strong on the mystical aspects, but you can ignore those parts.

Detailed illustrated instructions for making 27 origami polyhedra, ranging from elementary to advanced, each from a single sheet of paper, i.e., not a modular origami method.

Two dozen cut-and-paste paper models of interesting crystallographic polyhedra, with chemical and geometric notes on each.

Discusses one of the tetrahedral stellations of the dodecahedron.

Illustrated with beautiful solid-edge figures of polyhedra by Leonardo da Vinci, Pacioli wrote in Italian about the beauty of symmetry, proportion,  the golden number, and polyhedra. (Available in German, Spanish, or French translation, but not English.)

Mathematical paper for computer people programming polyhedra.

How to cut a cube into 15 pieces and reassemble as a rhombic dodecahedron, or into 17 pieces and reassemble as a truncated octahedron.

Discusses and enumerates the stellations of the rhombic triacontahedron.

A substantial, profusely illustrated discussion of 2D and 3D geometry, and its relations to natural design and space structures.

Introductory presentation of tessellations, polyhedra, and space fillings, well illustrated with excellent line drawings (but not much text).

Nicely illustrated 50 page self-published book shows relations between the Platonic, Archimedean, and dual solids.

Summary of Piero's original contributions to the revival of polyhedral geometry.

A wonderful, comprehensive survey with many illustrations, and an inexpensive paperback as well.

Introduction to the mathematics and making of polyhedral tensegrity structures, with detailed instructions for several dozen models.

Great analysis of Escher's use of symmetry, including several pages on his use of polyhedra.

Combination of a book and a set of ready-to-glue-together Platonic solids and rings of tetrahedra, decorated with Escher graphics.

A terrific compendium from a wide-ranging 1984 polyhedra conference. The material ranges from introductory to advanced, covering many aspects of polyhedra. (The conference itself was outstanding, I might mention. Look for this happy participant on p. 73.)

Amazing presentation of a great variety of beautiful rhombic solids to be found nowhere else, including details on how to cut them from solid blocks of wood.

Construction manual for making dozens of paper polyhedral constructions from origami components.

Mathematical paper describing the computer analysis that proved the Coxeter et al. enumeration is complete.

Mathematical paper describing the 75 uniform compounds.

Describes and illustrates stellations of the triakis tetrahedron.

Describes and illustrates the compounds of two and five great dodecahedra and small stellated dodecahedra.

Same as the above, without construction notes.

Cut-and-glue cardboard book with nets of the Platonic solids and three of the Kepler-Poinsot solids.

A classic of recreational mathematics which presents brief interesting snippets about many topics including polyhedra.

This German classic, Lectures on the Theory of Polyhedra, concerns the combinatorial properties of polyhedra, largely avoiding symmetric polyhedra. Most of its content is covered in Grunbaum's Convex Polytopes. (I have a rare English translation by the U.S. Air Force.)

A delightful and very original hand-printed and illustrated exploration of "donuts" etc., made from regular polygons. Written mostly at an elementary level, this 256 page book describes a great many novel polyhedral constructions.

Presents methods of expanding a polyhedron about its edges, faces, etc., to generate related polyhedra. (Summarized in W.W.R. Ball.)

Small introduction with very nice line illustrations.

Compares Coxeter's definition of zonohedron (requiring centrally symmetric faces) with Fedorov's definition (allowing faces with opposite edges parallel but unequal).

Nicely illustrated self-published booklets which consider a slight generalization of the usual definition of uniform polyhedra, in a series titled "The Complete? Polyhedra."

History of polyhedra in art.

Illustrates the use of perspective polyhedra in Renaissance intarsia (wooden inlay panels).

Mathematical monograph on polyhedra and related topics, with nice packet of red/green stereo images.

Describes the geometry of several geodesic dome methods and includes diagrams which you cut out, score, fold, and glue to make paper models.

Analysis of Leonardo's writings and drawings concerning perspective and polyhedra.  Many images of renaissance polyhedra.

Enumerates all the ways that a cube can be transformed by a polyhedral symmetry group. While mainly full of mathematical group-theory notation, there are also pictures of many compounds not described elsewhere, and patterns for making ten of them out of paper.

Describes and illustrates (with photos of paper models) roughly 100 infinite uniform polyhedral structures composed of regular polygons.

Almost 2000-pages, this encyclopedia is written at an accessible level and includes many polyhedral topics.

Monograph on infinite polyhedra and space structures, with a crystallographic perspective.

Short entries on many geometric topics, including polyhedral. Superficial but gives references.

Illustrated directions for constructing paper models of 119 polyhedra, including all the uniform polyhedra and 19 of the stellations of the icosahedron. A classic, written by a true philomorph.

A sort of "Readers Digest" version of his later 1971 book; a short how-to for a range of basic models.

Instructions for constructing spherical models based on polyhedral symmetries. Dover version has new additions in an appendix.

In the same informative how-to spirit as the above three books, this is the only reference I know which discusses and illustrates the duals to all of the uniform polyhedra.

Several spherical paper sculptures (with polyhedral symmetry) and how they are designed.

Shows the "Theosophical" compound of five cubes (which has octahedral symmetry) and one (slightly distorted) compound of ten cubes.

Illustrates the compound of three octahedra and a related compound of four.

Classic treatment of symmetry in its many forms, as found in art, nature, and mathematics.

An original exploration of design principles using polyhedral structures.

Clear instructions for making beautiful modular origami orbs with polyhedral symmetries.

Absolutely, positively, without a doubt, the most difficult method of constructing a dodecahedron I have ever seen in my life. One of the 34 steps is: "Oblique lines are drawn to shade ten equilateral triangles (1P2, 3Q4, 5R6, 7S8, and 9T10 on the right moiety, and 11W12, 13M*14, 3*X15, 16R*S*, and N6*17 on the left moiety)."

A lengthy fastidious mathematical proof that Johnson's list of solids is complete. Has anyone ever really read this?

Excellent mathematical text on convex polyhedra and polytopes.

Novel method of combining two or more given polyhedra to create interesting new "blended" polyhedra.

Virtual Polyhedra, (c) 1996, 1997, 1998, 1999, 2000, 2001, George W. Hart