References

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Annotated Bibliography Here is a list of introductory and intermediate works on polyhedra, along with my brief personal annotations. Descriptions with the word mathematical in them indicate more advanced sources. (I haven't been updating this for several years.) 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) Note: If you seek references on some particular topic, try using the Find option in the Edit menu of your browser to search through this page for your keyword. 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). Benno Artmann, "Roman Dodecahedra", Mathematical Intelligencer, Vol. 15, pp. 52-53, 1993. Brief survey of ancient dodecahedral artifacts. Benno Artmann, "A Roman Icosahedron Discovered", Mathematical Intelligencer, Vol. 18, pp. 132-133, 1996. Describes the one hollow bronze Roman icosahedron reported. Benno Artmann, "Symmetry Through the Ages: Highlights from the History of Regular Polyhedra", in In Eves' Circles , Joby Milo Anthony (ed.), Mathematical Association of America, pp. 139-148, 1994. A short history. It includes references describing Platonic solids being carved in stone circa 2000 B.C. Steve Baer, Zome Primer , Zomeworks, 1970 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.) T. Bakos, "Octahedra inscribed in a Cube," Mathematical Gazette, Vol. 43, pp. 17-20, 1959. Describes compounds of 4 cubes and 4 octahedra. Walter William Rouse Ball, revised by H.S.M. Coxeter, Mathematical Recreations and Essays , New York, 1938; 11th ed., 1960, (Dover reprint). 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. Thomas F. Banchoff, Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, W. H. Freeman, 1990 . Gentle introduction to polytopes and the geometry of four or more dimensions. Nicely illustrated. Daniele Barbaro, La Pratica Della Perspettiva, 1569 (Arnaldo Forni reprint, 1980) . Perspective manual with many drawings of polyhedra, including several unusual "symmetrohedra." (in Italian). Robert Stanley Beard, Patterns in Space, Creative Publications, 1973. A miscellany of geometric drawings and tables, including polyhedral patterns. Martin Berman, "Regular-faced Convex Polyhedra," Journal of the Franklin Institute, Vol. 291 No. 5, pp. 329-352, May 1971 . Gives photographs and nets for constructing all of the Johnson solids. V. G. Boltyanskii, Equivalent and Equidecomposable Figures, Heath 1956 . 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. Max Brückner, Vielecke und Vielflache: Theorie und Geschichte , Teubner, 1900. 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. M. J. Buerger, Elementary Crystallography , Wiley, 1956, (MIT press reprint, 1978). Good source for crystallographic polyhedra and the 230 space groups. Vladimir Bulatov, "An Interactive Creation of Polyhedra Stellations with Various Symmetries," in Proceedings of Bridges: Mathematical Connections in Art, Music, and Science, 2001. Describes an excellent program for generating stellations. M. E. Catalan, "Memoire sur la Theorie des Polyedres," Journal de L'ecole Imperiale Polytechnique, Vol. 24, book 41, pp. 1-71 plus plates, 1865. Original presentation (in French) of the Catalan solids (the duals to the Archimedean solids) plus some combinatorics. Stewart T. Coffin, The Puzzling World of Polyhedral Dissections , Oxford, 1990. Great book about wooden take-apart puzzles based on polyhedral shapes, written by a most ingenious puzzle designer. It is available online . Robert Connelly, "Rigidity," Chapter 1.7 (pp. 223-271) of the Handbook of Convex Geometry , P.M. Gruber and J.M. Wills (editors), Elsevier, 1993. Mathematical summary of results about the rigidity of polyhedra and tensegrity structures. Robert Connelly and Allen Back, "Mathematics and Tensegrity," American Scientist, Vol. 86, pp. 142-151, March/April, 1998. Analysis of polyhedral tensegrity structures. John Lodge Cowley, Solid Geometry , London, 1752. Interesting version of Euclid, containing pop-up paper models of polyhedra, including rhombic dodecahedron of 2nd type. H.S.M. Coxeter, Introduction to Geometry , 2nd ed., Wiley, 1969. Broad presentation of geometry with sections on platonic solids, the golden ratio, polyhedral symmetry, and four-dimensional polytopes. H.S.M. Coxeter, The Beauty of Geometry: Twelve Essays , Dover 1999 (reprint, with new title, of Twelve Geometric Essays, S. Illinois U. Pr., 1968). Collection of mathematical essays; not elementary. H.S.M. Coxeter, Kaleidoscopes: Selected Writings of H.S.M. Coxeter , Wiley, 1995. 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!) H.S.M. Coxeter, Regular Polytopes, Macmillan, 1963, (Dover reprint, 1973). H.S.M. Coxeter, Regular Complex Polytopes, Cambridge, 1974, (2nd ed., 1991). Mathematical text describes a generalization of polyhedra based on complex numbers. H.S.M. Coxeter, "Virus Macromolecules and Geodesic Domes," in A Spectrum of Mathematics, J.C. Butcher (editor), Aukland, 1971. Analysis of the icosahedron-based forms of various geodesic domes and viruses. H.S.M. Coxeter, M.S. Longuet-Higgins, and J.C.P. Miller, "Uniform Polyhedra," Philosophical Transactions of the Royal Society, Ser. A, 246, pp. 401-449, 1953. The first complete list of the uniform polyhedra. The essential mathematical paper on the nonconvex uniform polyhedra. H.S.M. Coxeter, P. DuVal, H.T. Flather, and J.F. Petrie, The Fifty-Nine Icosahedra, U. Toronto Pr., 1938, (Springer-Verlag reprint, 1982), (Tarquin reprint 1999). 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. H.S.M. Coxeter, M. Emmer, R. Penrose, and M.L. Teuber (editors), M.C. Escher: Art and Science, North-Holland, 1986. Collection of papers on Escher's work, with analyses of his use of tessellations and polyhedra. K. Critchlow, Order in Space: a design source book , Viking, 1970. Polyhedra, space-fillers, tessellations, sphere packings, and their relationships, with lots of line drawings. Peter R. Cromwell, Polyhedra, Cambridge, 1997. 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. Akos Csaszar, "A polyhedron without diagonals", Acta Univ Szegendiensis, Acta Scient. Math , v. 13, pp 140-2, 1949. Describes the Csaszar polyhedron: fourteen triangular faces forming a torus. H. Martyn Cundy and A.P. Rollett, Mathematical Models , Oxford, 1961; third edition Tarquin publ., 1981. 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. H. Martyn Cundy and Magnus J. Wenninger, "A compound of five dodecahedra," Mathematical Gazette , pp. 216-218., 1976. Describes the compound of five dodecahedra. Margaret Daly Davis, Piero della Francesca's Mathematical Treatices: The "Trattato d'abaco" and "Libellus de quinque corporibus regularibus," Longo Editore,1977. Traces the effects of Piero's writings on renaissance polyhedral developments. Rene Descartes, De Solidorum Elementis, circa 1637. See Federico, below. Andreas W. M. Dress, "A combinatorial theory of Grunbaum's new regular polyhedra," Aequationes Mathematicae, "Part I," Vol. 23, pp. 252-265, (1981); "Part II," Vol. 29, pp. 222-243, (1985). Two-part article analyzing and enumerating "hollow-faced" regular polyhedra. Albrecht Durer, Underweysung der Messung, 1525,(translated to English as Painter's Manual, Abaris reprint, 1977). The earliest use of nets to represent polyhedra. John D. Ede, "Rhombic Triacontahedra," Mathematical Gazette, Vol. 42, pp. 98-100, 1958. Discusses the stellation of the rhombic triacontahedron. Aniela Ehrenfeucht, The Cube Made Interesting , Macmillan, 1964. 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. Michele Emmer (editor), The Visual Mind , MIT, 1993. 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. David Eppstein, "Zonohedra and Zonotopes," Mathematica in Education and Research, Vol 5, No. 4, pp. 15-21, 1996. Mathematica code for generating zonohedra. M.C. Escher, The Graphic Work of M.C. Escher , Ballantine, 1971. Art of Maurits Cornelis Escher with his own commentary. M.C. Escher, Bruno Ernst, The Magic Mirror of M.C. Escher , Ballantine, 1976, (Tarquin reprint, 1982). Art of Maurits Cornelis Escher with commentary by Ernst on Escher's life and art, including several pages on his use of polyhedra. Euclid, The Thirteen Books of the Elements , circa 300 BC, (Dover reprint in three volumes, Thomas L. Heath editor, 1956). The urtext. Just do it. P.J. Federico, Descartes on Polyhedra: A Study of the De Solidorum Elementis , Springer-Verlag, 1982. Translation and analysis of Descartes' 1637 book which includes his famous angle deficit theorem. E. S. Fedorov, Symmetry of Crystals , transl. David and Katherine Harker, American Crystallographic Assoc. reprint 1971. Description of zonohedra and their properties. J. V. Field, "Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Durer, Daniele Barbaro, and Johannes Kepler," Archive for History of Exact Sciences, vol. 50, no. 3, pp. 241-289, 1996. Renaissance history of the rediscovery of the Archimedean polyhedra. G. M. Fleurent, "Symmetry and Polyhedral Stellation I," Computers Math. Applic., Vol 17, p. 167-175, 1989. Stellates the chiral icositetrahedron. Lorraine L. Foster, Archimedean and Archimedean Dual Polyhedra , VHS video tape, 47 minutes, California State University, Northridge, Instructional Media Center, 1990. 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. Greg N. Frederickson, Dissections: Plane and Fancy, Cambridge, 1997. Methods of dissecting shapes and reassembling the pieces into other shapes, with three chapters on dissecting polyhedra. H. Fruedenthal and B.L.van der Waerden, "Over een Bewering van Euclides", Simon Stevin, vol. 25, pp. 114-121, 1947. Describes the convex equilateral deltahedra. (in Dutch) Tomoko Fuse, Unit Origami: Multidimensional Transformations , Japan Publications, 1990. Impressive assemblage of modular origami polyhedra, with photos and instructions. J. Francois Gabriel (editor), Beyond the Cube: The Architecture of Space Frames and Polyhedra, Wiley, 1997. Collection of sixteen articles by architects on applications of polyhedra in architecture. (See Hanaor, Tomlow) Martin Gardner, The Five Platonic Solids , Chapter 1 of his The 2nd Scientific American Book of Mathematical Puzzles and Diversions , Simon and Schuster, 1961. Some polyhedral puzzles and miscellany. Matila Ghyka, The Geometry of Art and Life , Sheed and Ward, 1946, (Dover reprint, 1977). Discusses the relations between polyhedra and art, stretching things a bit far in places. J. R. Gott, "Pseudopolyhedrons," American Mathematical Monthly , Vol 74, p. 497, 1967. Illustrates a number of infinite polyhedra constructed of regular polygons. Ugo Adriano Graziotti, Polyhedra: The Realm of Geometric Beauty, self-published, 1962. Curious, nicely illustrated, 38 page booklet with original constructions for the Archimedean duals. Watch for errors. Robert Grip, Tensegrity: Introductory Theory and Model Construction , Fuller, 1978. Brief, well illustrated, 18 page booklet. Branko Grunbaum, Convex Polytopes , Interscience, 1967. Mathematical text focusing on combinatorial issues. Branko Grunbaum, "Regular Polyhedra --- Old and New," Aequationes Mathematicae, Vol 15, pp. 118-120, 1977. Short note pointing out that a consistent set of definitions allows for more regular polyhedra than are standardly counted. Branko Grunbaum, "Uniform Tilings of 3-Space," Geombinatorics 4, 1994, pp. 49-56. Lists all uniform ways to pack uniform polyhedra in 3-space. Includes several omitted in similar lists by Andreini, Critchlow, and Williams. Branko Grunbaum, "Polyhedra with hollow faces," in T. Bisztriczky (ed.) Polytopes: Abstract, Convex and Computational , Kluwer, 1994, pp. 43-70. 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. Branko Grunbaum and G. C. Shephard, "Duality of Polyhedra," in Senechal and Fleck (eds.) Shaping Space , Birkhauser, 1988. Mathematical paper discusses subtle inconsistencies in naive notions of duality. Rona Gurkewitz, Bennet Arnstein, 3-D Geometric Origami: Modular Polyhedra, Dover, 1995. An illustrated, step-by-step, how-to-fold-and-construct guide. Rona Gurkewitz, Bennet Arnstein, Multimodular Origami: Polyhedra, Dover, 2003. A follow-on guide with Archimedeans, Buckyballs, and dual models. Ernst Haeckel, Art Forms in Nature, 1904, Dover reprint, 1974. Not about polyhedra, but a beautiful book with 100 plates by Haeckel, starting with an icosahedral radiolarian. Ariel Hanaor, "Tensegrity: Theory and Application," in Gabriel, Beyond the Cube, 1997. Discusses a variety of tensegrity structures, including single and double layer polyhedral forms. Zvi Har'El, "Uniform Solution for Uniform Polyhedra," Geometriae Dedicata 47, 1993. Mathematical description of an algorithm (used here) for computing the descriptions of the uniform polyhedra. Istvan Hargittai (editor), Fivefold Symmetry, World Scientific, 1992 . 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"). Istvan Hargittai and Magdolna Hargittai, Symmetry: A Unifying Concept, Shelter Publications, 1994 . A rich compendium of hundreds of photos and drawings, this introduction to symmetry illustrates many geometric patterns in nature, art, and architecture. Istvan Hargittai and Magdolna Hargittai, Symmetry through the Eyes of a Chemist, Plenum, 2nd ed., 1995 . Introduction to symmetry in many forms, especially its role in chemistry, with many examples of polyhedral chemical structures. Michael G. Harman, "Polyhedral Compounds", unpublished manuscript, circa 1974 . Unpublished mathematical paper presents many interesting compounds not previously described. George W. Hart, "Calculating Canonical Polyhedra", Mathematica in Education and Research, Vol 6 No. 3, Summer 1997, pp. 5-10 . ( online Mathematica version ) ( supplement ) An algorithm for finding the canonical form of any given polyhedron. George W. Hart, "Zonish Polyhedra," Proceedings of Mathematics and Design '98 , San Sebastian, Spain, June 1-4,1998 . ( online version ) ( order proceedings ) A class of polyhedra with zones , which generalizes zonohedra, and their use in sculpture. George W. Hart, "Icosahedral Constructions," in Bridges: Mathematical Connections in Art, Music, and Science, Reza Sarhangi (editor), 1998, pp. 195-202, ISBN 0-9665201-0-6 . ( online version ) History of icosahedral symmetry and its use in sculpture. George W. Hart, "Zonohedrification," Mathematica Journal, Vol. 7 no. 3, 1999 . Zonohedra construction algorithm. ( online version for subscribers ) George W. Hart, "Reticulated Geodesic Constructions," Computers and Graphics 24(6), Dec. 2000, pp. 907-910 . ( online version ) Construction algorithm for icosahedral geodesic domes and related polyhedra. George W. Hart, "Sculpture based on Propellorized Polyhedra," Proceedings of MOSAIC 2000 , Seattle. ( online version ) Properties of the 'propello-Platonic' polyhedra, and their use in sculpture. George W. Hart, "The Millennium Bookball," Proceedings of Bridges 2000: Mathematical Connections in Art, Music and Science, Southwestern College, Winfield, Kansas, July 28-30, 2000, and in Visual Mathematics , Vol. 2, no. 3, 2000. ( US copy ) Use of the rhombic triacontahedron and other polyhedra in a large public sculpture. George W. Hart, "Loopy," Humanistic Mathematics , June, 2002. ( online version ) Describes a sculpture, and a paper model of it, related to the compound of five tetrahedra. George W. Hart, "Solid-Segment Sculptures,"Proceedings of Colloquium on Math and Arts , Maubeuge, France, 20-22 Sept., 2000, and in Mathematics and Art , Claude Brute ed., Springer-Verlag, 2002, pp. 17-27. ( online version ). A technique for making a polyhedron which envelops a given set of segments, plus its application to projections of polytopes. George W. Hart, "Computational Geometry for Sculpture", Proceedings of ACM Symposium on Computational Geometry, Tufts University, June 2001, pp.284-287. ( PDF version ) Illustrates and briefly describes sculpture with polyhedral symmetries. George W. Hart, "Rapid Prototyping of Geometric Models," Proceedings of Canadian Conference on Computational Geometry, August 2001. ( online version ) Illustrates and briefly describes 3D printings of various polyhedral models. George W. Hart, "In the Palm of Leonardo's Hand," Nexus Network Journal , vol. 4, no. 2, Spring 2002; reprinted in Symmetry: Culture and Science, vol. 11, 2000 (appeared 2003), pp. 17-25. ( online journal ). Discusses Leonardo's polyhedra models and shows 3D printings of them. George W. Hart, "A Color-Matching Dissection of the Rhombic Enneacontahedron", Symmetry: Culture and Science, vol. 11, 2000 (appeared in 2003), pp. 183-199. ( online version ) Does for the rhombic enneacontahedron what Kowalewski did for the rhombic triacontahedron. George W. Hart, "Sculpture from Symmetrically Arranged Planar Components", in Meeting Alhambra (Proceedings of ISAMA-Bridges 2003), University of Granada, Granada, Spain, pp. 315-322. ( online version ) Sculpture based on polyhedral stellations, make by rapid pro