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,
- Peter R. Cromwell, Polyhedra. (very comprehensive)
- George W. Hart and Henri Picciotto, Zome Geometry.
models throughout) (more
- H. Martyn Cundy and A.P. Rollett, Mathematical Models,
great polyhedra section)
- Alan Holden, Shapes, Spaces and Symmetry. (very visual
- Peter Pearce, Structure in Nature is a Strategy for Design.
- Magnus Wenninger, Polyhedron Models. (most
comprehensive instructions for paper models)
Note: If you seek references
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
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
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
architectural system and the Zometool plastic polyhedral construction
(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,
Gentle introduction to polytopes and the geometry
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:
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:
Connections in Art, Music, and Science, 2001.
Describes an excellent program for generating
M. E. Catalan, "Memoire sur la Theorie des
Polyedres," Journal de L'ecole Imperiale Polytechnique, Vol.
24, book 41, pp. 1-71 plus plates,
Original presentation (in French) of the Catalan
solids (the duals to the Archimedean solids) plus some combinatorics.
Stewart T. Coffin, The Puzzling World of
Great book about wooden take-apart puzzles based on
polyhedral shapes, written by a most ingenious puzzle designer.
It is available
Robert Connelly, "Rigidity," Chapter 1.7 (pp.
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,
Analysis of polyhedral tensegrity structures.
John Lodge Cowley, Solid Geometry, London,
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
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
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).
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),
- 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,
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
his use of tessellations and polyhedra.
K. Critchlow, Order in Space: a design source book,
Polyhedra, space-fillers, tessellations, sphere
packings, and their relationships, with lots of line drawings.
Peter R. Cromwell, Polyhedra, Cambridge,
A must-see for anyone interested in polyhedra. Much
art, history, and math, in a well illustrated book with lots of nice
450 pages, with many references, this is by far the most comprehensive
on polyhedra yet printed.
Akos Csaszar, "A polyhedron without diagonals", Acta
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
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
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
Aniela Ehrenfeucht, The Cube Made Interesting,
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
Michele Emmer (editor), The Visual Mind, MIT,
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
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,
Art of Maurits Cornelis Escher with his own
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
Euclid, The Thirteen Books of the Elements,
circa 300 BC, (Dover reprint in three volumes, Thomas L. Heath editor,
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
Description of zonohedra and their properties.
J. V. Field, "Rediscovering the Archimedean
della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Durer,
Barbaro, and Johannes Kepler," Archive for History of Exact
vol. 50, no. 3, pp. 241-289, 1996.
Renaissance history of the rediscovery of the
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
Instructional Media Center, 1990.
Video describing the Platonic and Archimedean
their duals, showing many models, some computer animations, and a few
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
H. Fruedenthal and B.L.van der Waerden, "Over een
Bewering van Euclides", Simon Stevin, vol. 25, pp. 114-121,
Describes the convex equilateral deltahedra. (in
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,
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
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,
1994, pp. 43-70.
Detailed framework for a general notion of
which the faces are basically a path of edges, and so may be nonplanar,
the edges may go around more than once, or may be infinite, e.g., a
Branko Grunbaum and G. C. Shephard, "Duality of
Polyhedra," in Senechal and Fleck (eds.) Shaping Space,
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,
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
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
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
Istvan Hargittai and Magdolna Hargittai, Symmetry:
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
Michael G. Harman, "Polyhedral Compounds",
Unpublished mathematical paper presents many
interesting compounds not previously described.
George W. Hart, "Calculating Canonical Polyhedra", Mathematica
Education and Research, Vol 6 No. 3, Summer 1997, pp. 5-10.
George W. Hart, "Zonish Polyhedra," Proceedings
of Mathematics and Design '98, San Sebastian, Spain, June
(online version) (order proceedings)
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
George W. Hart, "Zonohedrification," Mathematica
Journal, Vol. 7 no. 3, 1999.
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,
use in sculpture.
George W. Hart, "The Millennium
of Bridges 2000: Mathematical Connections in Art, Music and
College, Winfield, Kansas, July 28-30, 2000, and in Visual Mathematics,
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
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,
Brute ed., Springer-Verlag, 2002, pp. 17-27. (online version).
A technique for making a polyhedron which envelops
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
George W. Hart, "Rapid Prototyping of Geometric
Models," Proceedings of Canadian Conference on Computational Geometry,
August 2001. (online
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.
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
University of Granada, Granada, Spain, pp. 315-322. (online version)
Sculpture based on polyhedral stellations, make by
rapid prototyping or laser-cut acrylic.
George W. Hart and Henri Picciotto, Zome
Geometry: Hands-on Learning with Zome Models, Key Curriculum
Press, 2001. (more information)
Learn about geometry and polyhedra by making
beautiful 3D models.
Peter Hilton and Jean Pedersen, Build Your Own
Polyhedra, Addison Wesley, 1988.
Outstanding step-by-step construction manual for
folding paper strips into polyhedra. Includes a nice introduction to
Alan Holden, Shapes, Spaces and Symmetry,
Columbia Univ. Pr, 1971, (Dover reprint, 1991).
Well illustrated popular overview. Excellent
introduction to polyhedra.
Alan Holden, Orderly Tangles: Cloverleafs,
Gordian Knots, and Regular Polylinks, Columbia Univ. Pr, 1983.
Interesting constructions with polyhedral
John Jacob Holtzapffel, Hand or Simple Turning:
Principles and Practice, 1881, (Dover Reprint, 1976).
One chapter explains how to make nested ivory
Chinese balls with polyhedral symmeties, and how to turn a dodecahedron
or icosahedron on a lathe.
J. L. Hudson and J. G. Kingston, "Stellating
Polyhedra," The Mathematical Intelligencer, Vol. 10, No. 3, p.
Describes the general stellation process, and shows
various unusual stellations.
Andrew Hume, Exact Descriptions of Regular and
Semi-Regular Polyhedra and their Duals, Computing Science Technical
Report #130, AT&T
Bell Laboratories, Murray Hill, 1986.
Computational method for locating vertex
coordinates, with exact formulas for angles.
Wentzel Jamnitzer, Perspectiva Corporum
Regularium, Nuremberg, 1568, (Gutenberg reprint, Paris, 1981;
Siruela reprint, Madrid, 1993).
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.
Gerald Jenkins & Magdalen Bear, Compound
Polyhedra, Tarquin Publications, 1997.
A book which you cut up, score, fold, and glue to
make polyhedra: the compound of ten tetrahedra and the compound of five
Gerald Jenkins & Magdalen Bear, Stellated
Polyhedra, Tarquin Publications, 1997.
Another book which you cut up, score, fold, and
glue. This one makes two stellations of the icosahedron: the final
the one called Fg1 in Coxeter et al. (numbered sixth in
Gerald Jenkins & Anne Wild, Make Shapes 1, 2
and 3, Tarquin Publications, 1978.
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.
Gerald Jenkins & Anne Wild, Mathematical
Curiosities 1, 2 and 3, Tarquin Publications, 1981.
Another series of three books which you cut up,
score, fold, and glue, which include some folding and flexing
Norman W. Johnson, "Convex Solids with Regular
Faces," Canadian Journal of Mathematics, 18, 1966, pp. 169-200.
Definitive enumeration of the "Johnson solids".
Scott Johnson and Hans Walser, "Pop-up Polyhedra," Mathematical
Gazette, vol 81, Nov., 1997, pp. 364-380.
Plans for various collapsible polyhedra which
3D from flatness, using rubber bands, clever hinges, and rotating
R. Hughes Jones, "The pseudo-great
rhombicuboctahedron," Mathematical Scientist, Vol. 19, No. 1,
June, 1994, pp. 60-63.
Presents the polyhedron which is related to the
great rhombicuboctahedron in the same way that the pseudo
related to the rhombicuboctahedron.
Craig S. Kaplan and George W. Hart, "Symmetrohedra:
Polyhedra from Symmetric Placement of Regular Polygons," in Proceedings
of Bridges: Mathematical Connections in Art, Music, and Science,
Many images of attractive polyhedra formed by
convex hull of symmetrically placed polygons.
Jay Kappraff, Connections: The Geometric Bridge
Between Art and Science, McGraw Hill, 1990.
A wide-ranging smorgasbord of topics, with
considerable discussion of polyhedra and many references.
Hugh Kenner, Geodesic Math and How to Use It,
Univ. Cal Pr., 1976.
Non-rigorous computations of strut lengths in
tensegrity structures and geodesic domes.
Johannes Kepler, The Harmony of the World,
1625, (transl. E.J. Aiton, A.M. Duncan, and J.V. Field, 1997, American
Finally translated into English after 372 years,
Kepler presents the Archimedean solids, the small and great stellated
dodecahedra, and the rhombic dodecahedron and triacontahedron.
Felix Klein, The Icosahedron and the Solution to
Equations of the Fifth Degree, Dover reprint, 1956.
English translation of challenging 1884 German
monograph, with accessible introductory chapters on symmetry groups of
Gerhard Kowalewski, Der Keplersche Korper und
andere Bauspiele, Koehlers, Leipzig, 1938. (In English
translation as Construction Games with Kepler's Solid, tr.
David Booth, Parker Courtney Press, 2001.)
Wonderful book shows how to dissect the
triacontahedron into twenty three-colored rhombic parallelepipeds. Also
the thirty six-colored cubes. (Watch for mathematical typos in
English translation.) (Sold by Zometool
Imre Lakatos, Proofs and Refutations,
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
Haresh Lalvani, Transpolyhedra: Dual
Transformations by Explosion-Implosion, 1977; Patterns in
Hyperspaces, 1982; Structures on Hyperstructures:
Multidimensional Periodic Arrangements of Transforming Space Structures,
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
graphic arrangements of various polyhedra to illustrate their
(akin to the Periodic Table of the Elements.)
Mary Laycock, Straw Polyhedra/Gr 4-12.
One I read of, but haven't seen personally, on my
list to look up.
L. Lines, Solid Geometry, 1935, (Dover 1965).
Classic text with a good discussion of polyhedra.
Arthur L. Loeb, Space Structures: Their Harmony
Addison-Wesley, 1976, 1991.
An original of design science; emphasizes counting.
Arthur L. Loeb, "Polyhedra in the Work of M.C.
Escher," in Coxeter et al. (eds.), M.C. Escher: Art and Science,
Shows a study for Escher's Stars featuring
the first stellation of the rhombic dodecahedron instead of the
compound of three
Dorman Luke, "Stellations of the Rhombic
Dodecahedron," Mathematical Gazette, Vol. 41, pp. 189-194,1957.
Discusses some properties of the stellations of the
L. A. Lyusternik, Convex Figures and Polyhedra,
co., 1966. (translated by Donald Barnett from the 1956 Russian
Mathematical text which focuses on issues related
Nick MacKinnon, "The Portrait of Fra Luca Pacioli," Mathematical
Gazette, vol. 77, pp. 130-219, 1993.
Analysis of the famous painting and its
Speculative in parts.
Roman Maeder, "Uniform Polyhedra," Mathematica
Describes a port of Zvi Har'El's algorithm to the
Mathematica software environment (which I used here).
Roman Maeder, "The Stellated Icosahedra," Mathematica
Education, Vol 3, pp. 5-11, 1994.
Describes Mathematica software (which I used here)
to construct the 59 stellations.
Joseph Malkevitch, "Milestones in the History of
Polyhedra," in Senechal & Fleck (eds.), Shaping Space, pp.
80-92, Birkhauser, 1988.
Authoritative brief history with many references.
Peter W. Messer, "Stellations of the Rhombic
Triacontahedron and Beyond," Structural Topology 21, pp. 25-46,
General technique of analysis and nomenclature for
stellating polyhedra, applied to the triacontahedron. Describes the 226
fully supported stellations, with photos of some models.
Peter W. Messer, "Closed Form Expressions for Uniform
Polyhedra and Their Duals," Discrete and Computational Geometry
Exact formulas for constructing all 77 kinds of
uniform polyhedra---finally, fifty years after they were first
Peter W. Messer and Magnus J. Wenninger, "Symmetry
Stellation II," Computers Math. Applic. Vol. 17, pp. 195-201, 1989.
Stellates the trapezoidal hexacontahedron.
David Mitchell, Mathematical Origami: Geometrical
Shapes by Paper Folding, Tarquin Publ., 1997.
Instructions for sixteen different modular origami
Koji Miyazaki, An Adventure in Multidimensional
Space: The Art and Geometry of Polygons, Polyhedra, and Polytopes,
Beautifully crafted, but very expensive, book with
photographs of thousands of models, emphasizing the author's artistic
bit strong on the mystical aspects, but you can ignore those parts.
John Montrol, A Plethora of Polyhedra in Origami,
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.
Robert E. Newnham and Steven A. Markgraf, Classic
Crystals: A Book of Models, Materials Research Laboratory,
Pennsylvania State University,
Two dozen cut-and-paste paper models of interesting
crystallographic polyhedra, with chemical and geometric notes on each.
John Ounsted, "An Unfamiliar Dodecahedron," Mathematics
Teaching, Vol 83, pp. 46-47, 1978.
Discusses one of the tetrahedral stellations of the
Luca Pacioli, De Divine Proportione, 1509,
(Ambrosiana fascimile reproduction, 1956; Silvana fascimile
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.)
Alan W. Paeth, "Exact Dihedral Metrics for Common
Polyhedra," in Graphic Gems II, James Arvo (ed.), Academic,
Mathematical paper for computer people programming
David Paterson, "Two Dissections in 3-D," Journal of
Recreational Mathematics, Vol. 20, p. 257-270, 1988.
How to cut a cube into 15 pieces and reassemble as
dodecahedron, or into 17 pieces and reassemble as a truncated
G. S. Pawley, "The 227 Triacontahedra," Geometriae
Dedicata, Vol 4, pp. 221-232, 1975.
Discusses and enumerates the stellations of the
Peter Pearce, Structure in Nature is a Strategy
A substantial, profusely illustrated discussion of
2D and 3D geometry, and its relations to natural design and space
Peter Pearce and Susan Pearce, Polyhedra Primer, Van
Reinhold, (reprinted by Dale Seymour Publications) 1978.
Introductory presentation of tessellations,
polyhedra, and space fillings, well illustrated with excellent line
drawings (but not
Charles E. Peck, A Taxonomy of Fundamental
Polyhedra and Tessellations, P.O. Box 47186, Wichita, KS, USA
Nicely illustrated 50 page self-published book
between the Platonic, Archimedean, and dual solids.
Mark A. Peterson, "The Geometry of Piero della
Francesca," Mathematical Intelligencer, v. 19, No. 3, pp.
Summary of Piero's original contributions to the
revival of polyhedral geometry.
Anthony Pugh, Polyhedra: A Visual Approach,
U. Cal. pr., 1976, (reprinted by Dale Seymour Publ., 1990; $15.95,
A wonderful, comprehensive survey with many
illustrations, and an inexpensive paperback as well.
Anthony Pugh, An Introduction to Tensegrity,
Introduction to the mathematics and making of
polyhedral tensegrity structures, with detailed instructions for
several dozen models.
Doris Schattschneider, M.C. Escher: Visions of
Symmetry, Freeman, 1990.
Great analysis of Escher's use of symmetry,
including several pages on his use of polyhedra.
Doris Schattschneider and Wallace Walker, M.C.
Escher Kaleidocycles, Ballantine, 1977, (Tarquin reprint, 1982).
Combination of a book and a set of
ready-to-glue-together Platonic solids and rings of tetrahedra,
decorated with Escher graphics.
Marjorie Senechal and George Fleck (editors), Shaping
A Polyhedral Approach, Birkhauser, 1988, (available from the
Association of America; $23.00, phone: 1-800-331-1622).
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
Look for this happy participant on p. 73.)
Abraham Sharp, Geometry Improv'd, London,
Amazing presentation of a great variety of
solids to be found nowhere else, including details on how to cut them
solid blocks of wood.
Lewis Simon, Bennett Arnstein, and Rona Gurkewitz, Modular
Polyhedra, Dover, 1999 (Revised and enlarged from 1989
Construction manual for making dozens of paper
polyhedral constructions from origami components.
J. Skilling, "The Complete Set of Uniform Polyhedra,"
Philosophical Transactions of the Royal Society, Ser. A, 278,
pp. 111-135, 1975.
Mathematical paper describing the computer analysis
that proved the Coxeter et al. enumeration is complete.
J. Skilling, "Uniform Compounds of Uniform
Polyhedra," Mathematical Proceedings of the Cambridge Philosophical
Society, Vol. 79, pp. 447-457, 1976.
Mathematical paper describing the 75 uniform
Anthony Smith, "Stellations of the Triakis
Tetrahedron," Mathematical Gazette, Vol. 49, pp. 135-143, 1965.
Describes and illustrates stellations of the
Anthony Smith, "Some Regular Compounds of
Star-Polyhedra," and (with A.C. Norman) "Computer Drawings of Compounds
of Star Polyhedra," Mathematical Gazette, Vol. 57, pp. 39-46
and 303-306, 1973.
Describes and illustrates the compounds of two and
five great dodecahedra and small stellated dodecahedra.
Anthony Smith, "Uniform Compounds and the Group A4," Proceedings
the Cambridge Philosophical Society, Vol. 75, pp. 115-117,
Same as the above, without construction notes.
A. G. Smith, Cut and Assemble 3-D Geometrical
Shapes: 10 Models in Full Color, Dover, 1986.
Cut-and-glue cardboard book with nets of the
Platonic solids and three of the Kepler-Poinsot solids.
H. Steinhaus, Mathematical Snapshots, Oxford,
3rd ed., 1969, (Dover reprint).
A classic of recreational mathematics which
interesting snippets about many topics including polyhedra.
Ernst Steinitz and Hans Rademacher, Vorlesungen
uber die Theorie der Polyeder, Springer, Berlin, 1934.
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.)
Bonnie Madison Stewart, Adventures Among the
Toroids: a study of quasi-convex, aplanar, tunnelled orientable
polyhedra of positive genus having regular faces with disjoint interiors,
2nd ed. 1980.
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
Alicia Boole Stott, "Geometrical deduction of
semiregular from regular polytopes and space fillings," Verhandelingen
der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste
sectie), Vol. 11, No. 1,
pp. 1-24 plus 3 plates, 1910.
Presents methods of expanding a polyhedron about
faces, etc., to generate related polyhedra. (Summarized in W.W.R. Ball.)
Daud Sutton, Platonic & Archimedean Solids,
Small introduction with very nice line
Jean E. Taylor, "Zonohedra and Generalized
Zonohedra," American Mathematical Monthly, February 1992, pp.
Compares Coxeter's definition of zonohedron
(requiring centrally symmetric faces) with Fedorov's definition
(allowing faces with opposite edges parallel but unequal).
Patrick Taylor, "The Simpler? Polyhedra,"
"Additions to the Uniform Polyhedra," and "The Star and Cross
Polyhedra," 1995, 1999, 2000.
Nicely illustrated self-published booklets which
consider a slight generalization of the usual definition of uniform
polyhedra, in a
series titled "The Complete? Polyhedra."
Jos Tomlow, "Polyhedra, from Pythagoras to Alexander
Graham Bell," in Gabriel, Beyond the Cube, 1997.
History of polyhedra in art.
Alan Tormey and Judith Farr Tormey, "Renaissance
Intarsia: The Art of Geometry," Scientific American, vol. 247,
pp. 136-143, July, 1982.
Illustrates the use of perspective polyhedra in
Renaissance intarsia (wooden inlay panels).
L. Fejes Toth, Regular Figures, McMillan,
Mathematical monograph on polyhedra and related
topics, with nice packet of red/green stereo images.
Borin Van Loon, Geodesic Domes, Tarquin
Describes the geometry of several geodesic dome
methods and includes diagrams which you cut out, score, fold, and glue
to make paper
Kim H. Veltman, with Kenneth D. Keele, Linear
Perspective and the Visual Dimensions of Science and Art,
Analysis of Leonardo's writings and drawings
concerning perspective and polyhedra. Many images of renaissance
Hugo F. Verheyen, Symmetry Orbits,
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.
A. Wachman, M. Burt, and M. Kleinmann, Infinite
Polyhedra, Technion, Haifa, 1974.
Describes and illustrates (with photos of paper
models) roughly 100 infinite uniform polyhedral structures composed of
Eric W. Weisstein, The CRC Concise Encyclopedia
of Mathematics, CRC Press, 1998.
Almost 2000-pages, this encyclopedia is written at
an accessible level and includes many polyhedral topics.
A. F. Wells, Three-dimensional Nets and Polyhedra,
Monograph on infinite polyhedra and space
structures, with a crystallographic perspective.
David Wells, The Penguin Dictionary of Curious
and Interesting Geometry, Penguin, 1991.
Short entries on many geometric topics, including
polyhedral. Superficial but gives references.
Magnus J. Wenninger, Polyhedron Models,
Illustrated directions for constructing paper
119 polyhedra, including all the uniform polyhedra and 19 of the
stellations of the icosahedron. A classic, written by a true philomorph.
Magnus J. Wenninger, Polyhedron Models for the
Classroom, National Council of Teachers of Mathematics, 1966.
A sort of "Readers Digest" version of his later
a short how-to for a range of basic models.
Magnus J. Wenninger, Spherical Models,
Cambridge, 1979, (1999 Dover reprint).
Instructions for constructing spherical models
polyhedral symmetries. Dover version has new additions in an appendix.
Magnus J. Wenninger, Dual Models, Cambridge,
In the same informative how-to spirit as the above
three books, this is the only reference I know which discusses and
duals to all of the uniform polyhedra.
Magnus J. Wenninger and Peter W. Messer, "Patterns on
the Spherical Surface," International Journal of Space Structures,
Vol. 11, pp. 183-192, 1996.
Several spherical paper sculptures (with polyhedral
symmetry) and how they are designed.
Magnus J. Wenninger, "Polyhedra and the Golden
Number," Symmetry, Vol. 1, No. 1, pp. 37-40, 1990.
Shows the "Theosophical" compound of five cubes
(which has octahedral symmetry) and one (slightly distorted) compound
of ten cubes.
Magnus J. Wenninger, "Some Interesting Octahedral
Compounds," Mathematical Gazette, pp. 16-23, Feb., 1968.
Illustrates the compound of three octahedra and a
related compound of four.
Hermann Weyl, Symmetry, Princeton U. Pr.,
Classic treatment of symmetry in its many forms, as
found in art, nature, and mathematics.
Robert Williams, Natural Structure: Toward a Form
Language, Eudaemon Pr., 1972. Reprinted with corrections as The
Foundation of Natural Structure: A Source Book of Design,
An original exploration of design principles using
Makoto Yamaguchi, Kusudama: Ball Origami,
Shufunotomo, Tokyo, 1990.
Clear instructions for making beautiful modular
origami orbs with polyhedral symmetries.
Shukichi Yamana, "An Easily Constructed Dodecahedron
Model," Journal of Chemical Education, Vol 61, pp.
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)."
V.A. Zalgaller, Convex Polyhedra with Regular
Faces, Consultants Bureau, 1969.
A lengthy fastidious mathematical proof that
Johnson's list of solids is complete. Has anyone ever really read this?
Gunter M. Ziegler, Lectures on Polytopes,
Excellent mathematical text on convex polyhedra and
Douglas Zongker and George W. Hart, "Blending
Polyhedra with Overlays," Proceedings of Bridges: Mathematical
Connections in Art,
Music, and Science, 2001, pp. 167-174. (online pdf version)
Novel method of combining two or more given
polyhedra to create interesting new "blended" polyhedra.