Regular Polytopes

H. S. M. Coxeter's standard treatise on regular polytopes. A polytope is "a geometrical figure bounded by portions of lines, planes, or hyperplanes; e.g., in two dimensions it is a polygon, in three a polyhedron." The book builds dimensional analogy systematically — from regular polygons and the five Platonic solids, through symmetry/rotation groups and the kaleidoscope, to the regular polytopes and star-polytopes of four and higher dimensions. The cited Dover edition (1973) reprints the 1963 Macmillan second edition with a new preface.

Overview and editions

The first edition appeared in 1948; the work "grew out of an essay on 'Dimensional Analogy,' begun in February 1923," and represents, in Coxeter's words, "the fulfilment of 24 years' work." The Dover edition (1973) is an unabridged, corrected republication of the 1963 second edition and adds a third-edition preface noting "more than twenty small improvements." Coxeter declined to modernize terminology (keeping "congruent transformation" rather than "isometry") and pointed to the field's continued vitality through works by Fejes Tóth, Grünbaum, and Wenninger.

Coxeter positions the book as "a sequel to Euclid's Elements": the foundations were "laid by the Greeks over two thousand years ago," but the more elaborate developments (roughly Chapter V onward) are less than a century old, revived partly by the discovery that polyhedra occur in nature as crystals. He frames higher-dimensional figures with characteristic lyricism — in attempting to comprehend four-dimensional polytopes "we seem to peep through a chink in the wall of our physical limitations, into a new world of dazzling beauty."

Structure

The treatise runs fourteen chapters plus an epilogue, symbol tables, bibliography, and index. The first six chapters use only "ordinary solid geometry"; the remainder ascends into higher dimensions.

• I. Polygons and Polyhedra — regular p-gons (denoted {p}), Euclid's proof that there are at most five regular solids, polyhedra as maps and configurations, a topological route to Euler's formula.
• II. Regular and Quasi-Regular Solids
• III. Rotation Groups — an introduction to group theory via symmetry.
• IV. Tessellations and Honeycombs
• V. The Kaleidoscope — reflection groups.
• VI. Star-Polyhedra
• VII. Ordinary Polytopes in Higher Space — the rediscovery of Schläfli's regular polytopes.
• VIII. Truncation
• IX. Poincaré's Proof of Euler's Formula
• X. Forms, Vectors, and Coordinates
• XI. The Generalized Kaleidoscope — (which Coxeter warns "is likely to be found harder than the subsequent chapters").
• XII. The Generalized Petrie Polygon
• XIII. Sections and Projections
• XIV. Star-Polytopes — Hess's star-polytopes; followed by the Epilogue.

Core ideas

Notation and method. Coxeter introduces the Schläfli symbol ({p}, {p,q}, {p,q,r}, ...) for regular figures, and considers his "own best contribution" to be the invention of the "graphical" notation (the Coxeter diagram, §5·6) which "facilitates the enumeration of groups generated by reflections," of the polytopes derived from them by Wythoff's construction, and of their elements. He freely mixes synthetic and coordinate methods, choosing whichever clarifies a given chapter.

Why study regular figures. Coxeter notes a law of symmetry (4·32) prohibiting the inorganic occurrence of any pentagonal figure such as the regular dodecahedron, so "the chief reason for studying regular polyhedra is still the same as in the time of the Pythagoreans, namely, that their symmetrical shapes appeal to one's artistic sense." He cites Lobachevsky that no branch of mathematics, however abstract, may not someday be applied to the real world — a prophecy borne out (the third-edition preface adds that icosahedral symmetry appears in virus macromolecules and in the boron B12 molecule).

Internationalism. Coxeter observes that the bibliography spans German, British, American, French, Dutch, Swiss, and other contributors, taking the history of polytope theory as "an instance of the essential unity of our western civilization, and the consequent absurdity of international strife."

Significance

Regular Polytopes is the canonical modern treatment of its subject and the technical foundation for the symmetry mathematics relevant to Buckminster Fuller's synergetics and geodesic structures. Fuller dedicated Synergetics to Coxeter, and the icosahedral and Platonic symmetries this book systematizes underpin the geometry of geodesic domes and the close-packing transformations (the "jitterbug") associated with Fuller's work.

See Also

• The Man Who Saved Geometry — Siobhan Roberts's biography of Coxeter, the human and historical counterpart to this treatise
• Alicia Boole Stott (Alicia Boole Stott) — pioneer who named the "polytope" and modeled the regular 4-polytopes
• George W. Hart (George W. Hart) — contemporary sculptor-geometer of these polyhedra
• Martin Gardner (Martin Gardner) — popularizer of these higher-dimensional figures for a broad audience
• Other Fuller-corpus topics in this wiki
• H. S. M. Coxeter (H. S. M. Coxeter) — author of this treatise
• Johannes Kepler (Johannes Kepler) — early theorist of the polyhedral geometry this treatise formalizes
• Euclid (Euclid) — systematizer of the classical geometry these polytopes extend
• Plato (Plato) — the regular ('Platonic') solids in their modern treatment

Sources

• regular_polytopes/ — book project directory (repo-local source tree)
• regular_polytopes/index.md — project index (full extracted text)