Sources page biographical material

6.AA. NETS OF POLYHEDRA
Albrecht Dürer. Underweysung der messung mit dem zirckel uň [NOTE: ň denotes an n with an overbar.] richtscheyt, in Linien ebnen unnd gantzen corporen. Nürnberg, 1525, revised 1538. Facsimile of the 1525 edition by Verlag Dr. Alfons Uhl, Nördlingen, 1983. German facsimile with English translation of the 1525 edition, with notes about the 1538 edition: The Painter's Manual; trans. by Walter L. Strauss; Abaris Books, NY, 1977. Figures 29 43 (erroneously printed 34) (pp. 316-347 in The Painter's Manual, Dürer's 1525 ff. M-iii-v - N-v-r) show nets and pictures of the regular polyhedra, an approximate sphere (16 sectors by 8 zones), truncated tetrahedron, truncated cube, cubo-octahedron, truncated octahedron, rhombi cubo-octahedron, snub cube, great rhombi-cubo-octahedron, truncated cubo octahedron (having a pattern of four triangles replacing each triangle of the cubo octahedron -- not an Archimedean solid) and an elongated hexagonal bipyramid (not even regular faced). (See 6.AT.3 for more details.) (Panofsky's biography of Dürer asserts that Dürer invented the concept of a net -- this is excerpted in The World of Mathematics I 618 619.) In the revised version of 1538, figure 43 is replaced by the icosi-dodecahedron and great rhombi-cubo-octahedron (figures 43 & 43a, pp. 414 419 of The Painter's Manual) to make 9 of the Archimedean polyhedra.

Albrecht Dürer. Elementorum Geometricorum (?) -- the copy of this that I saw at the Turner Collection, Keele, has the title page missing, but Elementorum Geometricorum is the heading of the first text page and appears to be the book's title. This is a Latin translation of Unterweysung der Messung .... Christianus Wechelus, Paris, 1532. This has the same figures as the 1525 edition, but also has page numbers. Liber quartus, fig. 29-43, pp. 145-158 shows the same material as in the 1525 edition.

Cardan. De Rerum Varietate. 1557, ??NYS = Opera Omnia, vol. III, pp. 246-247. Liber XIII. Corpora, qua regularia diei solent, quomodo in plano formentur. Shows nets of the regular solids, except the two halves of the dodecahedron have been separated to fit into one column of the text.

Barbaro, Daniele. La Practica della Perspectiva. Camillo & Rutilio Borgominieri, Venice, (1569); facsimile by Arnaldo Forni, 1980, HB. [The facsimile's TP doesn't have the publication details, but they are given in the colophon. Various catalogues say there are several versions with dates on the TP and colophon varying independently between 1568 and 1569. A version has both dates being 1568, so this is presumed to be the first appearance. Another version has an undated title in an elaborate border and this facsimile must be from that version.] Pp. 45-104 give nets and drawings of the regular polyhedra and 11 of the 13 Archimedean polyhedra -- he omits the two snub solids.

E. Welper. Elementa geometrica, in usum geometriae studiosorum ex variis Authoribus collecta. J. Reppius, Strassburg, 1620. ??NYS -- cited, with an illustration of the nets of the octahedron, icosahedron and dodecahedron, in Lange & Springer Katalog 163 -- Mathematik & Informatik, Oct 1994, item 1350 & illustration on back cover, but the entry gives Trassburg.

Athanasius Kircher. Ars Magna, Lucis et Umbrae. Rome, 1646. ??NX. Has net of a rhombi-cuboctahedron.

Pike. Arithmetic. 1788. Pp. 458-459. "As the figures of some of these bodies would give but a confused idea of them, I have omitted them; but the following figures, cut out in pasteboard, and the lines cut half through, will fold up into the several bodies." Gives the regular polyhedra.

Dudeney. MP. 1926. Prob. 146: The cardboard box, pp. 58 & 149 (= 536, prob. 316, pp. 109 & 310). All 11 nets of a cube.

Perelman. FMP. c1935? To develop a cube, pp. 179 & 182 183. Asserts there are 10 nets and draws them, but two "can be turned upside down and this will add two more ...." One shape is missing. Of the two marked as reversible, one is symmetric, hence equal to its reverse, but the other isn't.

C. Hope. The nets of the regular star faced and star pointed polyhedra. MG 35 (1951) 8 11. Rather technical.

H. Steinhaus. One Hundred Problems in Elementary Mathematics. (As: Sto Zadań, PWN -- Polish Scientific Publishers, Warsaw, 1958.) Pergamon Press, 1963. With a Foreword by M. Gardner; Basic Books, NY, 1964. Problem 34: Diagrams of the cube, pp. 20 & 95 96. (Gives all 11 nets.) Gardner (pp. 5 6) refers to Dudeney and suggests the four dimensional version of the problem should be easy.

M. Gardner. SA (Nov 1966) c= Carnival, pp. 41 54. Discusses the nets of the cube and the Answers show all 11 of them. He asks what shapes these 11 hexominoes will form -- they cannot form any rectangles. He poses the four dimensional problem; the Addendum says he got several answers, no two agreeing.

Charles J. Cooke. Nets of the regular polyhedra. MTg 40 (Aut 1967) 48 52. Erroneously finds 13 nets of the octahedron.

Joyce E. Harris. Nets of the regular polyhedra. MTg 41 (Winter 1967) 29. Corrects Cooke's number to 11.

A. Sanders & D. V. Smith. Nets of the octahedron and the cube. MTg 42 (Spring 1968) 60 63. Finds 11 nets for the octahedron and shows a duality with the cube.

Peter Turney. Unfolding the tesseract. JRM 17 (1984 85) 1 16. Finds 261 nets of the 4 cube. (I don't believe this has ever been confirmed.)

Peter Light & David Singmaster. The nets of the regular polyhedra. Presented at New York Acad. Sci. Graph Theory Day X, 213 Nov 1985. In Notes from New York Graph Theory Day X, 23 Nov 1985; ed. by J. W. Kennedy & L. V. Quintas; New York Acad. Sci., 1986, p. 26. Based on Light's BSc project in 1984-1984 under my supervision. Shows there are 43,380 nets for the dodecahedron and icosahedron. I may organize this into a paper, but several others have since verified the result.