Sources page biographical material

6.Q. KNOTTING A STRIP TO MAKE A REGULAR PENTAGON
Urbano d'Aviso. Trattato della Sfera e Pratiche per Uso di Essa. Col modeo di fare la figura celeste, opera cavata dalli manoscritti del. P. Bonaventura Cavalieri. Rome, 1682. ??NYS cited by Lucas (1895) and Fourrey.

Dictionary of Representative Crests. Nihon Seishi Monshō Sōran (A Comprehensive Survey of Names and Crests in Japan), Special issue of Rekishi Dokuhon (Readings in History), Shin Jinbutsu Oraisha, Tokyo, 1989, pp. 271-484. Photocopies of relevant pages kindly sent by Takao Hayashi.

Crests 3504 and 3506 clearly show a strip knotted to make a pentagon. 3507 has two such knots and 3508 has five. I don't know the dates, but most of these crests are several centuries old.

Lucas. RM2, 1883, pp. 202 203.

Tom Tit.

Vol. 2, 1892. L'Étoile à cinq branches, pp. 153-154. = K, no. 5: The pentagon and the five pointed star, pp. 20 21. He adds that folding over the free end and holding the knot up to the light shows the pentagram.

Vol. 3, 1893. Construire d'un coup de poing un hexagone régulier, pp. 159-161. = K, no. 17: To construct a hexagon by finger pressure, pp. 49 51. Pressing an appropriate size Möbius strip flat gives a regular hexagon.

Vol. 3, 1893. Les sept pentagones, pp. 165-166. = K, no. 19: The seven pentagons, pp. 54 55. By tying five pentagons in a strip, one gets a larger pentagon with a pentagonal hole in the middle.

Somerville Gibney. So simple! The hexagon, the enlarged ring, and the handcuffs. The Boy's Own Paper 20 (No. 1012) (4 Jun 1898) 573-574. As in Tom Tit, vol. 3, pp. 159-161.

Lucas. L'Arithmétique Amusante. 1895. Note IV: Section II: Les Jeux de Ruban, Nos. 1 & 2: Le nœud de cravate & Le nœud marin, pp. 220-222. Cites d'Aviso and says he does both the pentagonal and hexagonal knots, but Lucas only shows the pentagonal one.

E. Fourrey. Procédés Originaux de Constructions Géométriques. Vuibert, Paris, 1924. Pp. 113 & 135 139. Cites Lucas and cites d'Aviso as Traité de la Sphère and says he gives the pentagonal and hexagonal knots. Fourrey shows and describes both, also giving the pictures on his title page.

F. V. Morley. A note on knots. AMM 31 (1924) 237-239. Cites Knott's translation of Tom Tit. Says the process generalizes to (2n+3) gons by using n loops. Gets even-gons by using two strips. Discusses using twisted strips.

Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge, 1941); revised ed., Educational Publishers, St. Louis, 1949. Pp. 64-65 gives square (a bit trivial), pentagon, hexagon, heptagon and octagon. Even case need two strips.

Donovan A. Johnson. Paper Folding for the Mathematics Class. NCTM, 1957, pp. 16-17: Polygons constructed by tying paper knots. Shows how to tie square, pentagon, hexagon, heptagon and octagon.

James K. Brunton. Polygonal knots. MG 45 (No. 354) (Dec 1961) 299 302. All regular n gons, n > 4, can be obtained, except n = 6 which needs two strips. Discusses which can be made without central holes.

Marius Cleyet-Michaud. Le Nombre d'Or. Presses Universitaires de France, Paris, 1973. On pp. 47-48, he calls this the 'golden knot' (Le "nœud doré") and describes how to make it.