Sources page biographical material

6.D. FLEXAGONS
These were discovered by Arthur H. Stone, an English graduate student at Princeton in 1939. American paper was a bit wider than English and would not fit into his notebooks, so he trimmed the edge off and had a pile of long paper strips which he played with and discovered the basic flexagon. Fellow graduate students Richard P. Feynman, Bryant Tuckerman and John W. Tukey joined in the investigation and developed a considerable theory. One of their fathers was a patent attorney and they planned to patent the idea and began to draw up an application, but the exigencies of the 1940s led to its being put aside, though knowledge of it spread as mathematical folklore. E.g. Tuckerman's father, Louis B. Tuckerman, lectured on it at the Westinghouse Science Talent Search in the mid 1950s.

S&B, pp. 148 149, show several versions. Most square versions (tetraflexagons or magic books) don't fold very far and are really just extended versions of the Jacob's Ladder -- see 11.L

Martin Gardner. Cherchez la Femme [magic trick]. Montandon Magic Co., Tulsa, Okla., 1946. Reproduced in: Martin Gardner Presents; Richard Kaufman and Alan Greenberg, 1993, pp. 361-363. [In: Martin Gardner Presents, p. 404, this is attributed to Gardner, but Gardner told me that Roger Montandon had the copyright -- ?? I have learned a little more about Gardner's early life -- he supported himself by inventing and selling magic tricks about this time, so it may be that Gardner devised the idea and sold it to Montandon.]. A hexatetraflexagon.

"Willane". Willane's Wizardry. Academy of Recorded Crafts, Arts and Sciences, Croydon, 1947. A trick book, pp. 42-43. Same hexatetraflexagon.

Sidney Melmore. A single sided doubly collapsible tessellation. MG 31 (No. 294) (1947) 106. Forms a Möbius strip of three triangles and three rhombi. He sees it has two distinct forms, but doesn't see the flexing property!!

Margaret Joseph. Hexahexaflexagrams. MTr 44 (Apr 1951) 247 248. No history.

William R. Ransom. A six sided hexagon. SSM 52 (1952) 94. Shows how to number the 6 faces. No history.

F. G. Maunsell. Note 2449: The flexagon and the hexahexaflexagram. MG 38 (No. 325) (Sep 1954) 213 214. States that Joseph is first article in the field and that this is first description of the flexagon. Gives inventors' names, but with Tulsey for Tukey.

R. E. Rogers & Leonard L. D'Andrea. US Patent 2,883,195 -- Changeable Amusement Devices and the Like. Applied: 11 Feb 1955; patented: 21 Apr 1959. 2pp + 1p correction + 2pp diagrams. Clearly shows the 9 and 18 triangle cases and notes that one can trim the triangles into hexagons so the resulting object looks like six small hexagons in a ring.

M. Gardner. Hexa hexa flexagon and Cherchez la femme. Hugard's MAGIC Monthly 13:9 (Feb 1956) 391. Reproduced in his: Encyclopedia of Impromptu Magic; Magic Inc., Chicago, 1978, pp. 439-442. Describes hexahexa and the hexatetra of Gardner/Montandon & Willane.

M. Gardner. SA (Dec 1956) = 1st Book, chap. 1. His first article in SA!!

Joan Crampin. Note 2672: On note 2449. MG 41 (No. 335) (Feb 1957) 55 56. Extends to a general case having 9n triangles of 3n colours.

C. O. Oakley & R. J. Wisner. Flexagons. AMM 64:3 (Mar 1957) 143 154.

Donovan A. Johnson. Paper Folding for the Mathematics Class. NCTM, 1957, section 61, pp. 24-25: Hexaflexagons. Describes the simplest case, citing Joseph.

Roger F. Wheeler. The flexagon family. MG 42 (No. 339) (Feb 1958) 1 6. Improved methods of folding and colouring.

M. Gardner. SA (May 1958) = 2nd Book, chap. 2. Tetraflexagons and flexatube.

P. B. Chapman. Square flexagons. MG 45 (1961) 192 194. Tetraflexagons.

Anthony S. Conrad & Daniel K. Hartline. Flexagons. TR 62-11, RIAS, (7212 Bellona Avenue, Baltimore 12, Maryland,) 1962, 376pp. This began as a Science Fair project in 1956 and was then expanded into a long report. The authors were students of Harold V. McIntosh who kindly sent me one of the remaining copies in 1996. They discover how to make any chain of polygons into a flexagon, provided certain relations among angles are satisfied. The bibliography includes almost all the preceding items and adds the references to the Rogers & D'Andrea patent, some other patents (??NYS) and a number of ephemeral items: Conrad produced an earlier RIAS report, TR 60-24, in 1960; Allan Phillips wrote a mimeographed paper on hexaflexagons; McIntosh wrote an unpublished paper on flexagons; Mike Schlesinger wrote an unpublished paper on Tuckerman tree theory.

Sidney H. Scott. How to construct hexaflexagons. RMM 12 (Dec 1962) 43 49.

William R. Ransom. Protean shapes with flexagons. RMM 13 (Feb 1963) 35 37. Describes 3 D shapes that can be formed. c= Madachy, below.

Robert Harbin. Party Lines. Op. cit. in 5.B.1. 1963. The magic book, pp. 124-125. As in Gardner's Cherchez la Femme and Willane.

Pamela Liebeck. The construction of flexagons. MG 48 (No. 366) (Dec 1964) 397 402.

Joseph S. Madachy. Mathematics on Vacation. Op. cit. in 5.O, (1966), 1979. Other flexagon diversions, pp. 76 81. Describes 3 D shapes that one can form. Based on Ransom, RMM 13.

Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. Pp. 66-75. Describes various tetra- and hexa-flexagons.

Douglas A. Engel. Hexaflexagon + HFG = slipagon! JRM 25:3 (1993) 161-166. Describes his slipagons, which are linked flexagons.

Robert E. Neale (154 Prospect Parkway, Burlington, Vermont, 05401, USA). Self-designing tetraflexagons. 12pp document received in 1996 describing several ways of making tetraflexagons without having to tape or paste. He starts with a creased square sheet, then makes some internal tears or cuts and then folds things through to miraculously obtain a flexagon! A slightly rearranged version appeared in: Elwyn R. Berlekamp & Tom Rodgers, eds.; The Mathemagician and Pied Puzzler A Collection in Tribute to Martin Gardner; A. K. Peters, Natick, Massachusetts, 1999, pp. 117-126.

Jose R. Matos. US Patent 5,735,520 -- Fold-Through Picture Puzzle. Applied: 7 Feb 1997; patented: 7 Apr 1998. Front page + 6pp diagrams + 13pp text. Robert Byrnes sent an example of the puzzle. This is a square in thin plastic, 100mm on an edge. Imagine a 2 x 2 array of squares with their diagonals drawn. Fold along all the diagonals and between the squares. This gives an array of 16 isosceles right triangles. Now cut from the centres of the four squares to the centre of the whole array. This produces an X cut in the middle. This object can now be folded through itself in various ways to produce a double thickness square of half the area with various logos. The example is 100mm along the edge of the large square and has four logos advertising Beanoland (at Chessington, 3 versions) and Strip Cheese. The patent is assigned to Lulirama International, but Byrnes says it has not been a commercial success as it is too complicated. The patent cites 19 earlier patents, back to 1881, and discusses the history of such puzzles. It also says the puzzle can form three dimensional objects.