Sources page biographical material

6.X. ROTATING RINGS OF POLYHEDRA
Generally, these have edge to edge joints. 'Jacob's ladder' joints are used by Engel -- see 11.L for other forms of this joint.
I am told these may appear in Fedorov (??NYS).

Max Brückner. Vielecke und Vielfläche. Teubner, Leipzig, 1900. Section 162, pp. 215 216 and Tafel VIII, fig. 4. Describes rings of 2n tetrahedra joined edge to edge, called stephanoids of the second order. The figure shows the case n = 5.

Paul Schatz. UK Patent 406,680 -- Improvements in or relating to Boxes or Containers. Convention date (Germany): 10 Dec 1931; application date (in UK): 19 Jul 1932; accepted: 19 Feb 1934. 6pp + 6pp diagrams. Six and four piece rings of prisms which fold into a box.

Paul Schatz. UK Patent 411,125 -- Improvements in Linkwork comprising Jointed Rods or the like. Convention Date (Germany): 31 Aug 1931; application Date (in UK): 31 Aug 1932; accepted: 31 May 1934. 3p + 6pp diagrams. Rotating rings of six tetrahedra and linkwork versions of the same idea, similar to Flowerday's Hexyflex.

Ralph M. Stalker. US Patent 1,997,022 -- Advertising Medium or Toy. Applied: 27 Apr 1933; patented: 9 Apr 1935. 3pp + 2pp diagrams. "... a plurality of tetrahedron members or bodies flexibly connected together." Shows six tetrahedra in a ring and an unfolded pattern for such objects. Shows a linear form with 14 tetrahedra of decreasing sizes.

Sidney Melmore. A single sided doubly collapsible tessellation. MG 31 (No. 294) (1947) 106. Forms a Möbius strip of three triangles and three rhombi, which is basically a flexagon (cf 6.D). He sees it has two distinct forms, but doesn't see the flexing property!! He describes how to extend these hexagons into a tessellation which has some resemblance to other items in this section.

Alexander M. Shemet. US Patent 2,688,820 -- Changeable Display Amusement Device. Applied: 25 Jul 1950; patented: 14 Sep 1954. 2pp + 2pp diagrams. Basically a rotating ring of six tetrahedra, but says 'at least six'. Gives an unfolded version or net for making it and a mechanism for flexing it continually. Cites Stalker.

Wallace G. Walker invented his "IsoAxis" ® in 1958 while a student at Cranbrook Academy of Art, Michigan. This is approximately a ring of ten tetrahedra. He obtained a US Patent for it in 1967 -- see below. In 1973(?) he sent an example to Doris Schattschneider who soon realised that the basic idea was a ring of tetrahedra and that Escher tessellations could be adapted to it. They developed the idea into "M. C. Escher Kaleidocycles", published by Ballantine in 1977 and reprinted several times since.

Douglas Engel. Flexahedrons. RMM 11 (Oct 1962) 3 5. These have 'Jacob's ladder' hinges, not edge to edge hinges. He says he invented these in Fall, 1961. He formed rings of 4, 6, 7, 8 tetrahedra and used a diagonal joining to make rings of 4 and 6 cubes.

Wallace G. Walker. US Patent 3,302,321 -- Foldable Structure. Filed: 16 Aug 1963; issued: 7 Feb 1967. 2pp + 6pp diagrams.

Joseph S. Madachy. Mathematics on Vacation. Op. cit. in 5.O, (1966), 1979. Solid Flexagons, pp. 81 84. Based on Engel, but only gives the ring of 6 tetrahedra.

D. Engel. Flexing rings of regular tetrahedra. Pentagon 26 (Spring 1967) 106 108. ??NYS -- cited in Schaaf II 89 -- write Engel.

Paul Bethell. More Mathematical Puzzles. Encyclopædia Britannica International, London, 1967. The magic ring, pp. 12-13. Gives diagram for a ten-tetrahedra ring, all tetrahedra being regular.

Jan Slothouber & William Graatsma. Cubics. Octopus Press, Deventer, Holland, 1970. ??NYS. Presents versions of the flexing cubes and the 'Shinsei Mystery'. [Jan de Geus has sent a photocopy of some of this but it does not cover this topic.]

Jan Slothouber. Flexicubes -- reversible cubic shapes. JRM 6 (1973) 39 46. As above.

Frederick George Flowerday. US Patent 3,916,559 -- Vortex Linkages. Filed: 12 Aug 1974 (23 Aug 1973 in UK); issued: 4 Nov 1975. Abstract + 2pp + 3pp diagrams. Mostly shows his Hexyflex, essentially a six piece ring of tetrahedra, but with just four edges of each tetrahedron present. He also shows his Octyflex which has eight pieces. Text refers to any even number  6.

Naoki Yoshimoto. Two stars in a cube (= Shinsei Mystery). Described in Japanese in: Itsuo Sakane; A Museum of Fun; Asahi Shimbun, Tokyo, 1977, pp. 208 210. Shown and pictured as Exhibit V 1 with date 1972 in: The Expanding Visual World -- A Museum of Fun; Exhibition Catalogue, Asahi Shimbun, Tokyo, 1979, pp. 102 & 170 171. (In Japanese). ??get translated??

Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. Pp. 63-66. Describes Walkers IsoAxis and rotating rings of six and eight tetrahedra.