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By George D. Birkhoff, Ralph Beatley

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All first order fundamental even edge flexagons can be flexed by using a pinch flex. This is their characteristic flex. 3 Flexagon figures for first order fundamental pentagon even edge flexagons. (a) S〈5, 5〉. 5 Flexagon figures for first order fundamental heptagon even edge flexagons. (a) S〈7, 7〉. (b) S〈7, 7/2〉. 6 Flexagon figures for first order fundamental octagon even edge flexagons. (a) S〈8, 8〉 (Les Pook, Flexagons inside out, 2003, © Cambridge University Press 2003, reprinted with permission) (b) S〈8, 8/3〉 position, pinch pairs of pats together to reach an intermediate position.

A flex sequence that transforms one form into another is a transformation between flexagons. 2). 3. 13. As assembled, the flexagon is in intermediate position 1. This is, in appearance, an equilateral triangle edge pair. From here it can be opened up into either principal main position 1(2) or principal main position 1(3). These are slant regular even edge rings of four equilateral triangles. 1) because the principal 3-cycle shown in the intermediate position map (Fig. 7) cannot be traversed without disconnecting a hinge, refolding the flexagon, and reconnecting the hinge.

In both main and intermediate positions, a first order fundamental even edge flexagon can be divided into S identical sectors; so its net also has S identical sectors. 1). The number of leaves in a sector is the same as the number of edges on the constituent polygons. All first order fundamental even edge flexagons are simple twisted bands. The torsion, T, of a flexagon is the number of half twists (180° twists) around the band. If T is odd then a band has only have one side and it is a Möbius band (Gardner 1965, 1978, 2008; Pedersen and Pedersen 1973).

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