Just under a year ago we had an exchange of messages in which JHC outlined his ideas and constructions for making tetraflexagons, with the proviso that they were all applicable to any kind of polygon. It is instructive to follow out his recipe for four-square flexagons, being about as short as they come. The first requirement is a frieze, generated from an initial polygon by reflecting the last polygon in one of its edges; JHC said N, S, E, W, but I prefer +, ++, - or numbers if there are lots of edges. The best way to follow this out is to orient the polygon, and advance so many edges from the last on to make the new reflection. The next requirement is a permutation, which says where a given polygon on the frieze ends up on the stack which is gotten by folding the frieze about its connecting edges. Since the polygon gets turned over with each new fold, it is convenient to enumerate the faces in a 2xn array, distinguishing top and bottom. The image, that is the corresponding permutation, of the ith polygon is written in the ith cell, alternating top and bottom row. That number plus 1 is written in the paired cell. JHC didn't say this; it is the Tukey triangle [n-gon] strip, and the matrix is mentioned in some of the flexagon articles. Actually it has three rows, because the turn in passing from one polygon to the next is noted in the extra row; that is where +'s, -'s, or whatever, go. A useful picture is to draw a polygon with enough edges for the length of the frieze and to draw labelled diagonals to show the permutation. This isn't the Tuckerman Tree, but it is related. One difference is that the labels are the turns. As JHC remarked, a basic requirement is that the sum of the turns be 360 degrees so that the final edge is parallel to the starting edge. That ``adding 1'' rule needs justifying, but presumably comes into play later on. When the assembled flexagon is opened up there are two consecutive members of the stack being separated, and they had better have the same colors to give a uniform appearance to the laid out flexagon. Anyway, if we accept this preparation, there are essentially two four-cycle permutations for a four square frieze: (1 2 3 4) and (1 2 4 3), the reasoning being that we can start where we want and go off in the direction we want. Turning sequences which add up to zero (ie. full circle) are (+ + + +). (+ + - -), (+ ++ -- -), (+ ++ -- -). There are others, such as {++ ++ ++ ++), but as JHC says, ``there are some additional restrictions.'' Some eliminate silly combinations, others eliminate impossible combinations such as hard knots. These combinations give sign sequence ---------------------------------------------------------- permutation | + + + + | + + - - | + ++ -- - | + ++ - -- | ---------------------------------------------------------- (1 2 3 4) | Natural | flaps |flap-scroll| Good f'gon| ---------------------------------------------------------- (1 2 4 3) |tubulate |tubulating| scrolling | crossed | ---------------------------------------------------------- The point here is that there are lots of things which can either be called flexagons or near flexagons. What I would call natural flexagons are only a part of the panorama; even so there is a second quite reasonable full flexagon, and four more objects with reasonable behavior. The scroll, in particular, admits of generalization and was more or less known before flexagons came around. So I guess we can see that many things can be promoted in the flexagon or origami books as ``flexagons.'' In a way, they are. - hvm ------------------------------------------------- Obtén tu correo en www.correo.unam.mx UNAMonos Comunicándonos