## Hexaflexagons

Hexaflexagons are paper hexagons folded from strips of
equilateral triangles, that show a number of different faces when
folded. See my links area for more detailed discussion of their
basic characteristics.

Here is a scanned copy of a "catalog" that I once
compiled (~1973) of all possible hexaflexagons up to order 10.

The pictures show the forms of the strips and the several
Tuckerman traverses (TT) that can be obtained from each strip, as
diagrams of interconnected triangles that describe how the faces
appear in the hexaflexagon. The numbers adjacent to most of the
TTs indicate the folding sequence in the strip that produce them.
The strip is to be folded at the triangle joints indicated by the
numbers, starting from the arrow, or from the left, always in the
same direction, with the sequence repeated 3 times at equally
spaced points of the strip. When there is more than one line, the
initial folding results in a straight strip, that is to be folded
as the strip of the hexaflexagons of order 6 or 9, starting from
the listed joints. Once obtained a strip with 9 triangles and a
tab, it is folded as the trihexaflexagon.

Note that the number of possible hexaflexagons of order N is
precisely the number of possible TTs, or the number of ways where
N-2 triangles can be connected by their corners, with only two
triangles per connection and without forming loops.

###### The background image shows the TTs of the 12 hexaflexagons of
order 8

I wrote a program named HexaFind
(rewriting an Algol program that I wrote by 1977) that finds all
the possible TTs for given orders of hexaflexagons. In the
present version it can also show the face numbers corresponding
to the nodes, and, using the "reflectocloning"
method developed by David
King, show the strips that when folded result in
hexaflexagons with those state diagrams.

Example: The Tuckerman traverse of the last hexaflexagon of order
12, and the strip that produces it (to be replicated 3 times).
The upper numbers correspond to the frontal face, and the ones
below to the back face.

See my links about hexaflexagons
and other subjects

Established: 31/01/1999

Last update: 23/02/2004

Antonio Carlos M. de Queiroz