## 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: 12/04/2011

Antonio Carlos M. de Queiroz