Hexaflexagons
Old collection of hexaflexagons. All up to order 10 and some more.
Hexaflexagons are paper hexagons folded from strips of equilateral
triangles, that show a number of different faces when folded by opening
them at the center.
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. The program runs in Windows (2019). There is also an
old DOS version (1999) here.
Example: The Tuckerman traverses of the last hexaflexagon of order 10
(#82), and the strip that produces it, front side and back side:

#82 of order 10.
Strip, front side and back side (with
horizontal flip).
There are 82 possible hexaflexagons of order 10. This is the last one. The
drawings of the strips generated by the program can be processed in any
drawing program and printed. With some adjustment the images of the front
and back sides, as the ones above, can be printed aligned at both sides of a
paper sheet. The lines can be deleted from one side to simplify the
alignment (the program can generate just the numbers too). In this example
the strip is planar, but some have overlaps, complicating the printing and
requiring some assembly. The program can also generate just 1/3 of the
strip, to be replicated 3 times, what avoids overlaps in many cases, but not
all. The parts can be joined with glued paper tabs, that may also be used to
close the folded flexagon. Avoid tape, as it will decay in little
time. The assembly is done by first folding back and forth all the joints,
and then folding together adjacent identical numbers, including pairs that
appear in the folded strip. There are always three pairs of each. Paper
clips (three) can be used to hold the folded strip during the folding. It's
not easy to make hexaflexagons of high order, greater than 12, because
the faces accumulate in several positions (the blue faces) deforming the
flexagon and making it difficult to fold. After the assembly, the outer
sides of single sheet triangle faces (they appear in the blue corners in the
map) can be trimmed with scissors, turning the folding easier.
Example: The last hexaflexagon of order 11, #228, and the 1/3 strip that
generates it. In this way a larger flexagon, easier to fold, can be
generated. The back face is represented by just the numbers, to simplify
alignment with both images printed in both sides of a paper sheet. A tab was
added to join the strips with glue. Three copies are needed, with the
central copy mounted with the back side up. In the final flexagon, some
faces appear with black numbers (1, 5, 9) others with red numbers (4, 7, 11)
and some mixed (2, 3, 6, 8, 10).
#228 of order 11.
1/3 strip, to be replicated 3 times.
Example: The dodecahexaflexagon, with 12 faces, has 733
possibilities, and admits a straight strip with 36 triangles. This strip
can be folded in 4 different ways, producing 4 different maps, shown
below. A template for the four possibilities was assembled from the
drawings made by the program. The two sides shall be printed well centered
in both sides of the same sheet, in three copies to be joined without
inversions. A portrait print requires no adjustment in most printers. A
landscape printing, for larger size, requires centering in the longer
direction of the sheet for correct alignment. The most well known version
is #632, that is folded twice as an hexahexaflexagon. It is the only one
of the four where the numbers in all the faces appear with the same color.
#92 and #412.
#595 and # 632.
Front faces, for the dodecahexaflexagons
#92, #412, #595, and #632, 1/3 strips.
Back faces of the 1/3 strips, #92, #412,
#595, and #632, (with horizontal flip).
All assembled and tested.
See my links about other subjects, including
hexaflexagons
Established: 31/01/1999
Last update: 06/08/2019
Antonio Carlos M. de Queiroz
Lamento informar que o Prof. Antonio Carlos Moreirão de Queiroz
faleceu há algum tempo.
Sei que esta página é visitada constantemente. Assim, gostaria de
saber se temos algum visitante (interessado) que seja da UFRJ. Se for,
por favor, envie um e-mail para watanabe@coe.ufrj.br.
Comento que é impressionante ver o que Moreirão foi capaz de fazer.
Ele não só projetou os circuitos, mas também fez todo o trabalho de
marceneiro (melhor que muitos que já vi e eram profissionais).
Segundo Moreirão contou em uma palestra, ele só levou choque uma vez.
Sem querer encostou o dedo médio em um capacitor com alta tensão que
se descarregou através do dedo. A corrente ao passar por uma das
articulações a danificou e doía sempre que dobrava esse dedo. Mas,
segundo ele, já tinha acostumado.
E. Watanabe (ELEPOT)