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Symmetries in the Stomachion tilings

There is a 64-fold symmetry in the solution space of Stomachion tilings.

That is how we can reduce the 17152 (=64 x 268) different tilings to merely 268.

**
An obvious factor of 4**

Two of the pieces with purple dots have the same shape.
So are the two with yellow dots.
By interchanging the pieces with same colored dots,
This accounts for a
factor of 4.

**
Another factor of 2
**
is due to a right triangle (having sides 3 and 6) that is the shape of
an original Stomachion piece (with an orange dot)
but it can also be formed
by joining the two original tiles with the green dots.

**
The dihedral group D**_{4} of order
8,
illustrated below.