# Polygem Oddities

A *polygem*
is a plane polyform whose cells are gems—figures formed by
dividing a regular hexagon into thirds using lines from the center to
the midpoints of the sides.
A gem is one of the two dikites.
You can see more about polygems at
Abaroth's Polygem Page.
An *oddity*
is a figure with binary symmetry formed by an odd number of copies of
a polyform.
Here are the minimal known oddities for polygems
of orders 1–4.
The black trigem has no solution; the gray trigems and most of
the tetragems are unsolved.
Please write if you find a smaller solution or solve an unsolved case.

See also Galvagni Figures for Polygems.

## Monogem

## Digems

## Trigems

## Tetragems

### Unsolved or Impossible

David Smith, Joseph Samuel Myers,
Craig S. Kaplan, and Chaim Goodman-Strauss
proved in 2023 that the third tetragem in the
top row can tile the plane,
and cannot tile the plane periodically.
Thanks to Ed Pegg for letting me know of this.

Last revised 2023-03-23.

Back to Polyform Oddities
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Polyform Curiosities

Col. George Sicherman
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