# Polyomino and Polynar Tetrads

A *tetrad* is a plane figure made of four congruent shapes,
joined so that each shares a boundary with each.
Here I show various minimal tetrads for polyominoes and polynars.
## Polyominoes

The smallest polyomino tetrads are made from octominoes:

The fifth tetrad was reported by Olexandr Ravsky in 2005.

### Symmetric Tiles

The smallest tetrad for a polyomino with mirror symmetry uses 13-ominoes:

The smallest tetrad for a polyomino with birotary symmetry also uses 13-ominoes:

The smallest tetrads for polyominoes with birotary symmetry about an edge
use 14-ominoes:

The smallest tetrads for polyominoes with mirror symmetry about an
edge use 18-ominoes:

The smallest tetrads for polyominoes with birotary symmetry
about a vertex also use 18-ominoes:

Juris Čerņenoks found the smallest tetrads for polyominoes
with diagonal symmetry, which use 19-ominoes:

### Restricted Motion

These octominoes form tetrads without being reflected:

The smallest polyominoes that form tetrads without 90° rotation
are 13-ominoes:

### Holeless

The smallest holeless polyomino tetrad, discovered by
Walter Trump, uses 11-ominoes:

The smallest known holeless tetrad for a symmetric polyomino
was found independently by Frank Rubin and Karl Scherer.
It uses 34-ominoes:

## Polynars

A *polynar* is a plane figure formed by joining equal squares
along edges or half edges.
The smallest polynar tetrads use pentanars:

*Last revised 2020-01-14.*

Back to Polyform Tetrads
< Polyform Curiosities

Col. George Sicherman
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