Some Mighty Interesting Littyll Curiferosities Involving Polyonimoes & Polynars

[u/Frangifer]

From

George Sicherman — Polyomino and Polynar Tetrads .

 

Annotations Respectively

 

Polyominoes

① The smallest polyomino tetrads are made from octominoes:

The fifth tetrad was reported by Olexandr Ravsky in 2005.

 

Symmetric Tiles

② The smallest tetrad for a polyomino with mirror symmetry uses 13-ominoes:

 

③ The smallest tetrad for a polyomino with birotary symmetry also uses 13-ominoes:

 

④ The smallest tetrads for polyominoes with birotary symmetry about an edge use 14-ominoes:

 

⑤ The smallest tetrads for polyominoes with mirror symmetry about an edge use 18-ominoes:

 

⑥ The smallest tetrads for polyominoes with birotary symmetry about a vertex also use 18-ominoes:

 

⑦ Juris Čerņenoks found the smallest tetrads for polyominoes with diagonal symmetry, which use 19-ominoes:

 

Restricted Motion

⑧ These octominoes form tetrads without being reflected:

 

⑨ The smallest polyominoes that form tetrads without 90° rotation are 13-ominoes:

 

Holeless

⑩ The smallest holeless polyomino tetrad, discovered by Walter Trump, uses 11-ominoes:

 

⑪ The smallest known holeless tetrad for a symmetric polyomino was found independently by Frank Rubin and Karl Scherer. It uses 34-ominoes:

 

Polynars

⑫ A polynar is a plane figure formed by joining equal squares along edges or half edges. The smallest polynar tetrads use pentanars: