Catalogue of Polyominoids

Introduction

A polyominoid is a polyform consisting of edge-connected squares in the polycube grid. It is a 3-dimensional analogue of the polyomino.

Polyominoids may be generalized to other dimensions. An (n,k)-polyominoid is a set of k-dimensional cells in the n-dimensional grid, connected at (k−1)-dimensional cells. Examples:

nkName
21polyline or polystick
22polyomino
313D polystick
32ordinary polyominoid
33polycube

One can further generalize polyominoids by specifying the dimension of their cell connections. For example, a polyking may be regarded as a (2,2,0)-polyominoid.

Here I show all (3,2)-polyominoids with at most 4 cells. Like polycubes, polyominoids may be one-sided or two-sided. One-sided means that distinct mirror images are counted as different polyominoids. Two-sided means that distinct mirror images are counted as the same polyominoid.

Enumeration

Cells Two-Sided
A075679
One-Sided
A056846
111
222
3911
45480
5448780
646508781
753611104828

The diagrams below show the two-sided polyominoids.

Monominoid

Dominoids

Trominoids

Tetrominoids

Last revised 2022-05-31.


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Col. George Sicherman [ HOME | MAIL ]