P | O | L | .. Y | A |

A | ... O | M | I | |

N | O | E | S | |

Till now, there is no formula for calculating the number
of different polyominoes, but only some results
(for smaller values of *n*), obtained by empirical derivation.

In every border square cell of a polyomino we could introduce two-sided mirrors perpendicular to the internal edges in their mid-points. After series of reflections, the ray of light will "describe" the shape: a closed Dragon curve. If we denote a reflection in a border mirror by 0, and a reflection in an internal mirror by 1, we have 0-1 words (or symbols) for polyominoes, where these words are cyclicaly-equivalent (this means, could be readed starting from any sign 0 or 1 and ending in it):

From their symbols we could directly make conclusions about the symmetry: every reversible word denotes polyominoes with a sense-reversing symmetry (they don't have "left" and "right" form); irreversible symbols correspond to the polyominos appearing in the "left" and "right" form (e.g. 4.4 and 4.5).

That symbols (or binary numbers) we could translate
into hexidecimal numbers and to every
polyomino asign exactly one such number.
For example, this could be the minimum of
all such cyclic-equivalent symbols (e.g. to the
polyomino 2.1 correspond cyclicaly-equivalent
symbols 00010001, 00100010, 01000100, 10001000
and the minimum of them is 00010001 = 11 in hexidecimal
system. Hence, we have the notation for polyominoes
where to every polyomino corresponds exactly one such
number, and *vice versa*.
(Opened question: find the general algebraic form
of number determinimg a polyomino?
Namely, some numbers will determine "opened polyominoes",
"holow polyominoes" or "overlaping polyominoes",
that are not included in our definition,
and other will determine "real" polyominoes.)

Every (*n*+1)-omino we derive from some *n*-omino
by adding to it a single square. Certainly, the addition
operation is positional one, this means, the result depends
from the position where we add the new square.

Here we have the following addition rules:

This "algebra" could be successfuly used for the
computer enumeration of polyominoes. In each
step we need to derive (*n*+1)-minoes from *n*-minoes by
adding a square, to test the equality of the
(large number) of polyominoes obtained and
to make a complete list of the different (*n*+1)-minoes obtained.

Every curvilinear shape (equivalent to a topological disk),
placed in a regular plane tiling (4^{4}) (i.e. in a
regular square grid) is enclosed between a maximal
polyomino contained in it and minimal polyomino containing
it. It could be approximated by this
maximal and minimal polyomino. The precision of
this approximation depends from the choice of the
grid, and could be

always rafined.

Polyominoes (either black or white) appearing in Lunda
designs will be called * Lunda polyominoes*.
The possible shape of Lunda polyominoes is conditioned by
the local equilibrium condition for Lunda designs.
Therefore, some polyominoes are inadmissible (e.g. 4.3,
or the following 10-minoes) as Lunda polyominoes.

On the other hand, in Lunda polyominoes are included also "holow" polyominoes.

By introducing the concept of Lunda animals, P.Gerdes in
his book "Lunda geometry: Designs, Polyominoes, Patterns,
Symmetries" obtained the first approximation for the total
number of different Lunda *n*-ominoes.