Monogram

Gen. 1	Gen. 2
Gen. 3 is a rotated version of Gen 1. Gen. 4 is a rotated version of Gen 2. ( rotation 90°)

Per.4

IS-osc.

Monogram has 4 stator cells and 14 rotor cells. The numbering is explained in the cycle matrix discussion.

Jenny Axén K05

Cycle matrix

Det = 0

See the 18x18 cycle matrix in a new window!

Note that the determinant is 0. Hence it is not possible to recover the initial colouring by means of matrix inversions..

A closer look at the cycle matrix reveals that the rotor cells 5-12 form a communicating set (each cell in the set contributes to the colouring of any other cell in the set.)
The cells 13-18 however do not belong to any commutating set (except their singletons), since they do not affect any other set.
The cells 15 and 16 can be called sterile since they never get any offspring. (This is reflected in their 0-columns).
The cells 13,14,17,18 (the 4 corner cells) get offspring (in the third generation) but their impact is restricted to the cells in the corner positions in the fifth generation , i.e. to themselves. (This is also visible in the columns: nonzero components only in the diagonal positions).
The above properties explain the chosen numbering.

Limit matrix

Greyscale discussion in new window

Here below is the G* part of the limit matrix in rational and floating form.

	Since there are 4 stator cells the number in the i^th row and j^th column reflect the genetic impact of the j^th stator cell on the rotor cell with number 4+i. The total genetic impact of each stator cell is necessarily 3.5. (In view of the symmetry of the pattern, each stator takes a 4^th part of the available total impact 14 (= number of rotor cells).
Also, due to the symmetry, each stator has in principle the same genetic effect on the pattern as any other. Hence each column contains the same set of numbers in different permutations