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Spiralsymmetry (in a very small and discrete world)

This is a nice example of naive mathematics. A long time I played with spiralisation of prime numbers and related things. On the other side I wrote a a little permutation game programm which brought me to think about permutations and so about group theory.

I read abaut concepts like symmetric permutation groups and so on, but as naive mathematican I don’t go to deep in real mathematics, just so deep, that I can imagine, some relations to other concepts.

So if you go on reading, don’t forget, for the naive mathematican fun and fascination are the priority. So be patient, if there are misconcepts and naive errors ;-)

Ok, some definitions (wow how formal)

spiralisationalbe board:= odd quadratic matrix (it has to fit)

standard interpretation:= read the board left to right and top down (ok, i am german and our books are read like this. But I think, all cases are handled, because all: )

possible spiralisations: ur, rr, dr, lr, ul, rl, dl, ll with for example ur means: take standard interpretation and the first element (index=0) becomes the center of the spiral, the next element (index=1) is placed one cell up (u) next element (index=2) one cell right ( r). Go on clockwise ( r) until the spiral is ready.

So ur (up-right) means,

0	1	2
3	4	5
6	7	8

becomes

8	1	2
7	0	3
6	5	4

and he others right-rigth, down-right, left-right, up-left (left at the end means against the clock), right-left, down-left and as last example left-left:

0	1	2
3	4	5
6	7	8

becomes

8	7	6
1	0	5
2	3	4

as we see, this could be seen as eight different permutations of (0,1,2,3,4,5,6,7,8) and they are all cyclic, so if applicated iterated, the original permutation comes back.

3×3

the list gives the cycles:

["ur", 3]
["rr", 12]
["dr", 6]
["lr", 6]
["ul", 10]
["rl", 6]
["dl", 20]
["ll", 6]

5×5

the list gives the cycles:

["ur", 66]
["rr", 210]
["dr", 36]
["lr", 60]
["ul", 132]
["rl", 10]
["dl", 420]
["ll", 140]

And now for the naive experiment

I wonderd how many bit patterns applied to the board are spiral symmetric? So I applied every possible bit pattern and tested against every of the 8 spiralisations.

You can see in the pictures at the top of the page one example of 3×3 (dl) with the 4 possible bit patterns that fit 3×3-dl. A blue field means 0 a red field 1.

The simplest case is also there: 1×1-* is always spiral symmetric. (the blue quadrat, and the red one interpreted as 1×1)

Results

Number of spiral bit patterns that are invarant for 3×3 and 5×5 for all orientations

orientation	3×3	5×5
ur	32	64
rr	8	256
dr	32	64
lr	32	1024
ul	16	64
rl	64	256
dl	4	256
ll	16	64

Downloads

You can download the resulting pictures as zipfiles here: 3×3.zip (130 KB) and 5×5.zip (1.8MB over 2000 pics)

here some 5×5 examples (name of he pic means orientation+bit pattern):

ur0101010101010100010000010.png

rl1000101000001000100000101.png

lr0000000000000000000011111.png

Because every orientation check for a bxb board needs to check 2^(b*b) permutations its not possible for me to check the 7×7 or greater boards.