visualisations of primes / cycle representations

[bhelmes]

A peaceful night for you,

this is a visualisation for the exponentation of 2*2 matrixs with determant 1 mod f.
I think it is mathematical correct, but perhaps there might be a more beautiful visualisation possible.

http://devalco.de/matrix_with_det_1/...with_det_1.php

For mathematicians, who likes visualisations of prime pattern I suggest proudly my other implementations in php:

http://devalco.de/System/system_natural.php
http://devalco.de/unit_circle/system_complex.php
http://devalco.de/unit_circle/system_unit_circle.php
http://devalco.de/unit_circle/system_tangens.php

All my friends ask me daily, if there is a new Mp arrived.
Some friends asked me also, if the amound of Mp is not limited.
The problem is, if I try to explain them some Number theory, they shut down immediatly.
I am glad, that spring is arriving and removes some depressiv moods.

[bhelmes]

Originally Posted by bhelmes

this is a visualisation for the exponentation of 2*2 matrixs with determant 1 mod f.
I think it is mathematical correct, but perhaps there might be a more beautiful visualisation possible.

After removing some bugs and some duplicates it looks nicer:

http://devalco.de/matrix_with_det_1/...with_det_1.php

The matrixs mod p, with p=3 mod 4 look o.k.

but the other matrixs mod p, with p=1 mod 4 contain elements which do not "lead" to the identity matrix:
For example: p:=37 (16, 23, 23, 33)^(36*38)=/=(1,0,0,1).

Why ? or is it a bug ?

There might be a more mathematically accurate title for this thread:
something like : visualisation of different fields mod p
Perhaps a mod could improve the title.

[Dr Sardonicus]

Originally Posted by bhelmes

<snip>
but the other matrixs mod p, with p=1 mod 4 contain elements which do not "lead" to the identity matrix:
For example: p:=37 (16, 23, 23, 33)^(36*38)=/=(1,0,0,1).

Why ? or is it a bug ?
<snip>

It's a bug. I note also that the determinant of your matrix isn't 1 (mod 37), but is rather Mod(24,37).

? M=Mod(1,37)*[16,23;23,23]
%1 = 
[Mod(16, 37) Mod(23, 37)]

[Mod(23, 37) Mod(23, 37)]

? M^(36*38)
%2 = 
[Mod(1, 37) Mod(0, 37)]

[Mod(0, 37) Mod(1, 37)]

?