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Rings on a matrix

  • To: mathgroup at smc.vnet.net
  • Subject: [mg108512] Rings on a matrix
  • From: mokambo <alexandrepassosalmeida at gmail.com>
  • Date: Sun, 21 Mar 2010 02:05:52 -0500 (EST)

I'm having a problem trying to find a procedure to generate rings in a
matrix. Here are 3 steps of the algorithm (if it exists):
Use ArrayPlot[%, Mesh -> True] for quick visualization.

1 ring at iteration 1:
{{0, 0, 0, 0}, {0, 1, 1, 0}, {0, 1, 1, 0}, {0, 0, 0, 0}}

2 rings at iteration 2:
{{0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 8, 8, 0, 0, 0},
{0, 0, 8, 0, 0, 8, 0, 0}, {0, 8, 0, 8, 8, 0, 8, 0},
{0, 8, 0, 8, 8, 0, 8, 0}, {0, 0, 8, 0, 0, 8, 0, 0},
{0, 0, 0, 8, 8, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}}

4 rings at iteration 3:
{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 16,
   16, 16, 16, 16, 16, 0, 0, 0, 0, 0}, {0, 0, 0, 16, 16, 16, 0, 0, 0,
  0, 16, 16, 16, 0, 0, 0}, {0, 0, 16, 16, 16, 0, 0, 16, 16, 0, 0, 16,
  16, 16, 0, 0}, {0, 0, 16, 16, 0, 16, 16, 0, 0, 16, 16, 0, 16, 16, 0,
   0}, {0, 16, 16, 0, 16, 16, 0, 16, 16, 0, 16, 16, 0, 16, 16, 0}, {0,
   16, 0, 0, 16, 0, 16, 0, 0, 16, 0, 16, 0, 0, 16, 0}, {0, 16, 0, 16,
  0, 16, 0, 16, 16, 0, 16, 0, 16, 0, 16, 0}, {0, 16, 0, 16, 0, 16, 0,
  16, 16, 0, 16, 0, 16, 0, 16, 0}, {0, 16, 0, 0, 16, 0, 16, 0, 0, 16,
  0, 16, 0, 0, 16, 0}, {0, 16, 16, 0, 16, 16, 0, 16, 16, 0, 16, 16, 0,
   16, 16, 0}, {0, 0, 16, 16, 0, 16, 16, 0, 0, 16, 16, 0, 16, 16, 0,
  0}, {0, 0, 16, 16, 16, 0, 0, 16, 16, 0, 0, 16, 16, 16, 0, 0}, {0, 0,
   0, 16, 16, 16, 0, 0, 0, 0, 16, 16, 16, 0, 0, 0}, {0, 0, 0, 0, 0,
  16, 16, 16, 16, 16, 16, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0}}

I've tried just 1/4 of the problem (due to symmetries) and playing
with the circle equation and measuring distances from
points in a lattice. I've tried DiskMatrix but can't find a recursion
to generate the examples. Any ideas, hints?
Is there a way to solve this problem (Congruence equations perhaps?)

Alex


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