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A Pythagorean triple example of a non-halting tiling system:

• a ( 2 / 3 )^n scale symmetry log function;

• a 12-by-18 parent rectangle; and

• a repeating rectangle with an n = 0 of 12-by-10.

Printed by this alchemy repository.

• Approximate 1 + 7^( 1 / 2 ) with a natural number sequence.

• Approximate 1 + 2 * 2^( 1 / 2 ) with a natural number sequence.

• Approximate 1 + 10^( 1 / 2 ) with a natural number sequence.

• Approximate 1 + 11^( 1 / 2 ) with a natural number sequence.

• Approximate 1 + 2 * 3^( 1 / 2 ) with a natural number sequence.

• Approximate 1 + 3 * 2^( 1 / 2 ) with a natural number sequence.

Context:

• 1 + sqrt(#):

  • #02 ~ A001333,

  • #03 ~ A002531,

  • #04 ~ A000244,

  • #05 ~ A000045 Fibonacci sequence,

  • #06 ~ A123011,

  • #07 ~ A297189,

  • #08 ~ A164544,

  • #09 ~ A000079 powers of 2,

  • #10 ~ A328604,

  • #11 ~ A328605,

  • #12 ~ A328606,

  • #13 ~ A105476.

• 1 + #sqrt(2):

  • #2 ~ A164544,

  • #3 ~ A330390,

  • #4 ~ A330391,

  • #5 ~ A330392.

• 1 + #sqrt(3):

  • #2 ~ A328606,

  • #3 ~ A330393,

  • #4 ~ A330394.

• 1 + #sqrt(4):

  • #2 ~ A330395 ~ 5,

  • #3 ~ A330396 ~ 7,

  • #4 ~ A330397 ~ 9.

• 1 + #sqrt(5):

  • #2 ~ A330398,

  • #3 ~ A330399,

  • #4 ~ A330400.

• Inchoate conjecture:

  • These { this, this } YouTube videos, namely how

  • the unique parts { cos(#) & +0,-2,... "oscillating even", sin(#) & +1,-3,... "oscillating odd" }, and

  • the unique parts { cos(#)*cos(#) & 1 + "Sum", sin(#)*sin(#) & - "Sum" }, rhyme with

    • { (c + b) ~ 2b*x^0 + 2b*x^2 + ..., a ~ 2b*x^1 + 2b*x^3 + ... },

    • given a^2 + b^2 = c^2 and (c - b) / a = x, which is

    • context for x^0 = 1 as well as 2b = 1; 2b*x^0 = 1;

  • and also rhyme with

    • the A099603 Fibonacci sequence approximations of 1 + sqrt5 and (1 + sqrt5) / 2 = Golden ratio, which is

    • context for why the geometric series and Silver 日本 ratio are uniquely non-halting sequences.

• ( c - b ) / a = x^1 moves toward the vanishing point. (Myself).

• x^0 = 1 is the current position. (Me).

• a / ( c - b ) = x^-1 moves away from the vanishing point. (I).