Grid Spirals

A Grid Spiral consist of a set of connected segments starting at the center and curling counter-clockwise. Segments are diagonals of the square grid, being the first ones the square root of the sequence:
1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18...
In order to obtain the tightest spiral, the angles between consecutive segments must to be the smallest after the last segment don't intersect any previous. A series of data defines a spiral, for instance
1A-2-9
means the spiral starts with segment 1 at direction A, followed by a segment 2, then segment 9, then segment 1 again and so on, so we have the three order snake: 1, 2, 9, 1, 2, 9, 1, 2, 9...
The 1A-2-9 spiral likely diverges (is infinite) while others clearly don't. For instance, try in the applet at the left, the spiral 1A-2-8 and you'll notice it converges or halts after 17 segments, because the "snake" reaches a dead end, so is finite.
Other examples: 5A-1 converges in 5 steps while 5B-1 clearly diverges.
Most of the combinations seems to generate finite small spirals while large ones are rare like 9A-1-2-4 with 911 segments! What are the conditions for convergence / divergence and how to prove it seems to be rather dificult.
Applet at the left has controls to visualize up to six-order spirals with diagonals sizes up to sqrt(18).

GridSpiralsApplet.java
Next some results found with a hidden tool in the program.
"Infinite candidates" are the spirals which don't reach a dead-end before 1000 segments.
"Greatest finite" are the sorted spirals which reach a dead-end before 1000 segments. Some are shown as images.
Note these spirals have as largest diagonal segment sqrt(18), so larger sizes would give other results.

One order spirals

Infinite candidates:
1A, 2A, 4A, 5A, 8A, 9A, 10B, 16A, 18A

Greatest finite:
13A -> 21
17A -> 16
17B ->  8
13B ->  6
5B  ->  4
10A ->  4

13A with 21 segments

17A with 16 segments

17B with 8 segments

Two order spirals

Infinite candidates:
1A-2, 1A-5, 1A-13
2A-1, 2A-4, 2A-10, 2A-17
4A-1, 4A-2, 4A-8
5B-1, 5A-5, 5A-9, 5B-9, 5B-13
8A-2, 8A-4, 8A-16
9A-4, 9A-5, 9A-18
10A-2, 10B-10, 10A-18, 10B-18
13B-1
16A-4, 16A-8, 16A-9
17A-2
18A-8, 18A-9, 18A-10

Greatest finite:
13B-16 -> 214
17A-18 ->  92
13B-9  ->  73
13A-16 ->  72
5B-18  ->  65
18A-5  ->  59
18A-17 ->  51

13B-16 with 214 segments

17A-18 with 92 segments

13B-9 with 73 segments

Three order spirals

Infinite candidates:
1A-1-4, 1A-2-9, 1A-2-13, 1A-5-18
2A-2-8, 2A-4-18, 2A-5-1, 2A-9-1
4A-1-1, 4A-4-9, 4A-4-16, 4A-10-2, 4A-18-2
5A-1-2, 5B-1-2, 5A-5-5, 5A-18-1, 5B-18-1
8A-2-2, 8A-8-18
9A-1-1, 9A-1-2, 9A-4-4, 9A-9-16
10A-2-4, 10B-2-4, 10B-10-10
16A-4-4, 16A-9-9
18A-1-5, 18A-2-2, 18A-2-4, 18A-8-8

Greatest finite:
1A-2-4 -> 911
2A-4-8 -> 911
4A-8-16 -> 911
1A-5-16 -> 434
5A-16-1 -> 433
5B-16-1 -> 355
16A-1-5 -> 354
2A-5-16 -> 227

1A-2-4 with 911 segments

1A-5-16 with 434 segments

5A-16-1 with 433 segments

Four order spirals

Infinite candidates:
1A-1-4-1, 1A-2-1-2, 1A-5-1-5, 1A-5-16-16, 1A-13-1-13
2A-1-2-1, 2A-2-8-2, 2A-4-2-4, 2A-10-2-10, 2A-10-2-17, 2A-17-2-17
4A-1-4-1, 4A-2-4-2, 4A-4-9-4, 4A-4-16-4, 4A-8-4-8
5B-1-5-1, 5A-9-5-9, 5B-9-5-9, 5B-13-5-13, 5A-16-16-1
8A-2-8-2, 8A-4-8-4, 8A-8-18-8, 8A-16-8-16
9A-4-4-4, 9A-4-9-4, 9A-5-9-5, 9A-9-4-4, 9A-9-4-9, 9A-9-16-9, 9A-18-9-18
10A-2-10-2, 10A-18-10-18, 10B-18-10-18
13B-1-13-1
16A-4-9-4, 16A-4-16-4, 16A-8-16-8, 16A-9-9-9, 16A-9-16-9, 16A-16-9-9, 16A-16-9-16
17A-2-17-2
18A-8-8-8, 18A-8-18-8, 18A-9-18-9, 18A-10-18-10, 18A-18-8-8, 18A-18-8-18

Greatest finite:
9A-1-2-4   -> 638
18A-2-4-8  -> 638
4A-8-2-9   -> 320
8A-16-4-18 -> 320
8A-2-9-4   -> 319
16A-4-18-8 -> 319
2A-9-4-8   -> 319
4A-18-8-16 -> 319
9A-4-8-2   -> 318
18A-8-16-4 -> 318

9A-1-2-4 with 638 segments

4A-8-2-9 with 320 segments

8A-2-9-4 with 319 segments

Five order spirals

Infinite candidates
1A-1-1-1-4, 1A-1-4-1-1, 1A-1-4-4-4, 1A-4-4-4-1, 1A-4-8-2-9, 1A-9-4-1-4
2A-2-2-2-8, 2A-2-8-2-2, 2A-2-8-8-8, 2A-8-8-8-2, 2A-8-16-4-18, 2A-18-8-2-8
4A-1-1-1-1, 4A-1-9-4-1, 4A-4-1-1-4, 4A-4-4-4-9, 4A-4-4-4-16, 4A-4-9-4-4, 4A-4-9-9-4, 4A-4-9-9-9, 4A-4-9-9-16
4A-4-9-16-4, 4A-4-9-16-9, 4A-4-16-4-4, 4A-4-16-9-4, 4A-4-16-16-9, 4A-4-16-16-16, 4A-8-2-9-1, 4A-9-1-2-4
4A-9-9-9-4, 4A-9-9-16-4, 4A-9-16-9-4, 4A-16-16-16-4
8A-2-2-2-2, 8A-2-18-8-2, 8A-8-2-2-8, 8A-8-8-8-18, 8A-8-18-8-8
8A-8-18-18-8, 8A-8-18-18-18, 8A-16-4-18-2, 8A-18-2-4-8, 8A-18-18-18-8
9A-4-16-4-4, 9A-9-4-4-4, 9A-9-4-4-9, 9A-9-9-9-16
9A-9-16-9-9, 9A-9-16-16-9, 9A-9-16-16-16, 9A-16-4-4-4, 9A-16-16-16-9
16A-4-4-4-4, 16A-16-4-4-16, 16A-16-9-9-9, 16A-16-9-9-16, 16A-16-16-16-16
18A-18-8-8-8, 18A-18-8-8-18, 18A-18-18-18-18

Greatest finite:
2A-4-4-9-1     -> 717
4A-8-8-18-2    -> 717
4A-4-9-1-2     -> 716
8A-8-18-2-4    -> 716
1A-5-8-2-8     -> 599
2A-10-16-4-16  -> 599
1A-2-4-4-9     -> 506
2A-4-8-8-18    -> 506
5A-18-10-18-1  -> 420
18A-10-18-1-5  -> 419
1A-5-9-5-9     -> 343
2A-10-18-10-18 -> 343

2A-4-4-9-1 with 717 segments	4A-4-9-1-2 with 716 segments
1A-5-8-2-8 with 598 segments	1A-2-4-4-9 with 506 segments

Jorge Mireles
Feb 14 2004

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