Heptagons Spiral Tiling

Hello Everyone,

Does anyone have any ideas or can guide me in how to make a tiled spiral heptagon ? Kinda like the image below. I am thinking of using fibonacci sequence but not sure how to go about it mathematically in grasshopper.

This image below is the work of Joe Bartholomew

Seems more an involute of a circle.

Draw a circle passing through the center of the second row of eptagons, make the involute of that circle, starting from the center of an eptagon.

Make attempts from the last made eptagon (6 attempts, but first two and last two almost always are unsuitable):
for each side of the eptagon draw the outgoing “normal” and pick the side with the normal having the closest angle relative to te local tangent of the involute.
Iterate.


The two green arrows are the local involute tangent and the best “segment normal” candidate.

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The always good Anemone!

eptagons_involute_V1.gh (21.5 KB)

eptagons_involute_2
Interesting to see: (speculations after seeing a large spiral made) apart from the very start, this spiral-involute have straight segments!
It make straight sequence of eptagons doing left-right-left-right-etc turns, you can “detect” the change of segment when it does two right turns one after the other.

Apart from the start, the segments “lengths” are:
3,5,5,5,7,7,7,9,9,9,11,11,11,13,13,13,15,15,15, …
So… a sequence of odd numbers with each number triple time.

This could become a new, very light and fast method to build the spiral.

It would be really interesting if some math expert/graduate come up here explaining what is going on …
i have no clue XD.

EDIT:
the sequence actually works from the very start:
1,1,1,3,3,3,5,5,5, …

EDIT 2:
Just for completion:



Light-weight script: 37k eptagons in istants.
eptagons_involute_V2.gh (27.0 KB)

The “fillings” can be made with other 2 repeatable shapes, which follow again a predictable pattern.

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