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This blew my mind:

https://youtu.be/aAD5yGaHunw

enter image description here

(Take a square-ended dowel and make two curvy cuts. Re-arrange the pieces, and a seemingly impossibly smooth curved surface results.)

Can we describe this topologically or geometrically? Can we predict "good" shapes? Is there some other algorithm that's isomorphic to this, for comparison?

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Very Interesting & Nice technique !

Mathematically , it is very Simple !

The Critical Part is that the Cross Section CS of the Wood Piece WP should have 2 Pairs of Matching Sides. It may be Square or rectangle (It can be even more Exotic !) such that the cuts can be re-arranged.

2 CUTS

Here , we have the rectangle Cross Section with corners named $ABCD$. The video says to cut it with that machine , which will give a line in the Cross Section CS.
That gives 2 Pieces which will align Correctly because the rectangle has matching lengths.

Variation 1 : That machine in the video can cut it in straight lines , but with other machines (& even manually) we can cut it like shown next where the CS is cut with jagged line. We can see mathematically , that still the 2 new pieces can align because the rectangle has 2 more matching lengths.

Thus we get the 4 Corners in the Center.

We can maintain this cutting though-out the height of the Wood Piece WP or we can alter the Cutting continuously on the height. Each CS will have to align.
Over-all, the 4 Pieces will have to align due to the rectangle having matching sides.

Variation 2 : We do not have to use Square or rectangle CS. It can even be a curve. The Criteria is the the 2 sides must match.

Here is such a CS , which can be cut & re-arranged to get Exotic Designs :
Exotic CS
The sides of the Same Colour are Exact Matches , hence we have a guarantee about the rearrangement.

Variation 3 : The Cut , which was a Straight line in the video , can be jagged line. It can even be a smooth curve !
Here , we can see such Cuttings :
Curve Cut
The matching sides will ensure that we can rearrange to get even more Exotic Designs , though It might be more Difficult with more manual Effort.

The Height of the Wood Piece WP is mathematically immaterial , though Practically , we require that it should be long enough to get nice Patterns. It must be sufficiently long to allow variations in the Cross Section CS of the Cuttings.

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