# What's happening — mathematically — in this bandsaw technique? [closed]

This blew my mind:

(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?

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.

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 :

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 :

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.