@johncarlosbaez@mathstodon.xyz
Post #3917452
2026-07-18 15:23 UTC
But suppose we measure distance between points in the plane in the usual way. What's the average distance between two points in the Sierpiński triangle with side length 1? I'm getting
0.4226884 ± 0.00004
This is close to 41/(56√3). But is that the answer, or just a coincidence? 🤷♂️
Maybe someone here can compute more decimal places and settle this question!
(3/n)
Replies (1)
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@johncarlosbaez@mathstodon.xyz 2026-07-18 15:36
Why do paths through the nth approximation to the Sierpiński triangle correspond to allowed sequences of moves in the n-disc Tower of Hanoi? Well, suppose you have 3 discs. Draw the allowed states of the Tower of Hanoi puzzle as below. For example, (3,2,1) means "biggest disc on post 1, second biggest on post 2, third biggest on post 3". Draw edges for allowed moves between states. You get the 3rd approximation to the Sierpiński triangle! I got this picture from an article with more details: • Alexander Bogomolny, Sierpinski Gasket and Tower of Hanoi, https://www.cut-the-knot.org/triangle/Hanoi.shtml