A conjecture on iterated circumcentres

Importance: Medium ✭✭
Author(s): Goddyn, Luis A.
Subject: Geometry
Recomm. for undergrads: yes
Posted by: mdevos
on: June 8th, 2007
Conjecture   Let $ p_1,p_2,p_3,\ldots $ be a sequence of points in $ {\mathbb R}^d $ with the property that for every $ i \ge d+2 $, the points $ p_{i-1}, p_{i-2}, \ldots p_{i-d-1} $ are distinct, lie on a unique sphere, and further, $ p_i $ is the center of this sphere. If this sequence is periodic, must its period be $ 2d+4 $?

Luis Goddyn discovered this curiosity, and proved the above conjecture for $ d \le 5 $. He also studied related sequences, for instance, the sequence in $ {\mathbb R}^2 $ where the $ i^{th} $ point is the circumcentre of the points with index $ i-2 $, $ i-3 $, and $ i-4 $. See Iterated Circumcenters for a delightful and interactive discussion of this problem.


*[G] Luis Goddyn, Iterated Circumcenters

* indicates original appearance(s) of problem.