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

\begin{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 $2d+4$? \end{conjecture}

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 \href[Iterated Circumcenters]{http://www.math.sfu.ca/~goddyn/Circles} for a delightful and interactive discussion of this problem.


% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries) *[G] Luis Goddyn, \href[Iterated Circumcenters]{http://www.math.sfu.ca/~goddyn/Circles}

* indicates original appearance(s) of problem.