A conjecture on iterated circumcentres

 Importance: Medium ✭✭
 Author(s): Goddyn, Luis A.
 Subject: Geometry
 Keywords: periodic plane geometry sequence
\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.