A conjecture on iterated circumcentres

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $d\geq1$ be an integer, and let $(p_i)_{i\geq1}$ be points in $\mathbb R^d$. A sphere is a set $\{x:\|x-c\|_2=R\}$ with centre $c\in\mathbb R^d$ and radius $R>0$. Suppose that for each $i\geq d+2$ the preceding $d+1$ points $p_{i-1},\ldots,p_{i-d-1}$ are distinct and lie on a unique sphere, with centre $p_i$. The least period of a periodic sequence is the least $T\geq1$ such that $p_{i+T}=p_i$ for every $i\geq1$.

Statement recorded in the supplied source.

If the sequence is periodic, is its least period necessarily $2d+4$? [S2]

Short English statement

If repeatedly taking the circumcentre of the previous d+1 points eventually returns to the original sequence, must its least period be 2d+4?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3073 (OPG-357). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Goddyn, Luis A., A conjecture on iterated circumcentres, Open Problem Garden, node 357. Complete problem, discussion and bibliography (including mathematical image alt text). https://www.openproblemgarden.org/op/a_conjecture_on_iterated_circumcentres.