TITLE: Is it possible to partition $\mathbb R^3$ into unit circles?
QUESTION [51 upvotes]: Is it possible to partition $\mathbb R^3$ into unit circles?

REPLY [13 votes]: A very nice, explicit, elementary partition (without the Axiom of Choice) of $\mathbb{R}^3$ into geometric circles of variable radii is given in: [MR0719756] Szulkin, Andrzej. $\mathbb{R}^3$ is the union of disjoint circles. Amer. Math. Monthly 90 (1983), no. 9, 640–641.