A four-dimensional 4-reptile simplex

A $d$-dimensional simplex is a $k$-reptile if it can be tiled by $k$ mutually congruent simplices similar to it. Matoušek and Safernová conjectured that for $d\ge3$ a $k$-reptile $d$-simplex exists only when $k=m^d$ for an integer $m$, and proved this for $d=3$. Kynčl and Patáková proved that in dimension four $k$ must be a square, noted that every $k$-reptile simplex known in dimension $d\ge3$ has $k=m^d$, and asked whether a four-dimensional $k$-reptile simplex exists with $k$ a square but not a fourth power. We show that the simplex with vertices $(0,0,0,0)$, $(1,0,0,0)$, $(\tfrac12,\tfrac12,0,0)$, $(\tfrac23,\tfrac13,\tfrac13,0)$ and $(\tfrac34,\tfrac14,\tfrac14,\tfrac14)$, the fundamental alcove of the affine Weyl group of type $F_4$, is a $4$-reptile: it is tiled by four congruent copies of itself scaled by $1/\sqrt2$, two of them mirror images. This answers the question of Kynčl and Patáková with $k=4$ and disproves the conjecture of Matoušek and Safernová in dimension four. The subdivision itself is due to Felikson and Tumarkin, who found in 2004 that the affine reflection group of type $F_4$ has a reflection subgroup of index $4$ whose fundamental simplex is similar to the alcove. What we add is the observation that this settles the reptile question, a short direct proof of the tiling, and exact computer checks of it. This work has not yet been reviewed by experts. The proofs, the computer checks and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction. Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.