2021/07/21 by Ryan Blair, Blair, Ryan, Zoe Marley +3
Engineering · #52C22 #57M50 #Advanced Materials and Mechanics #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2107.10216
openalex publication_date 2021/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A 3D rep-tile is a compact 3-manifold X in ℝ3 that can be decomposed into finitely many pieces, each of which are similar to X, and all of which are congruent to each other. In this paper we classify all 3D rep-tiles up to homeomorphism. In particular, we show that a 3-manifold is homeomorphic to a 3D rep-tile if and only if it is the exterior of a connected graph in S3.