{'fraction_non_alphanumeric': 0.018124507486209612, 'fraction_numerical': 0.02442868400315209, 'mean_word_length': 4.474137931034483, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Rolling Polyhedra on Tessellations

22 We study the possible moves and reachable space by rolling a 3D convex polyhedron on a 2D periodic 23 tessellation in the xy -plane, where at every step a face of the polyhedron must coincide exactly with 24 a tile of the tessellation it rests upon. We topple the polyhedron around one of the edges of the 25 grounded face toward a neighboring face until it hits the xy -plane on a neighboring tile, only if the 26 new face and the new tile also coincide. We observe the space that can be reached by succession of 27 such rolling moves. If the whole plane can be reached, we call the polyhedron a plane roller for 28 the given tessellation. We further classify polyhedra that only reach a limited strip or a bounded 29 area on given tessellations as band rollers and bounded rollers respectively. We present a 30 polynomial-time algorithm to determine the set of tiles reachable from a given starting position, 31 which in particular determines the roller type of the given polyhedron and periodic tessellation. 32 Using this algorithm, we compute the reachability for every regular-faced convex polyhedron on any 33 regular-tiled ( ≤ 4)-uniform tessellation. Finally, we suggest how to employ these ﬁndings in puzzle 34 games. 35