Editing Regular polyhedron (section)

=== Regular skew apeirohedra ===

{{Main|Regular skew apeirohedron}}

In the first decades, Coxeter and Petrie allowed "saddle" vertices with alternating ridges and valleys, enabling them to construct three infinite folded surfaces which they called [[regular skew polyhedron|regular skew polyhedra]].<ref>[[Coxeter]], ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, {{isbn|0-486-40919-8}} (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)</ref> Coxeter offered a modified [[Schläfli symbol]] {l,m|n} for these figures, with {l,m} implying the [[vertex figure]], with ''m'' regular ''l''-gons around a vertex. The ''n'' defines ''n''-gonal ''holes''. Their vertex figures are [[regular skew polygon]]s, vertices zig-zagging between two planes.

{| class="wikitable"
!colspan=3| Infinite regular skew polyhedra in 3-space (partially drawn)
|- align=center
||[[Image:mucube external.png|150px]]<br>{4,6|4}
|[[Image:muoctahedron external.png|150px]]<br>{6,4|4}
|[[Image:mutetrahedron external.png|150px]]<br>{6,6|3}
|}