In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.
[1] Coxeter and Petrie found three of these that filled 3-space: There also exist chiral skew apeirohedra of types {4,6}, {6,4}, and {6,6}.
These skew apeirohedra are vertex-transitive, edge-transitive, and face-transitive, but not mirror symmetric (Schulte 2004).
Beyond Euclidean 3-space, in 1967 C. W. L. Garner published a set of 31 regular skew polyhedra in hyperbolic 3-space.
[2] J. Richard Gott in 1967 published a larger set of seven infinite skew polyhedra which he called regular pseudopolyhedrons, including the three from Coxeter as {4,6}, {6,4}, and {6,6} and four new ones: {5,5}, {4,5}, {3,8}, {3,10}.