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We study chiral polyhedra in 3-dimensional geometries (Euclidean, hyperbolic, and projective) in a unified manner. This extends to hyperbolic and projective spaces some structural results in the classification of chiral polyhedra in Euclidean 3-space given in 2005 by Schulte. Then, we describe a way to produce examples with helical faces based on a classic Petrie–Coxeter construction, yielding a new family in $$\mathbb {S}^3$$ which is described exhaustively.

A maniplex of rank n is a connected, n-valent, edge-coloured graph that generalises abstract polytopes and maps. If the automorphism group of a maniplex M partitions the vertex-set of M into k distinct orbits, we say that M is a k-orbit n-maniplex. The symmetry type graph of M is the quotient pregraph obtained by contracting every orbit into a single vertex. Symmetry type graphs of maniplexes satisfy a series of very specific properties. The question arises whether any pregraph of order k satisfying these properties is the symmetry type graph of some k-orbit maniplex. We answer the question when k = 2 .

Maniplexes are combinatorial objects that generalize, simultaneously, maps on surfaces and abstract polytopes. We are interested on studying rotary maniplexes, that is, those having maximal 'rotational' symmetry. This note classifies rotary 4-dimensional maniplexes with the property of having exactly one facet, and gives examples and related results.

By an equivelar toroid we understand a map satisfying the requirements of an abstract polyhedron on the 2-torus where all faces have a fixed number p of edges, and all vertices belong to a fixed number q of edges. We establish that if every regular toroid of type $$\{4,4\}$${4,4} admits a faithful and symmetric realisation in a metric space $$\mathcal {S}$$S then every equivelar toroid of type $$\{4,4\}$${4,4} admits a faithful and symmetric realisation in $$\mathcal {S}$$S. We exemplify this by showing that every equivelar toroid of type $$\{4,4\}$${4,4} admits faithful and symmetric realisations in the 3-sphere and in 3-dimensional projective space.

Up to isomorphism, there are six fixed-point free crystallographic groups in Euclidean 3-space $$\mathbb {E}^3$$E3 generated by twists (screw motions). In each case, an orientable 3-manifold is obtained as the quotient of $$\mathbb {E}^3$$E3 by such a group. The cubic tessellation of $$\mathbb {E}^3$$E3 induces tessellations on each such manifold. The corresponding classification for the 3-torus and the didicosm were classified as `equivelar toroids' and `cubic tessellation of the didicosm' in previous works. This paper concludes the classification of cubic tessellations on the remaining four orientable manifolds.

In this paper, we give an example of a chiral 4-polytope in projective 3-space. This example naturally yields a finite chiral 4-polytope in euclidean 4-space, giving a counterexample to a theorem previously published in the literature [Theorem 11.2 of McMullen (Discrete Comput Geom 32(1):1---35, 2004)].

A k-orbit map is a map with k flag-orbits under the action of its automorphism group. We give a basic theory of k-orbit maps and classify them up to k⩽4. “Hurwitz-like” upper bounds for the cardinality of the automorphism groups of 2-orbit and 3-orbit maps on surfaces are given. Furthermore, we consider effects of operations like medial and truncation on k-orbit maps and use them in classifying 2-orbit and 3-orbit maps on surfaces of small genus.

Schulte classified the discrete chiral polyhedra in Euclidean 3-space and showed that they belong to six families. The polyhedra in three of the families have finite faces and the other three families consist of polyhedra with (infinite) helical faces. We show that all the chiral polyhedra with finite faces are combinatorially chiral. However, the chiral polyhedra with helical faces are combinatorially regular. Moreover, any two such polyhedra with helical faces in the same family are isomorphic.

We say that a (d+1)-polytope P is an extension of a polytope K if the facets or the vertex figures of P are isomorphic to K. The Schläfli symbol of any regular extension of a regular polytope is determined except for its first or last entry. For any regular polytope K we construct regular extensions with any even number as first entry of the Schläfli symbol. These extensions are lattices if K is a lattice. Moreover, using the so-called CPR graphs we provide a more general way of constructing extensions of polytopes.

This paper studies C-group permutation representation graphs, or for short, CPR graphs. C-groups are the groups of abstract regular polytopes. CPR graphs are shown to be a useful tool for studying such polytopes. We establish general properties of CPR graphs. Moreover, we illustrate their use by constructing regular polyhedra with alternating groups An as the automorphism groups.

This paper discusses consistent flag bicolorings of maps, in their own right and as generalizations of orientations and pseudo-orientations. Furthermore, a related doubling concept is introduced, and relationships between these ideas are explored.

We derive some general results on the symmetries of equivelar toroids and provide detailed analysis of the subgroup lattice structure of the dihedral group D4 and of the octahedral group to complete classification by symmetry type of those in ranks 3 and 4.

We discuss representations of non-finite polyhedra as quotients of regular polytopes. We provide some structural results about the minimal regular covers of non-finite polyhedra and about the stabilizer subgroups of their flags under the flag action of the automorphism group of the covering polytope. As motivating examples we discuss the minimal regular covers of the Archimedean tilings, and we construct explicit minimal regular covers for three of them.