Informally, faces do not intersect. More formally, an acoptic polyhedra is one where no edge, face or vertex intersects any other. All polyhedra on this website are acoptic.
Archimedean means that faces are of different types unlike the platonic polyhedra where all faces are of the same type.
A polyhedron is chiral if it is distinguishable from its mirror image; that is, it cannot be superimposed onto it.
Note that the 34.6 tiling is usually considered chiral, but not on this website: these are all three dimensional polyhedral. In three dimensions, a tiling can be turned over to match its mirror image.
The complement of a polyhedron swaps the inside-outside property. Many polyhedra are self-complementary: the complement is identical. Some are reverse-complementary: the complement is mirrored.
A polyhedron is dipleural if it remains isogonal even when the two surfaces of a vertex are marked differently, for example one surface could be black and the other white, or the internal volume could be solid.
Edges that are edge-transitive are all identical and cannot be distinguished from one another. More specifically, all faces must be not merely congruent but must be transitive. In other words, for any edges A and B, there must be a symmetry of the entire polyhedron by rotations and reflections that maps A onto B. For example, all edges of a cube are edge transitive: no edge can be distinguished from another. For another example, look at the edges of a rhombicuboctahedron. The triangular edges all form an edge-transitive set. Also the edges between square faces form a second edge-transitive set.
When a polyhedron is chiral then the enantiomorph is the mirrored polyhedron.
A polyhedron is equitransitive if each set of a face type forms a single transitivity class. So a face with n sides will be isohedral to every other face with n sides.
Faces that are face-transitive are all identical and cannot be distinguished from one another. More specifically, all faces must be not merely congruent but must be transitive. In other words, for any faces A and B, there must be a symmetry of the entire polyhedron by rotations and reflections that maps A onto B. For example, all faces of a cube are face transitive: no face can be distinguished from another. For another example, look at the faces of a rhombicuboctahedron. The triangular faces all form a face-transitive set. But the square faces form two face-transitive sets: one set is of those square adjacent to four more squares and the second set are those square faces adjacent to two triangles and two squares.
Some polyhedra are flexible: they can be deformed while retaining their isogonal property.
In layman’s terms, this is the number of “holes” a polyhedron has. Since the majority of the polyhedra of this website are infinite, then this value is defined for each translation unit. Its value is given by the formula 1+(E-V-F)/2 where E, V and F are the number of edges, vertices and faces of each translation unit.
A honeycomb is a tessellation of polyhedra which fills space with non-overlapping convex polyhedral cells. A uniform honeycomb is isogonal.
A polyhedron is isogonal when all its vertices are the same. More specifically, all vertices must be not merely congruent but must be vertex-transitive. All polyhedra on this website are isogonal.
A polyhedron is isohedral when all its faces are the same. More specifically, all faces must be not merely congruent but must be face-transitive.
A polyhedron is isopleural when all its surfaces are the same. The polyhedra of this website always have exactly two surfaces: the internal and external surfaces. Non-acoptic polyhedra can have a single surface. A cube is not isopleural: the inside and outside surfaces are different: you can tell the difference between being inside or outside a cube. But a square tiling is isopleural: both surfaces are identical.
A polyhedron is isotoxal when all its edges are the same. More specifically, all edges must be not merely congruent but must be edge-transitive.
Platonic means that all faces have the same number of sides unlike the archimedean polyhedra. Although faces all have the same number of sides, the polyhedron may still not be isohedral.
A regular polyhedron is highly symmetrical, being isotoxal, isogonal and isohedral. It may not be isopleural.
A polyhedron is a reverse complement if the internal volume is the mirror image of the external volume. It will also be isopleural and will have a volume of ½. Note that a polyhedron can be reverse complementary but still be achiral: chirality ignores the internal/external property.
A polyhedron is self complementary if the internal volume is identical to the external volume. It will also be isopleural and will have a volume of ½.
The space group of a polyhedron defines its symmetries. See this page for further information.
A polyhedron has symmetry if there is a transformation (such as translation, scaling, rotation or reflection) that maps the polyhedron onto itself.
This is the smallest part of the polyhedron that, by translation, can be repeated to form the whole polyhedron while preserving the internal/external property.
A uniform polyhedron has regular polygons as faces and is isogonal.
This is the number of faces at each vertex. Alternatively, it is the number of edges that meet at each vertex.
Vertices that are vertex-transitive are all identical and cannot be distinguished from one another. More specifically, all vertices must be not merely congruent but must be transitive. In other words, for any vertices A and B, there must be a symmetry of the entire polyhedron by rotations and reflections that maps A onto B. For example, all edges of a cube are vertex transitive: no vertex can be distinguished from another. For another example, look at the edges of a square pyramid. The square vertices of the base all form an vertex-transitive set but are not vertex transtive with the apex. No amount of rotation and reflection will make the apex coincide with a base vertex.