Near-miss Johnson solid

A near-miss Johnson solid is any polyhedron that visually approximates a Johnson solid, but which does not meet all of the requirements to be one. It can have slightly irregular faces, usually close enough to regular that one can make a physical model with regular faces and not notice a problem, by having slightly differing edges. This also includes those that can be made equilateral, but with slightly different angle sizes. Others may not be strictly convex yet have fully regular faces, meaning they might have coplanar faces or a dihedral angle of 180°. While the definition of near-miss is rather subjective, a shape is usually considered as such if the faces almost appear as regular.

Unlike the Johnson solids, the near-misses can have regular polyhedral symmetries that would otherwise be found in the regular and uniform polyhedra, and they can have unusual faces with odd numbers of edges.

There are three types of near-miss Johnson solids: the first with slightly irregular faces (including those that are equilateral), the second with regular but coplanar faces, and third with regular faces but not forming a convex hull.

Another way of categorizing near-miss Johnson solids is by properties of their vertices:

If a near-miss has any vertices that would be planar (if the polygons surrounding those vertices were regular), the polyhedron is considered locally Euclidean.

If all of the vertices of a near-miss have polyhedral angles less than 2π (if the polygons surrounding those vertices were regular), the polyhedron is considered locally spherical.

The Goldberg polyhedra and their dual geodesic polyhedra are good examples of locally Eucledean near-miss Johnson solids, since they approach the regular hexagonal tiling or the triangular tiling as the number of faces increase.