miss is a polyhedron that's almost a Johnson solid.
Without formalizing what that means, we might ask that one should be
able to construct the polyhedron out of paper (or some construction
toy), using only regular faces, without noticing the discrepancy.
To my knowledge, there has never been a rigorous study of near misses,
though Norman Johnson did encounter (and reject!) a number of them in
his classication of Johnson solids. Evelyn Lamb wrote a
fun article on this subject in 2017.
The hendecagonal near-miss at the bottom of the page ultimately played
an important role in describing the structure of a protein cage in a
and spurred Agnieszka Kowalczyk to pursue some research on finding
polyhedral cages, which are related to solids but somewhat more
flexible. I occasionally talk about new near-misses on
Click on the image for a larger version.
O(*,3,*,): 8 enneagons, 6 squares, 24 almost equilateral triangles.
The notation given is from the 2001 Symmetrohedra paper.
I(1,2,*,): 12 pentagons, 20 hexagons, 60 almost equilateral
I(2,*,3,e): 12 decagons, 30 hexagons, 20 equilateral
triangles, 60 almost squares.
I(2,4,*,e): 12 decagons, 20 dodecagons, 60 almost
equilateral triangles (these are actually trapezoids with very
narrow tops), 30 very narrow rectangles.
This solid is very similar to a truncated buckyball. One imagines
that by fiddling with it a bit, one could sacrifice the perfect
regularity of the decagons and dodecagons in exchange for eliminating
the narrow rectangular faces, resulting in a near miss.
A solid constructed by inscribing a regular 11-gon in every
face of a pentagonal icositetrahedron. The result is a little
loose, but there's room for improvement by starting with a
differently-parameterized base solid.
Alex Doskey was kind enough to contribute WRL files for this
solid: Solid, Leonardo-style.