Near-miss Johnson solid

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Short description: Convex polyhedron whose faces are almost regular polygons

In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces.[1] The precise number of near-misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons.

Some near-misses with high symmetry are also symmetrohedra with some truly regular polygon faces.

Some near-misses are also zonohedra.

Examples

Name
Conway name		 Image Vertex
configurations		 V E F F3 F4 F5 F6 F8 F10 F12 Symmetry
Associahedron
t4dP3		 Associahedron.gif 2 (5.5.5)
12 (4.5.5)		 14 21 9 3 6 Dih3
order 12
Truncated triakis tetrahedron
t6kT 		Truncated triakis tetrahedron.png 4 (5.5.5)
24 (5.5.6) 		28 42 16     12 4       Td, [3,3]
order 24
Pentahexagonal pyritoheptacontatetrahedron Pyritohedral near-miss johnson.png 12 (3.5.3.6)
24 (3.3.5.6)
24 (3.3.3.3.5) 		60 132 74 56 12 6 Th, [3+,4]
order 24
Chamfered cube
cC 		Truncated rhombic dodecahedron.png 24 (4.6.6)
8 (6.6.6) 		32 48 18   6   12       Oh, [4,3]
order 48
-- Hexpenttri near-miss Johnson solid.png 12 (5.5.6)
6 (3.5.3.5)
12 (3.3.5.5) 		30 54 26 12   12 2       D6h, [6,2]
order 24
-- Dh3 symmetry dodecahedral nearmiss johnson.png 6 (5.5.5)
9 (3.5.3.5)
12 (3.3.5.5) 		27 51 26 14   12         D3h, [3,2]
order 12
Tetrated dodecahedron Tetrated dodecahedron.svg 4 (5.5.5)
12 (3.5.3.5)
12 (3.3.5.5) 		28 54 28 16   12         Td, [3,3]
order 24
Chamfered dodecahedron
cD 		Truncated rhombic triacontahedron.png 60 (5.6.6)
20 (6.6.6) 		80 120 42     12 30       Ih, [5,3]
order 120
Rectified truncated icosahedron
atI 		Rectified truncated icosahedron.png 60 (3.5.3.6)
30 (3.6.3.6) 		90 180 92 60   12 20       Ih, [5,3]
order 120
Truncated truncated icosahedron
ttI 		Truncated truncated icosahedron.png 120 (3.10.12)
60 (3.12.12) 		180 270 92 60         12 20 Ih, [5,3]
order 120
Expanded truncated icosahedron
etI 		Expanded truncated icosahedron.png 60 (3.4.5.4)
120 (3.4.6.4) 		180 360 182 60 90 12 20       Ih, [5,3]
order 120
Snub rectified truncated icosahedron
stI 		Snub rectified truncated icosahedron.png 60 (3.3.3.3.5)
120 (3.3.3.3.6) 		180 450 272 240   12 20       I, [5,3]+
order 60

Coplanar misses

Some failed Johnson solid candidates have coplanar faces. These polyhedra can be perturbed to become convex with faces that are arbitrarily close to regular polygons. These cases use 4.4.4.4 vertex figures of the square tiling, 3.3.3.3.3.3 vertex figure of the triangular tiling, as well as 60 degree rhombi divided double equilateral triangle faces, or a 60 degree trapezoid as three equilateral triangles. It is possible to take an infinite amount of distinct coplanar misses from sections of the cubic honeycomb (alternatively convex polycubes) or alternated cubic honeycomb, ignoring any obscured faces.

Examples: 3.3.3.3.3.3

4.4.4.4

  • Square icositetrahedron
    (Cube)

3.4.6.4:

  • Hexagonal cupola
    (Degenerate)

See also

References

  1. ↑ Kaplan, Craig S.; Hart, George W. (2001), "Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons", Bridges: Mathematical Connections in Art, Music and Science, https://cs.uwaterloo.ca/~csk/publications/Papers/kaplan_hart_2001.pdf .

External links



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