Near-miss Johnson solid

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In geometry, a near-miss Johnson solid is a strictly convex polyhedron in which the faces are close to being regular polygons but in which some or all of the faces are not precisely regular. They generalize the Johnson solids, polyhedra in which all faces are regular, and "can often be physically constructed without noticing the discrepancy" between their 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.

Examples[edit]

Name Image verfs V E F F3 F4 F5 F6 F8 F10 Symmetry
Truncated triakis tetrahedron Truncated triakis tetrahedron.png 4 (5.5.5)
24 (5.5.6)
28 42 16     12 4     Td
-- 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
Tetrated dodecahedron Tetrated Dodecahedron.gif 4 (5.5.5)
12 (3.5.3.5)
12 (3.3.5.5)
28 54 28 16   12       Td
-- 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

Coplanar misses[edit]

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.

Examples: 3.3...

4.4.4.4

3.4.6.4:

See also[edit]

References[edit]

  1. ^ Kaplan, Craig S.; Hart, George W. (2001), "Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons", Bridges: Mathematical Connections in Art, Music and Science .

External links[edit]