Modified Johnson Near Misses
By Ed Baumann
Johnson polyhedra are convex and have only regular n-gons as faces. There exists a list of 31 Johnson Near Miss polyhedra which are not perfect. See the list here. I try to change the angles in the n-gons, n>3, in order to achieve that all edges of the new polyhedra have same length and that those n-gons are perfectly plan. The new polyhedra aren't near misses anymore. Above the modified no 1 of the list is shown in cross eye stereo. You can see that the pentagons have angles different from 108 degree and that 8 triangles are merged to 4 rhombi. Releasing the internal angles of the n-gons is NOT allways sufficient to reach the aim.
There exists an infinite familly of equal edge polyhedra between the above symetric polyhedron and a asymmetric variant shown here.
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No 2 of the modified Johnson Near Misses (2021).
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No 3 of the modified Johnson Near Misses (2020).
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No 4 of the modified Johnson Near Misses (2020).
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No 5 of the modified Johnson Near Misses (2020).
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No 6a of the modified Johnson Near Misses. Acrohedron 554 Type A (2020)
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No 6b of the modified Johnson Near Misses. Acrohedron 554 Type B (2020)
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No 7 of the modified Johnson Near Misses (2020).
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No 9 of the modified Johnson Near Misses (2021).
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No 10 of the modified Johnson Near Misses (2021).
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No 11 of the modified Johnson Near Misses (2021).
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No 12 of the modified Johnson Near Misses (2021).
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No 13 of the modified Johnson Near Misses (2021).
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No 14 of the modified Johnson Near Misses (2021).
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No 15 of the modified Johnson Near Misses (2021).
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No 16 of the modified Johnson Near Misses (2021).
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No 17 of the modified Johnson Near Misses (2021).
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No 18 of the modified Johnson Near Misses (2021).
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No 19 of the modified Johnson Near Misses (2021).
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No 20 of the modified Johnson Near Misses (2021).
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No 21 of the modified Johnson Near Misses (2021).
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No 22 of the modified Johnson Near Misses (2021).
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No 23 of the modified Johnson Near Misses (2021).
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No 25 of the modified Johnson Near Misses (2021).
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No 26 of the modified Johnson Near Misses (2021).
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No 27 of the modified Johnson Near Misses (2021).
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No 28 of the modified Johnson Near Misses (2021).
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No 29 of the modified Johnson Near Misses (2021).
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No 30 of the modified Johnson Near Misses (2021).
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No 31 of the modified Johnson Near Misses (2021).
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Finally I found for ALL thirty-one Johnson Nearmisses an equilateral version (equal edge lengthes) sacrifying the regular angles in the polygons EXCEPTED for No 24. Perhaps it can be prooven that in this case no solution exists.
Here you can download zipped files with the vertex coordinates in off-Format.
Here you can download zipped files with the vertex coordinates in wrl-Format.