Johnson solid

In geometry, a Johnson solid is a convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism. There is no requirement that each face must be the same polygon. An example of a Johnson solid is the square-based pyramid with equilateral sides (J1); it has one square face and four triangular faces.

Related Topics:
Geometry - Convex - Polyhedron - Regular polygon - Platonic solid - Archimedean solid - Prism - Antiprism - Pyramid - ''J''₁

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As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid (J2) is an example that actually has a degree-5 vertex.

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Although there is no obvious restriction on any regular polygon's being a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3, 4, 5, 6, 8, or 10 sides.

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In 1966, Norman Johnson published a list which included all 92 solids, and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

Related Topics:
1966 - Norman Johnson - Victor Zalgaller - 1969

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Of the Johnson solids, the elongated square gyrobicupola (J37) is unique in being locally vertex-regular: there are four faces at each vertex, and their arrangement is always the same: three squares and one triangle.

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