Information AboutJohnson Solid |
| CATEGORIES ABOUT JOHNSON SOLID | |
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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 ( ''J''1 ); it has one square face and four triangular faces. 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 (''J''2) is an example that actually has a degree-5 vertex. 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. 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. Of the Johnson solids, the Elongated Square Gyrobicupola (''J''37) is unique in being vertex-uniform: there are four faces at each vertex, and their arrangement is always the same: three squares and one triangle. NAMES The names are listed below and are more descriptive than they sound. Most of the Johnson solids can be constructed from the first few ( Pyramids , Cupola e, and Rotunda e), together with the Platonic and Archimedean solids, Prism s, and Antiprism s.
The last three operations — augmentation, diminution, and gyration — can be performed more than once on a large enough solid. We add ''bi-'' to the name of the operation to indicate that it has been performed twice. (A ''bigyrate'' solid has had two of its cupolae rotated.) We add ''tri-'' to indicate that it has been performed three times. (A ''tridiminished'' solid has had three of its pyramids or cupolae removed.) Sometimes, ''bi-'' alone is not specific enough. We must distinguish between a solid that has had two parallel faces altered and one that has had two oblique faces altered. When the faces altered are parallel, we add ''para-'' to the name of the operation. (A ''parabiaugmented'' solid has had two parallel faces augmented.) When they are not, we add ''meta-'' to the name of the operation. (A ''metabiaugmented'' solid has had two oblique faces augmented.) NAMES AND JOHNSON NUMBERS J1 - J12 J13 - J24 J25 - J36 J37 - J48 J49 - J59 J61 - J72 J73 - J84 J85 - J92 SEE ALSO REFERENCES
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