Tridiminished icosahedron

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Tridiminished icosahedron
Tridiminished icosahedron.png
Type Johnson
J62J63J64
Faces 5 triangles
3 pentagons
Edges 15
Vertices 9
Vertex configuration
Symmetry group
Properties convex,
non-composite
Net
Johnson solid 63 net.png

In geometry, the tridiminished icosahedron is a Johnson solid that is constructed by removing three pentagonal pyramids from a regular icosahedron.

Contents

Construction

The tridiminished icosahedron can be constructed by removing three regular pentagonal pyramid from a regular icosahedron. [1] The aftereffect of such construction leaves five equilateral triangles and three regular pentagons. [2] Since all of its faces are regular polygons and the resulting polyhedron remains convex, the tridiminished icosahedron is a Johnson solid, and it is enumerated as the sixty-third Johnson solid . [3] This construction is similar to other Johnson solids as in gyroelongated pentagonal pyramid and metabidiminished icosahedron. [1]

The tridiminished icosahedron is a non-composite polyhedron: there is no plane that intersects its surface only in edges, so that it cannot be thereby divided into two or more regular or Johnson polyhedra. [4]

Properties

The surface area of a tridiminished icosahedron is the sum of all polygonal faces' area: five equilateral triangles and three regular pentagons. Its volume can be ascertained by subtracting the volume of a regular icosahedron from the volume of three pentagonal pyramids. Given that is the edge length of a tridiminished icosahedron, they are: [2]

A tridiminished icosahedron has three kinds of dihedral angles. These angles are between two triangles: 138.1°, triangle to pentagon: 100.8°, and two pentagons: 63.4°. [5]

The tridiminished icosahedron is a vertex figure of a 4-polytope, a snub 24-cell. [6]

See also

References

  1. 1 2 Gailiunas, Paul (2001), "A Polyhedral Byway" (PDF), in Sarhangi, Reza; Jablan, Slavik (eds.), Bridges: Mathematical Connections in Art, Music, and Science, Bridges Conference, pp. 115–122.
  2. 1 2 Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR   0290245 .
  3. Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177
  4. Timofeenko, A. V. (2009), "Convex Polyhedra with Parquet Faces" (PDF), Docklady Mathematics, 80 (2): 720–723, doi:10.1134/S1064562409050238 .
  5. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi: 10.4153/CJM-1966-021-8 , MR   0185507, S2CID   122006114 ; see Table III, line 63.
  6. Johnson (1966), p. 174.