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Icosododecahedron

Icosidodecahedron.svg

Icosododecahedron [1] [2] [3] is a semi-regular polyhedron consisting of 32 faces (12 regular pentagons and 20 regular triangles ). The icosododecahedron has 30 identical vertices in which two triangles and two pentagons converge, as well as 60 identical edges, each of which separates a triangle and a pentagon. The polyhedron dual to the icosododecahedron is the rhombotriacontahedron .

For icosododecahedron with rib lengtha {\ displaystyle a} a some quantitative characteristics can be expressed:

  • volume :V=a36(45+17five); {\ displaystyle V = {\ frac {a ^ {3}} {6}} \ left (45 + 17 {\ sqrt {5}} \ right);} V = \ frac {a ^ 3} {6} \ left (45 + 17 \ sqrt {5} \ right);
  • area :S=a2(five3+325+tenfive); {\ displaystyle S = a ^ {2} \ left (5 {\ sqrt {3}} + 3 {\ sqrt {25 + 10 {\ sqrt {5}}}} right);} S = a ^ 2 \ left (5 \ sqrt {3} + 3 \ sqrt {25+ 10 \ sqrt {5}} \ right);
  • dihedral angle :α=arccos⁡(-five+2five15)≈142,62∘. {\ displaystyle \ alpha = \ arccos \ left (- {\ sqrt {\ frac {5 + 2 {\ sqrt {5}}} {15}}} \ right) \ approx 142.62 ^ {\ circ}.} {\ displaystyle \ alpha = \ arccos \ left (- {\ sqrt {\ frac {5 + 2 {\ sqrt {5}}} {15}}} \ right) \ approx 142.62 ^ {\ circ}.}

Notes

  1. ↑ Wenninger, 1974 , p. 20, 36.
  2. ↑ Encyclopedia of Elementary Mathematics, 1963 , p. 437, 435.
  3. ↑ Lyusternik, 1956 , p. 183.

Literature

  • M. Wenninger . Polyhedron models. - World , 1974.
  • Polygons and polyhedra // Encyclopedia of Elementary Mathematics. Book Four. Geometry / Ed. P.S. Aleksandrova , A.I. Markushevich , A. Ya. Khinchin . - M .: State Publishing House of Physics and Mathematics , 1963. - S. 382-447. - 568 p. - 20,000 copies.
  • L.A. Lyusternik . Convex figures and polyhedra. - M .: State Publishing House of technical and theoretical literature , 1956.

Links

  • Weisstein, Eric W. Icosidodecahedron on the Wolfram MathWorld website.


Source - https://ru.wikipedia.org/w/index.php?title=Isosododecahedron&oldid=99826553


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