Tridiminished icosahedron
{{Short description|63rd Johnson solid}}
{{Infobox polyhedron
| image = Tridiminished icosahedron.png
| type = Johnson
{{math|metabidiminished icosahedron – J{{sub|63}} – augmented tridiminished icosahedron}}
| faces = 5 triangles
3 pentagons
| edges = 15
| vertices = 9
| symmetry =
| vertex_config =
| 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.
Construction
The tridiminished icosahedron can be constructed by removing three regular pentagonal pyramid from a regular icosahedron.{{r|gailiunas}} The aftereffect of such construction leaves five equilateral triangles and three regular pentagons.{{r|berman}} 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 .{{r|francis}} This construction is similar to other Johnson solids as in gyroelongated pentagonal pyramid and metabidiminished icosahedron.{{r|gailiunas}}
The tridiminished icosahedron is non-composite polyhedron, meaning it is convex polyhedron that cannot be separated by a plane into two or more regular polyhedrons.{{r|timofeenko-2009}}
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 with the volume of three pentagonal pyramids. Given that is the edge length of a tridiminished icosahedron, they are:{{r|berman}}
A &= \frac{5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}}{4} a^2 &\approx 7.3265a^2, \\
V &= \frac{15 + 7 \sqrt{5}}{24}a^3 &\approx 1.2772a^3.
\end{align}
See also
- Snub 24-cell, a 4-polytope whose vertex figure is a tridiminished icosahedron
References
{{reflist|refs=
| last = Berman | first = Martin
| year = 1971 | volume = 291 | issue = 5 | pages = 329–352
| journal = Journal of the Franklin Institute
| title = Regular-faced convex polyhedra
| mr = 290245
| doi = 10.1016/0016-0032(71)90071-8
}}.
| last = Francis | first = Darryl
| title = Johnson solids & their acronyms
| journal = Word Ways
| date = August 2013 | volume = 46 | issue = 3 | page = 177
| url = https://go.gale.com/ps/i.do?id=GALE%7CA340298118
}}
| last = Gailiunas | first = Paul
| contribution = A Polyhedral Byway
| contribution-url = https://archive.bridgesmathart.org/2001/bridges2001-115.pdf
| pages = 115–122
| title = Bridges: Mathematical Connections in Art, Music, and Science
| year = 2001
| editor-last1 = Sarhangi | editor-first1 = Reza
| editor-last2 = Jablan | editor-first2 = Slavik
| publisher = Bridges Conference
}}.
| last = Timofeenko | first = A. V.
| year = 2009
| title = Convex Polyhedra with Parquet Faces
| journal = Docklady Mathematics
| url = https://www.interocitors.com/tmp/papers/timo-parquet.pdf
| volume = 80 | issue = 2
| pages = 720–723
| doi = 10.1134/S1064562409050238
}}.
}}
External links
- {{MathWorld2 |urlname2=JohnsonSolid |title2=Johnson Solid|urlname=TridiminishedIcosahedron |title=Tridiminished icosahedron }}
{{Johnson solids navigator}}
{{DEFAULTSORT:Tridiminished Icosahedron}}