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Tridiminished icosahedron
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{{Short description|63rd Johnson solid}} {{Infobox polyhedron | image = Tridiminished icosahedron.png | type = [[Johnson solid|Johnson]]<br>{{math|[[metabidiminished icosahedron|''J''{{sub|62}}]] β '''''J''{{sub|63}}''' β [[augmented tridiminished icosahedron|''J''{{sub|64}}]]}} | faces = 5 [[triangle]]s<br>3 [[pentagon]]s | edges = 15 | vertices = 9 | symmetry = <math> C_{3 \mathrm{v}} </math> | vertex_config = <math> 2 \times 3 \times (3 \times 5^2) + 3 \times (3^3 \times 5) </math> | properties = [[convex polytope|convex]],<br>[[non-composite polyhedron|non-composite]] | net = Johnson solid 63 net.png }} In [[geometry]], the '''tridiminished icosahedron''' is a [[Johnson solid]] that is constructed by removing three [[pentagonal pyramid]]s 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 triangle]]s and three [[regular pentagon]]s.{{r|berman}} Since all of its faces are [[regular polygon]]s and the resulting polyhedron remains [[convex set|convex]], the tridiminished icosahedron is a [[Johnson solid]], and it is enumerated as the sixty-third Johnson solid <math> J_{63} </math>.{{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 polyhedron]]s.{{r|timofeenko-2009}} == Properties == The surface area of a tridiminished icosahedron <math> A </math> is the sum of all polygonal faces' area: five equilateral triangles and three regular pentagons. Its volume <math> V </math> can be ascertained by subtracting the volume of a regular icosahedron with the volume of three pentagonal pyramids. Given that <math> a </math> is the edge length of a tridiminished icosahedron, they are:{{r|berman}} <math display="block"> \begin{align} 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} </math> == See also == * [[Snub 24-cell]], a [[4-polytope]] whose [[vertex figure]] is a tridiminished icosahedron == References == {{reflist|refs= <ref name=berman>{{citation | last = Berman | first = Martin | year = 1971 | volume = 291 | issue = 5 | pages = 329β352 | journal = Journal of the Franklin Institute | title = Regular-faced convex polyhedra | mr = 290245 | doi = 10.1016/0016-0032(71)90071-8 }}.</ref> <ref name=francis>{{citation | 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 }}</ref> <ref name="gailiunas">{{citation | 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 }}.</ref> <ref name=timofeenko-2009>{{citation | 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 = 720β723 | doi = 10.1134/S1064562409050238 }}.</ref> }} ==External links== *{{MathWorld2 |urlname2=JohnsonSolid |title2=Johnson Solid|urlname=TridiminishedIcosahedron |title=Tridiminished icosahedron }} {{Johnson solids navigator}} {{DEFAULTSORT:Tridiminished Icosahedron}} [[Category:Johnson solids]]
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