Rhombicosidodecahedron: Difference between revisions

Revision as of 02:48, 18 June 2007

Revision as of 19:12, 27 June 2007

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{{Semireg polyhedra db|Semireg polyhedron stat table|lrID}}

The '''rhombicosidodecahedron''', (rom-bi-co-si-do-dec-a-he-dron) or '''small rhombicosidodecahedron''', is an [[Archimedean solid]]. It has 20 regular [[triangle (geometry)|triangular]] faces, 30 regular [[square (geometry)|square]] faces, 12 regular [[pentagon]]al faces, 60 vertices and 120 edges.

The name ''rhombicosidodecahedron'' refers to the fact that the 30 square faces lie in the same planes as the 30 faces of the [[rhombic triacontahedron]] which is dual to the [[icosidodecahedron]].

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The [[Zometool]] kits for making [[geodesic dome]]s and other polyhedra use slotted balls as connectors. The balls are "expanded" small rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are golden rectangles.

Kids at St. Germaine School in Bethel Park, PA built the worlds largest rhombicosidodecahedron, which stood at over 9 feet tall. The workers included, but are certainly not limited to (in alphabetical order) Bridget Bielich, Callie Hanua, Glenn Huetter, Carl Mitchell, Anthony "TJ" Serafini, and Leanna Talarico. Bielich's grandfather, Walter, donated the cardboard for the project. The shape stood on its own, although for several weeks it sat in a cold convent, haunted by dead nuns and supported by chairs. It was painted blue, purple, and pink. Most of the guys in the class did not like that it was purple and pink, because they are insecure and, sometimes, a little whiney. The building of the shape caused great unity among the 8th grade class, and it also took up a lot of math class time. The record-breaking construction was written about in the Post-Gazette.

In a recent ground-breaking poll of people who live in Bethel Park named Kelly Emmett, 100% of people consider this rhombicosidodecahedron to be the most amazing shape ever constructed in history.

== Area and volume ==

The area ''A'' and the [[volume]] ''V'' of a rhombicosidodecahedron of edge length ''a'' are:

:<math>\begin{align}

A & = \left \{ 30 + \sqrt{ 30 \left [ 10 + 3\sqrt{5} + \sqrt{15 (5 + 2\sqrt{5})} \right ] } \right \} a^2 \\

& \approx 59.3059828a^2 \\

V & = \frac{1}{3} (60+29\sqrt{5})a^3 \approx 41.6153238a^3 \\

\end{align}</math>

==Cartesian coordinates==

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: (0, &plusmn;&tau;², &plusmn;(2+&tau;)),

where &tau; = (1+&radic;5)/2 is the [[golden ratio]].

== Vertex arrangement ==

The rhombicosidodecahedron shares its [[vertex arrangement]] with 3 nonconvex [[uniform polyhedron]]s:

{| class="prettytable"

| [[Image:Small dodecicosidodecahedron.png|150px]]<BR>[[Small dodecicosidodecahedron]]

| [[Image:Small rhombidodecahedron.png|150px]]<BR>[[Small rhombidodecahedron]]

| [[Image:Small stellated truncated dodecahedron.png|150px]]<BR>[[Small stellated truncated dodecahedron]]

|}

==See also==

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*[[rhombicuboctahedron]]

*[[truncated icosidodecahedron]] (great rhombicosidodecahedron)

== References ==

* {{cite book | first=Robert | last=Williams | authorlink=Robert Williams | title=The Geometrical Foundation of Natural Structure: A Source Book of Design | publisher=Dover Publications, Inc | year=1979 | id=ISBN 0-486-23729-X }} (Section 3-9)

==External links==

* {{mathworld | urlname = SmallRhombicosidodecahedronRhombicosidodecahedron| title = Small Rhombicosidodecahedron}}

*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]

*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra

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[[es:Rombicosidodecaedro]]

[[fr:Petit rhombicosidodécaèdre]]

[[ko:부풀린 십이이십면체]]

[[it:Rombicosidodecaedro]]

[[nl:Romboëdrisch icosidodecaëder]]

[[ja:斜方二十・十二面体]]

[[pl:Dwunasto-dwudziestościan rombowy mały]]

[[pt:Rombicosidodecaedro]]

[[th:รอมบิโคซิโดเดคาฮีดรอน]]