Icosahedron (EntityTopic, 12)

From Hi.gher. Space

(Difference between revisions)

Latest revision as of 14:50, 26 March 2017

The icosahedron is one of the five Platonic solids. It contains 20 triangles joined five to a vertex. It can also be considered as a snubbed tetrahedron.

Coordinates

The coordinates of an icosahedron with edge length 2 are:

(0, ±1, ±φ)
(±1, ±φ, 0)
(±φ, 0, ±1)

Equations

The hypervolumes of an icosahedron with side length l are given by:

total edge length = 30l
surface area = 5√3 · l²
volume = ^5(3+√5)∕₁₂ · l³

Relation to other polyhedra

Non-convex polyhedra sharing vertex coordinates with the icosahedron can be morphed into J91 (bilbiro) and J92 (thawro) as seen in the following animations:

The above animations are not embedded into the page due to a wiki bug. Please leave them as links and do not embed them.

Incidence matrix

Dual: dodecahedron

#	TXID	Va	Ea	3a	Type	Name
0	Va				= point	;
1	Ea	2			= digon	;
2	3a	3	3		= triangle	;
3	C1a	12	30	20	= icosahedron	;

Usage as facets

6× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron

Notable Trishapes
Regular:	tetrahedron • cube • octahedron • dodecahedron • icosahedron
Direct truncates:	tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate
Mesotruncates:	stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron
Snubs:	snub staurohedron • snub rhodohedron
Curved:	sphere • torus • cylinder • cone • frustum • crind

@@ Line 1: / Line 1: @@
-{{Shape|Icosahedron|http://www.soulsofdistortion.nl/images/Top%20View%20Icosahedron.jpg|3|20, 30, 12|0|{[[Triangle|3,]]5}|<nowiki>5 | 2 3 </nowiki>|''Unknown''|N/A|Regular [[pentagon]], edge 1|Ike|[[Dodecahedron]]|N/A|N/A|N/A|none|''Unknown''|''Unknown''|[[Triangle|3]]<sup>5</sup>}}
+<[#ontology [kind topic] [cats 3D Regular Polytope] [alt [[freebase:03vyw]] [[wikipedia:Icosahedron]]]]>
-== Geometry ==
+{{STS Shape
-=== Equations ===
+| name=Icosahedron
-*Assumption: Icosahedron is centered at the origin.
+| image=<[#embed [hash 9F9FGF6HTQC8EZ51GC34JEKCKN] [width 150]]>
-*Variables:
+| dim=3
-<blockquote>''l'' ⇒ length of edges of the icosahedron</blockquote>
+| elements=20 [[triangle]]s, 30 [[digon]]s, 12 [[point]]s
+| sym=[[Rhodohedral symmetry|I<sub>h</sub>, H<sub>3</sub>, [5,3], (*532)]]
+| genus=0
+| ssc={G3<sup>5</sup>}
+| ssc2=Ki4
+| pv_circle=~0.6055
+| extra={{STS Polytope
+| flayout={{FLD|a3|er|e5}}
+| petrie=10,2
+| dual=[[Dodecahedron]]
+| bowers=Ike
+}}{{STS Uniform polytope
+| wythoff=<nowiki>5 | 2 3 </nowiki>
+| schlaefli={[[Triangle|3,]]5}
+| dynkin=o5o3x, s3s3s
+| conway=s[[Tetrahedron|Y3]]
+| vlayout=[[Triangle|3]]<sup>5</sup>
+| vfigure=Regular [[pentagon]], edge 1
+}}}}
+The icosahedron is one of the five Platonic solids. It contains 20 triangles joined five to a vertex. It can also be considered as a snubbed tetrahedron.
-*The [[hypervolume]]s of an icosahedron are given by:
+==Coordinates==
+The coordinates of an icosahedron with edge length 2 are:
+<blockquote>(0, ±1, ±φ)<br>(±1, ±φ, 0)<br>(±φ, 0, ±1)</blockquote>
+== Equations ==
+*The [[hypervolume]]s of an icosahedron with side length ''l'' are given by:
 <blockquote>total edge length = 30''l''<br>
-surface area = 5sqrt(3)''l''<sup>2</sup><br>
+surface area = 5√3 {{DotHV}}<br>
-volume = 5''l''<sup>3</sup>(3+sqrt(5))12<sup>-1</sup></blockquote>
+volume = {{Over|5(3+√5)|12}} {{DotHV|3}}</blockquote>
-*The [[planar]] [[cross-section]]s (''n'') of an icosahedron are:
+== Relation to other polyhedra ==
-<blockquote>''Unknown''</blockquote>
+Non-convex polyhedra sharing vertex coordinates with the icosahedron can be morphed into J91 ([[bilbiro]]) and J92 ([[thawro]]) as seen in the following animations:
-<br clear="all"><br>
+*[http://hddb.teamikaria.com/dl/VH65013KB0RBM2245X71MWVZY2.gif bilbiro]
-{{Trishapes}}
+*[http://hddb.teamikaria.com/dl/G6TYPETKGR822V4FPSQYPQ5PEF.gif thawro]
+{{Selfref|The above animations are not embedded into the page due to a wiki bug. Please leave them as links and do not embed them.}}
-[[Category:Regular polyhedra]]
+<[#polytope [id 5]]>
+{{Trishapes}}