Bilunabirotunda (EntityTopic, 14)

(Difference between revisions)

Latest revision as of 10:38, 2 January 2018

The bilunabirotunda is the 91st Johnson solid, J91. It suddenly became important when the castellated rhodoperihedral prism was discovered.

The following coordinates give an origin-centered bilunabirotunda with edge length 2:

<±1, 0, ±φ²>

<±φ, ±1, ±1>

<0, ±φ, 0>

where φ=(1+√5)/2 is the Golden Ratio.

The bilunabirotunda can be constructed from an icosahedron, as follows:

Firstly, write the icosahedron in [2,2,2]-symmetry:

   x2o2f
   f2x2o
   o2f2x

Then, apply a caleido-faceting to the last node:

   x2o2f -> x2o2f
   f2x2o -> f2x2o
   o2f2x -> o2f2(-x)

Finally, apply a Stott-expansion to the last node:

   x2o2f    -> x2o2F
   f2x2o    -> f2x2x
   o2f2(-x) -> o2f2o

The result is J91, written in [2,2,2]-symmetry: xfo2oxf2Fxo&#zx.

You can compare this representation with the coordinates as given above.

total edge length = 26l
surface area = (2 + 2√3 + √(25+10√5)) · l²
volume = ¹∕₆ · (4φ² + 5φ) · l³

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 4: / Line 4: @@
 | image=<[#embed [hash E8W2T39SHBA6TBWV9MR9AY87GA] [width 160]]>
 | elements=14, 26, 14
+| sym=[[Biprismatic symmetry|D<sub>2h</sub>]]
 | genus=0
 | extra={{STS Polytope
 | flayout={{FLD|a2tr|ti|asrtm|ditm|aqtm|editm|atlsl|cr}}
+| bowers=Bilbiro
 }}}}
 The '''bilunabirotunda''' is the 91st [[Johnson solid]], J91. It suddenly became important when the [[castellated rhodoperihedral prism]] was discovered.
+== Coordinates ==
+The following coordinates give an origin-centered bilunabirotunda with edge length 2:
+:<±1, 0, ±φ<sup>2</sup>>
+:<±φ, ±1, ±1>
+:<0, ±φ, 0>
+where φ=(1+√5)/2 is the Golden Ratio.
+== Construction from icosahedron ==
+The bilunabirotunda can be constructed from an icosahedron, as follows:
+Firstly, write the icosahedron in [2,2,2]-symmetry:
+    x2o2f
+    f2x2o
+    o2f2x
+Then, apply a caleido-faceting to the last node:
+    x2o2f -> x2o2f
+    f2x2o -> f2x2o
+    o2f2x -> o2f2(-x)
+Finally, apply a Stott-expansion to the last node:
+    x2o2f    -> x2o2F
+    f2x2o    -> f2x2x
+    o2f2(-x) -> o2f2o
+The result is J91, written in [2,2,2]-symmetry: xfo2oxf2Fxo&#zx.
+You can compare this representation with the coordinates as given above.
 == Equations ==
@@ Line 15: / Line 52: @@
 surface area = (2 + 2√3 + √(25+10√5)) {{DotHV}}<br>
 volume = {{Over|1|6}} &middot; (4φ<sup>2</sup> + 5φ) {{DotHV|3}}</blockquote>
+<[#polytope [id 66]]>
 {{Trishapes}}