Snub disphenoid (EntityTopic, 14)

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(Difference between revisions)
(Cartesian coordinates: Oops!!! Wrong ranges for the roots!)
(Cartesian coordinates: Ugh, the equations were transposed)
 
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where A=√u, B=√v, C=√w, and u, v, w are roots of the following cubic polynomials:
where A=√u, B=√v, C=√w, and u, v, w are roots of the following cubic polynomials:
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:<table><tr><td style="vertical-align:bottom">2u<sup>3</sup> + 11u<sup>2</sup> + 4u - 1 = 0</td><td style="padding-left:2em; vertical-align:bottom">0 &lt; u &lt; 1</td></tr><tr><td style="vertical-align:bottom">v<sup>3</sup> - 17v<sup>2</sup> + 64v - 64 = 0</td><td style="padding-left:2em; vertical-align:bottom">1 &lt; v &lt; 2</td></tr><tr><td style="vertical-align:bottom">2w<sup>3</sup> - w<sup>2</sup> - 8w - 4 = 0</td><td style="padding-left:2em; vertical-align:bottom">2 &lt; w &lt; 3</td></tr></table>
+
:<table><tr><td style="vertical-align:bottom">2u<sup>3</sup> - u<sup>2</sup> - 8u - 4 = 0</td><td style="padding-left:2em; vertical-align:bottom">2 &lt; u &lt; 3</td></tr><tr><td style="vertical-align:bottom">2v<sup>3</sup> + 11v<sup>2</sup> + 4v - 1 = 0</td><td style="padding-left:2em; vertical-align:bottom">0 &lt; v &lt; 1</td></tr><tr><td style="vertical-align:bottom">w<sup>3</sup> - 17w<sup>2</sup> + 64w - 64 = 0</td><td style="padding-left:2em; vertical-align:bottom">1 &lt; w &lt; 2</td></tr></table>
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The values of A, B, C are approximately:
 +
:A = 1.567861848465127
 +
:B = 0.411123131706519
 +
:C = 1.289168546448310
''See also:'' [[Derivation of snub disphenoid coordinates]]
''See also:'' [[Derivation of snub disphenoid coordinates]]

Latest revision as of 22:20, 14 July 2016

The snub disphenoid is the 84th Johnson solid, J84.

Cartesian coordinates

The Cartesian coordinates of the snub disphenoid, centered on the origin and with edge length 2, are:

(0, A, ±1)
(±C, B, 0)
(0, -B, ±C)
(±1, -A, 0)

where A=√u, B=√v, C=√w, and u, v, w are roots of the following cubic polynomials:

2u3 - u2 - 8u - 4 = 02 < u < 3
2v3 + 11v2 + 4v - 1 = 00 < v < 1
w3 - 17w2 + 64w - 64 = 01 < w < 2

The values of A, B, C are approximately:

A = 1.567861848465127
B = 0.411123131706519
C = 1.289168546448310

See also: Derivation of snub disphenoid coordinates

Images

(image)

Software models

Incidence matrix

Dual: digon-unpinched pentagonal prism

#TXIDVaVbEaEbEcEd3a3bTypeName
0 Va = point ;
1 Vb = point ;
2 Ea 20 = digon ;
3 Eb 11 = digon ;
4 Ec 11 = digon ;
5 Ed 02 = digon ;
6 3a 211110 = triangle ;
7 3b 120021 = triangle ;
8 C1a 44448284 = snub disphenoid ;

Usage as facets

This polytope does not currently appear as facets in any higher-dimensional polytopes in the database.


Notable Trishapes
Regular: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind