Snub disphenoid (EntityTopic, 14)

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(link to long and involved derivation)
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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:
-
:2u<sup>3</sup> + 11u<sup>2</sup> + 4u - 1 = 0, 2 u 3;
+
:2u<sup>3</sup> + 11u<sup>2</sup> + 4u - 1 = 0, 2 < u < 3;
-
:v<sup>3</sup> - 17v<sup>2</sup> + 64v - 64 = 0, 0 v 1;
+
:v<sup>3</sup> - 17v<sup>2</sup> + 64v - 64 = 0, 0 < v < 1;
-
:2w<sup>3</sup> - w<sup>2</sup> - 8w - 4 = 0, 1 w 2.
+
:2w<sup>3</sup> - w<sup>2</sup> - 8w - 4 = 0, 1 < w < 2.
''See also:'' [[Derivation of snub disphenoid coordinates]]
''See also:'' [[Derivation of snub disphenoid coordinates]]

Revision as of 07:09, 13 March 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 + 11u2 + 4u - 1 = 0, 2 < u < 3;
v3 - 17v2 + 64v - 64 = 0, 0 < v < 1;
2w3 - w2 - 8w - 4 = 0, 1 < w < 2.

See also: Derivation of snub disphenoid coordinates

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