General Deltahedron

Polyhedron (homeomorphic to a sphere) consisting of equilateral triangles only. If we define the valency of a sphere as the number of rods connected to it, and write the number of spheres with valencies 3, 4, 5, 6, ..., as n₃,n₄,n₅,n₆,... , this provides a rudimentary classification: (Note that there may be several shapes with the same signature)

(n₃,n₄,n₅,n₆,...)-deltahedron.

(go until the largest valency in your deltahedron)

This deltahedron then contains (n₃+n₄+n₅+n₆+...) spheres, and (3.n₃+4.n₄+5.n₅+6.n₆+...)/2 rods.

Also, using Euler's formula, it is easy to show that

$3\,n_3+2\,n_4+n_5-n_7-2\,n_8-3\,n_9-4\,n_{10}-\cdots -(k-6)\,n_k-\cdots=12$

Related Links

• Mathworld

Models

(0,0,12,10)-deltahedron ' Type Deltahedron Rods 60 × Spheres 22 × Author Amafirlian 23:00, 1 July 2007 (UTC)

(0,0,12,16)-deltahedron ' Type Deltahedron Rods 78 × Spheres 28 × Author Amafirlian 10:44, 12 July 2007 (UTC)

(0,0,12,17)-deltahedron ' Type Deltahedron Rods 81 × Spheres 29 × Author Amafirlian 18:43, 1 July 2007 (UTC)

(0,0,12,20)-deltahedron ' Type Deltahedron Rods 90 × Spheres 32 × Author Amafirlian 18:45, 1 July 2007 (UTC)

(0,0,12,42)-deltahedron ' Type Deltahedron Rods 156 × Spheres 54 × Author Amafirlian 14:16, 21 July 2007 (UTC)

(0,0,12,45)-deltahedron ' Type Deltahedron Rods 165 × Spheres 57 × Author Amafirlian 13:30, 15 July 2007 (UTC)

(0,0,12,8)-deltahedron ' Type Deltahedron Rods 54 × Spheres 20 × Author Amafirlian 12:25, 12 July 2007 (UTC)

(0,0,18,32,6)-deltahedron ' Type Deltahedron Rods 162 × Spheres 56 × Author Amafirlian 08:14, 22 July 2007 (UTC)

(0,12,0,36,6,3)-deltahedron ' Type Deltahedron Rods 165 × Spheres 57 × Author Amafirlian 23:41, 16 March 2008 (UTC)

(0,18,0,80,24)-deltahedron ' Type Deltahedron Rods 360 × Spheres 122 × Author Amafirlian 15:42, 26 July 2007 (UTC)

(0,24,12,0,0,0,8,6)-deltahedron Wedge-deltified rhombic dodecahedron Type Deltahedron Rods 144 × Spheres 50 × Author Amafirlian 20:13, 2 August 2007 (UTC)

128-hedron (0,6,60,0)-deltahedron Type Deltahedron Rods 192 × Spheres 66 × Author Amafirlian 00:25, 10 June 2007 (UTC)

Creased Platonic Deltahedra ' Type Polyhedron Rods 3F(3p2+3pf+3p2)/2 × Spheres 2+Rods/3 × Author --Leo Dorst 07:45, 30 August 2008 (UTC)

Diamond lattice cell ' Type Deltahedron? Rods (420Y+120G+96R+120B) × Spheres 224 × Author --Leo Dorst 17:24, 11 September 2008 (UTC) Diamond lattice cell ' Type Deltahedron? Rods (144Y+96R+48B) × Spheres (72+3) × Author --Leo Dorst 06:27, 30 August 2008 (UTC)

Extended Elongated Hexagonal Antiprism (0,0,12,32)-deltahedron Type Deltahedron Rods 126 × Spheres 44 × Author Amafirlian 21:36, 21 June 2007 (UTC) after input from Peter Jepsen

Lobel Frames

Pentagonal Deltahedron (0,0,12,45)-deltahedron Type Deltahedron Panels 110 × Rods 165 × Spheres 57 × Author Amafirlian 22:22, 27 June 2007 (UTC)

Pentagonal Torus ' Type Deltahedron Rods 120 × Spheres 40 × Author Amafirlian 08:29, 15 August 2007 (UTC)

Snub Disphenoid Johnson solid J84 Type Johnson Solid Rods 18 × Spheres 8 × Author Amafirlian 18:04, 23 July 2007 (UTC)

(0,0,12,24)-deltahedron ' Type Deltahedron Rods 102 × Spheres 36 × Author Amafirlian 23:12, 1 July 2007 (UTC)

Snub Expanded (0,0,12,17)-deltahedron (0,0,12,38)-deltahedron Type Rods 144 × Spheres 50 × Author Amafirlian 13:16, 10 July 2007 (UTC)

Snub Expanded Snub Disphenoid (0,4,4,36)-deltahedron Type Deltahedron Rods 126 × Spheres 44 × Author Amafirlian 19:55, 23 July 2007 (UTC)

Spiral Deltahedron (0,0,12,30)-deltahedron Type Deltahedron Panels 23 × Rods 120 × Spheres 42 × Author Amafirlian 08:44, 23 June 2007 (UTC)

Spiral sphere ' Type Polyhedron Rods 123+63 × Spheres 64 × Author --Leo Dorst 15:10, 14 June 2008 (UTC)L.Dorst

3m-fold Torus ' Type Polyhedron:Torus Rods 180 × Spheres 108 × Author Leo Dorst 06:58, 24 May 2008 (UTC) 5m-fold Torus ' Type Polyhedron:Torus Rods 300 × Spheres 180 × Author Leo Dorst 06:58, 24 May 2008 (UTC) Small Torus ' Type Polyhedron:Torus Rods 108 × Spheres 36 × Author Leo Dorst 05:59, 6 September 2008 (UTC) Squared Torus ' Type Polyhedron:Torus Rods 198 × Spheres 66 × Author --Leo Dorst 19:26, 20 September 2008 (UTC) Tiny Torus ' Type Polyhedron:Torus Rods 72 × Spheres 24 × Author ----Leo Dorst 19:26, 21 September 2008 (UTC)