Convex Triamond Regular Polyhedra

This project is an attempt to enumerate all convex solids with regular faces that also contain Triamonds. A Triamond is three equilateral triangles which are coplanar. The outer perimeter of the three triangles is a 1:1:1:2 trapezoid. These solids will always contain at least two Triamonds because the side of length 2 must be adjacent to another size 2 edge of a Triamond. This in effect forms solids with hexagons which are folded along a midline diagonal. In these solids, all other regular polygons must be of unit size and no other coplanar surfaces are allowed including that of two Triamonds.

As of January 16th, 2007 there have been discovered 40 such solids. None of these are chiral. If you find a polyhedra that is not currently in this list please E-mail me at polyhedra@bigfoot.com and I will list it with proper credit.

The Convex Triamond Polyhedra can either be made by standard means (augmentation or truncation) or by a process of stretching. Some can be made both ways or by a combination of the two processes. Some can only be made by stretching. There are instances such that a regular polyhedra can be stretched into more than one Convex Triamond Solid.

Many of the Convex Triamond Polyhedra can be built from component parts. For example those that are built on "TAPS" or Triamond Antiprisms.

There is a commentary box under the polyhedra name to note alternative methods of construction. The number of faces of each type are listed - Triamonds, Triangle, Square, Pentagon, Hexagon, Octagon and Decagon, as well as the total Face, Edge and Vertex counts. A (C) denotes that solid is chiral (none yet found).

The Octahedron and Icosahedron can be stretched into Convex Triamond Solids. The Triamond Stretched Octahedron can also be made by augmenting the Stretched J1 Wedge with itself. The Stretched Icosahedron has been modeled by Alex Doskey and can also be described as a J91 augmentation. The Stretched Icosahedron can also be made up of a TAP 5,5 and two Stretched J2 Wedges.

Platonics
Triamond Stretched Octahedron - R.K. (2)Octahedra+(2)Tetrahedra T 3 4 5 6 8 10 4 4 * * * * * F=8       E=14       V=8 stel     wrl     switch	Triamond Stretched Icosahedron - A.D. J91+(4)J2 T 3 4 5 6 8 10 4 16 2 * * * * F=22       E=36       V=16 stel     wrl     switch

Truncated Dipyramids, or Bifrusta. can be made by augmenting the Triamond Frusta together (see component parts).

Triamond Bifrusta
Triamond Triangular Bifrustum - R.K. (2)Octahedra+(6)Tetrahedra or Truncated Triangular Dipyrmid (J12) T 3 4 5 6 8 10 6 2 * * * * * F=8       E=15       V=9 stel     wrl     switch	Triamond Square Bifrustum - R.K. (4)Octahedra+(8)Tetrahedra+(2)J1 or Truncated Octahedron T 3 4 5 6 8 10 8 * 2 * * * * F=10       E=20       V=12 stel     wrl     switch	Triamond Pentagonal Bifrustum - R.A. Truncated Pentagonal Dipyramid (J13) T 3 4 5 6 8 10 10 * * 2 * * * F=12       E=25       V=15 stel     wrl     switch

The Triamond Cupolas (see component parts) can be self augmented to form Bicupolas. When the Bilateral Triamond Cupola is self augmented it results in a Hexagonal Prism with is regular.

Bicupolas
Triamond Triangular Bicupola - R.K. (6)Octahedra+(6)J1+(20)Tetrahedra or Stretched J27 T 3 4 5 6 8 10 6 * 6 * 2 * * F=14       E=30       V=18 stel     wrl     switch	Triamond Square Bicupola - R.K. Stretched J28 T 3 4 5 6 8 10 8 * 8 * * 2 * F=18       E=40       V=24 stel     wrl     switch	Triamond Pentagonal Bicupola - R.K. Stretched J29 T 3 4 5 6 8 10 10 * 10 * * * 2 F=22       E=50       V=30 stel     wrl     switch

The Bilateral Triamond Cupola can be augmented with the Stretched J1 Wedge. It can then be augmented with J1's for three variations. Three of these were discovered by Jim McNeill to be stretched Johnson Solids. One cannot be made by stretching.

Bilateral Triamond Cupola Augmentations
Bilateral Triamond Cupola+ Stretched J1 Wedge - R.K. (3)Triangular Prism+(2)J1+Tet or Stretched J49 T 3 4 5 6 8 10 4 2 3 * * * * F=9       E=17       V=10 stel     wrl     switch	Bilateral Triamond Cupola+ Stretched J1 Wedge+J1,1 - J.M. (3)Triangular Prism+(3)J1+Tet or Stretched J50 T 3 4 5 6 8 10 4 6 2 * * * * F=12       E=21       V=11 stel     wrl     switch	Bilateral Triamond Cupola+ Stretched J1 Wedge+J1,2 - R.K. (3)Triangular Prism+(3)J1+Tet T 3 4 5 6 8 10 4 6 2 * * * * F=12       E=21       V=11 stel     wrl     switch	Bilateral Triamond Cupola+ Stretched J1 Wedge+(2)J1 - R.K. (3)Triangular Prism+(4)J1+Tet or Stretched J51 T 3 4 5 6 8 10 4 10 1 * * * * F=15       E=25       V=12 stel     wrl     switch

The Rotundas (see component parts) can be augmented into Convex Triamond Solids.

Rotunda Augmentations
Pentagonal Triamond Rotunda,1+ (Self) - J.M. No Decomposition T 3 4 5 6 8 10 10 * * 12 10 * * F=32       E=80       V=50 stel     wrl     switch	Pentagonal Triamond Rotunda,1+ Triamond Pentagonal Bicupola - J.M. No Decomposition T 3 4 5 6 8 10 10 * 5 6 5 * 1 F=27       E=65       V=40 stel     wrl     switch	Pentagonal Triamond Rotunda,2+ Triamond Pentagonal Bicupola - J.M. No Decomposition T 3 4 5 6 8 10 10 * 5 6 10 * 1 F=32       E=80       V=50 stel     wrl     switch

The Tap 4,1 and Tap 5,1 of the Triamond Antiprisms of Order 1 can be augmented into Convex Triamond forms. The TAP 2,1 can be augmented into a Truncated Tetrahedron which is a regular solid. The TAP 3,1 can be augmented in the a Truncated Octahedron which is also a regular solid.

It is noteworthy that the Gyroelongated Triamond Pentagonal Bicupola is an alternative way to stretch the Icosahedron.

Augmentations of Triamond Antiprisms Order 1
Gyroelongated Triamond Square Bicupola - R.K. Stretched J17 T 3 4 5 6 8 10 16 * 8 * * 2 * F=26       E=56       V=32 stel     wrl     switch	Gyroelongated Triamond Pentagonal Bicupola - R.K. Also: Stretched Icosahedron T 3 4 5 6 8 10 20 * 10 * * * 2 F=32       E=70       V=40 stel     wrl     switch

The Triamond Prisms of Order 2 can be augmented into numerous forms. The TAP 3,2 cannot be augmented into a Convex Triamond Solid, but it can be augmented into the Truncated Tetrahedron which is a convex regular solid.

While the TAP 2,2 is a standalone Convex Triamond Solid, it can also be augmented to a copy of itself for another one. This new solid can also be described as a stretched J12.

Augmentations of the Triamond Antiprism TAP 2,2
TAP 2,2 - R.K. J1+(2)Tetrahedra T 3 4 5 6 8 10 2 2 1 * * * * F=5       E=9       V=6 stel     wrl     switch	TAP 2,2+(Self) - R.K. (2)J1+(4)Tetrahedra or Stretched J12 T 3 4 5 6 8 10 2 4 2 * * * * F=8       E=14       V=8 stel     wrl     switch

The TAP 4,2 can be augmented into 3 forms.

Augmentations of the Triamond Antiprism TAP 4,2
TAP 4,2+ Square Triamond Cupola - R.K. Stretched J10 T 3 4 5 6 8 10 8 4 1 * * 1 * F=14       E=28       V=16 stel     wrl     switch	TAP 4,2+Square Triamond Cupola+ J4,1 - R.K. Stretched J17 T 3 4 5 6 8 10 8 8 6 * * * * F=22       E=40       V=20 stel     wrl     switch	TAP 4,2+Square Triamond Cupola+ J4,2 - R.K. No Decomposition T 3 4 5 6 8 10 8 8 6 * * * * F=22       E=40       V=20 stel     wrl     switch

The TAP 5,2 can be augmented into 4 forms. TAP 5,2+Pentagonal Triamond Cupola+J5,2 is an alternative way of stretching an Icosahedron.

Augmentations of the Triamond Antiprism TAP 5,2
TAP 5,2+ Pentagonal Triamond Cupola - R.K. J6+(5)J2 or Stretched J11 T 3 4 5 6 8 10 10 5 * 1 * * 1 F=17       E=35       V=20 stel     wrl     switch	TAP 5,2+Pentagonal Triamond Cupola+ J5,1 - R.K. No Decomposition T 3 4 5 6 8 10 10 10 5 2 * * * F=27       E=50       V=25 stel     wrl     switch	TAP 5,2+Pentagonal Triamond Cupola+ J5,2 - R.K. Also: Stretched Icosahedron T 3 4 5 6 8 10 10 10 5 2 * * * F=27       E=50       V=25 stel     wrl     switch	TAP 5,2+Pentagonal Triamond Cupola+ J6,1 - R.K. No Decomposition T 3 4 5 6 8 10 10 15 * 7 * * * F=32       E=60       V=30 stel     wrl     switch

Some Convex Triamond Solids can be found by augmentation and truncation of J91 using J2's. When Four Triangular Pyramids (J2) are augmented to J91 the result is the Stretched Icosahedron displayed above (See Platonics).

J91 Augmentations
Triamond J91+(2)J2 - R.K. Also: Stretched J63+J2 Wedge T 3 4 5 6 8 10 2 12 2 2 * * * F=18       E=31       V=15 stel     wrl     switch	Triamond J91+(4)J2-(1)J2 - R.K. Also: Stretched J11 T 3 4 5 6 8 10 4 11 2 1 * * * F=18       E=31       V=15 stel     wrl     switch	Triamond J91+(4)J2-(2)J2,1 - R.K. Also: Stretched J62 T 3 4 5 6 8 10 4 6 2 2 * * * F=14       E=26       V=14 stel     wrl     switch	Triamond J91+(4)J2-(2)J2,2 - R.K. Also: Stretched Pentagonal Antiprism T 3 4 5 6 8 10 4 6 2 2 * * * F=14       E=26       V=14 stel     wrl     switch

Some Convex Triamond Solids can only be derived by stretching of Johnson Solids

Stretched Johnson Solids
Triamond Stretched J13 - R.K. (2)Stretched J2 Wedges T 3 4 5 6 8 10 2 8 2 * * * * F=12       E=20       V=10 stel     wrl     switch	Triamond Stretched J17 - J.M. No Decomposition T 3 4 5 6 8 10 2 12 4 * * * * F=18       E=30       V=14 stel     wrl     switch	Triamond Stretched J84 - J.M. No Decomposition T 3 4 5 6 8 10 2 10 3 * * * * F=15       E=25       V=12 stel     wrl     switch	Triamond Stretched J86 - J.M. No Decomposition T 3 4 5 6 8 10 2 10 5 * * * * F=17       E=29       V=14 stel     wrl     switch	Triamond Stretched J88 - J.M. No Decomposition T 3 4 5 6 8 10 2 14 7 * * * * F=23       E=39       V=18 stel     wrl     switch

An additional stretched Johnson Solids was found by Jim McNeill

Stretched Johnson Solids
Triamond Stretched J51 - J.M. No Decomposition T 3 4 5 6 8 10 6 8 * * * * * F=14       E=24       V=12 stel     wrl     switch

Some Convex Triamond Solids can be derived from sections of Johnson Solids J3 and J92 can be sectioned on a seam cutting the central hexagons bilaterally. The J92 section can be self augmented to create a new Convex Triamond Solid. The shape of J92 section cut is the same as that of the Stretched J2 Wedge thus another one can be created by capping.

Sectioned Johnson Solids
Triamond J3 Section - R.K. No Decomposition T 3 4 5 6 8 10 2 2 2 * * * * F=6       E=11       V=7 stel     wrl     switch	Triamond J92 Section+(Self) - R.K. No Decomposition T 3 4 5 6 8 10 2 10 4 2 * * * F=18       E=32       V=16 stel     wrl     switch	Triamond J92 Section+ Stretched J2 Wedge - R.K. No Decomposition T 3 4 5 6 8 10 2 9 3 1 * * * F=15       E=26       V=13 stel     wrl     switch

Try as we might there is still at least one example which can't be made from any of the components, stretching, truncating or sectioning. The Misc1 model was found using JovoToys.

Miscellaneous
Misc1 - R.K. No Decomposition T 3 4 5 6 8 10 2 12 * * * * * F=14       E=22       V=10 stel     wrl     switch

Note: The polyhedra on this page were catalogued/discovered by Alex Doskey, Robert Austin, and Roger Kaufman as identified by initials listed after the model's name.
J.M. = Jim McNeill   A.D. = Alex Doskey   R.A. = Robert Austin   R.K. = Roger Kaufman.

Generation of VRML models was expedited by the use of Robert Webb's Stella application, and the .Stel files available above to Stella users. The Hedron application by Jim McNeill was used extensively for model creation and to generate switch files.

I'd also like to thank Alex Doskey for his spreadsheet method which made the construction of this page much easier. I also use JovoToys in polyhedra contruction.

History:

2007-01-17 Initial Release
2007-01-16 Beta
2007-01-06 Alpha

Roger's Polyhedra, (c) 2006, Roger Kaufman