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.
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.
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 PolyhedraSmith@gmail.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).
Links below display various models
- Clicking on pictures opens an Interactive OFF Viewer
- OFF files open in an online browser. Downloaded files may be viewed with an application such as Antiprism
- STEL files may be saved and opened with Stella
- VRML files (wrl,switch,cyl) open in an online browser. Downloaded files may be viewed with a VRML browser such as FreeWRL
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 |  | | off 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 |  | | off 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 Tetrahedron | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 6 | 2 | * | * | * | * | * | | F=8 E=15 V=9 |  | | off stel wrl switch |
|
| Triamond Square Bifrustum - R.K. | (4)Octahedra+(8)Tetrahedra+(2)J1
or Truncated Square Pyramid (J1) | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 8 | * | 2 | * | * | * | * | | F=10 E=20 V=12 |  | | off stel wrl switch |
|
| Triamond Pentagonal Bifrustum - R.A. | Truncated Pentagonal Pyramid (J2)
| | T | 3 | 4 | 5 | 6 | 8 | 10 | | 10 | * | * | 2 | * | * | * | | F=12 E=25 V=15 |  | | off stel wrl switch |
|
The Triamond Cupolas, Stretched J3, Stretched J4, and Stretched J5 (see component parts) can be self augmented to form Bicupolas. When the Triamond Bilateral 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 |  | | off 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 |  | | off 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 |  | | off stel wrl switch |
|
The Triamond Bilateral 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.
| Triamond Bilateral Cupola Augmentations |
|---|
Triamond Bilateral 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 |  | | off stel wrl switch |
|
Triamond Bilateral 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 |  | | off stel wrl switch |
|
Triamond Bilateral 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 |  | | off stel wrl switch |
|
Triamond Bilateral 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 | J1.png) | | off stel wrl switch |
|
The Rotundas Triamond Pentagonal Rotunda,1 and Triamond Pentagonal Rotunda,2 (see component parts) can be augmented into Convex Triamond Solids.
| Rotunda Augmentations |
|---|
Triamond Pentagonal Rotunda,1+
(Self) - J.M. | | No Decomposition | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 10 | * | * | 12 | 10 | * | * | | F=32 E=80 V=50 | .png) | | off stel wrl switch |
|
Triamond Pentagonal Rotunda,1+
Triamond Pentagonal Cupola - J.M. | | Truncated U25 | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 10 | * | 5 | 6 | 5 | * | 1 | | F=27 E=65 V=40 |  | | off stel wrl switch |
|
Triamond Pentagonal Rotunda,2+
Triamond Pentagonal Cupola - J.M. | | Truncated U25 | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 10 | * | 5 | 6 | 10 | * | 1 | | F=32 E=80 V=50 |  | | off 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 using Triamond Cupolas (see component parts). 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. The TAP 6,1 when augmented with the Triamond Hexagonal Frustum has (coplanar faces).
It is noteworthy that the Triamond Gyroelongated Pentagonal Bicupola is an alternative way to stretch the Icosahedron.
| Augmentations of Triamond Antiprisms Order 1 |
|---|
Triamond Gyroelongated
Square Bicupola - R.K. | | Stretched J17 | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 16 | * | 8 | * | * | 2 | * | | F=26 E=56 V=32 |  | | off stel wrl switch |
|
Triamond Gyroelongated
Pentagonal Bicupola - R.K. | | Also: Stretched Icosahedron | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 20 | * | 10 | * | * | * | 2 | | F=32 E=70 V=40 |  | | off stel wrl switch |
|
The Triamond Antiprisms 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+(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 | .png) | | off stel wrl switch |
|
The TAP 4,2 can be augmented into 3 forms using the Triamond Square Frustum.
| Augmentations of the Triamond Antiprism TAP 4,2 |
|---|
TAP 4,2+
Triamond Square Frustum - R.K. | | Stretched J10 | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 8 | 4 | 1 | * | * | 1 | * | | F=14 E=28 V=16 |  | | off stel wrl switch |
|
TAP 4,2+
Triamond Square Frustum+J4,1 - R.K. | | Stretched J17 | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 8 | 8 | 6 | * | * | * | * | | F=22 E=40 V=20 |  | | off stel wrl switch |
|
TAP 4,2+
Triamond Square Frustum+J4,2 - R.K. | | No Decomposition | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 8 | 8 | 6 | * | * | * | * | | F=22 E=40 V=20 |  | | off stel wrl switch |
|
The TAP 5,2 can be augmented into 4 forms using Triamond Cupolas (see component parts). TAP 5,2+Triamond Pentagonal Cupola+J6 has only one form since when J6 is twisted it results in a model with coplanar triangles. TAP 5,2+Triamond Pentagonal Cupola+J5,2 is an alternative way of stretching an Icosahedron.
| Augmentations of the Triamond Antiprism TAP 5,2 |
|---|
TAP 5,2+Triamond Pentagonal 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 |  | | off stel wrl switch |
|
TAP 5,2+Triamond Pentagonal Cupola+
J5,1 - R.K. | | No Decomposition | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 10 | 10 | 5 | 2 | * | * | * | | F=27 E=50 V=25 |  | | off stel wrl switch |
|
TAP 5,2+Triamond Pentagonal 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 |  | | off stel wrl switch |
|
TAP 5,2+Triamond Pentagonal Cupola+
J6 - R.K. | | No Decomposition | | T | 3 | 4 | 5 | 6 | 8 | 10 | | 10 | 15 | * | 7 | * | * | * | | F=32 E=60 V=30 |  | | off 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 |  | | off 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 |  | | off 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 |  | | off 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 |  | | off 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 |  | | off 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 |  | | off stel wrl switch |
|
Some Convex Triamond Solids can be found by augmentation and truncation of J91 using J2's. When Four Pentagonal Pyramids (J2) are augmented to J91 the result is the Stretched Icosahedron (See Platonics above).
| 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 | J2.png) | | off 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 | J2-(1)J2.png) | | off 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 | J2-(2)J2,1.png) | | off 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 | J2-(2)J2,2.png) | | off 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 hexagon bilaterally. The J3 section forms two Triamonds. 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 |  | | off 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 | .png) | | off 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 |  | | off 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. If the Triamonds on this model are kept as coplanar triangles, the 18 faced deltahedron is produced.
Question or comments about the web page should be directed to PolyhedraSmith@gmail.com.
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:
2024-06-17 Use base.css
2024-06-13 Switch to Multi OFF Viewer
2023-09-05 Switch to Simple OFF Viewer and X_Ite VRML Viewer
2023-03-16 Standardized namings and fixed many errors
2023-03-11 Put underlines in file names
2023-02-23 Click on pictures goes to live model
2019-03-12 Changed email address from defunct bigfoot.com
2013-05-10 Added note about Misc1 model
2012-02-22 Change to index.html
2007-01-17 Initial Release
2007-01-16 Beta
2007-01-06 Alpha
Back to the main Polyhedron Page.
Link to this page as http://www.interocitors.com/polyhedra/Triamonds
