Convex Triamond Regular Polyhedra Components

Convex Triamond Regular Polyhedra Components

Roger's Polyhedra, (c) 2006-2024, Roger Kaufman

Many Convex Triamond Solids contain the following components. These components contain either irregular polygons of an 1:2:1:2: ... edge size ratio, or polygons of both edge size 1 and edge size 2. While these components are not Convex Triamond Solids themselves it is noteworthy to list them.

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A Tetrahedron, Square Pyramid (J1), or Pentgonal Pyramid (J2) can be truncated into Triamond Frusta. A Hexagonal Pyramid with regular faces is flat thus a Triamond Hexagonal Frustum is flat.

The Triamond Triangular Frustum is a section shown in this Truncated Tetrahedron. The Triamond Square Frustum is a section shown in this Truncated Octahedron. The Triamond Pentagonal Frustum is a section shown in this Truncated Icosahedron.

Triamond Frusta

There are four Cupolas consisting of alternating Triamond and squares radiating around a central polygon.

The Bilateral Triamond Cupola is a section shown in this Hexagonal Prism. The Triangular Triamond Cupola is a section shown in this Truncated Octahedron. The Square Triamond Cupola is a section shown in this Truncated Cuboctahedron. The Pentagonal Triamond Cupola is a section shown in this Truncated Icosidodecahedron.

Triamond Cupolas

There are two components that can be classed as wedges. The Stretched J1 Wedge is a section shown in this Truncated Tetrahedron.

There is a special wedge, the Stretched J2 Wedge. This wedge turns up on a few Convex Triamond solids. It contains a hexagon with edges of ratio 1:1:1:1:1:2. Its internal angles are 108°-108°-108°-108°-144°-144°. It is interesting that 144° is the internal angle of a decagon. This wedge will cover the case when stretching a pentagon bilaterally into a hexagon.

Wedges

There are two figures containing Triamonds which can be considered Rotundas. Both are derived of a Truncated Icosahedron and if the two are joined make up the entire Truncated Icosahedron.

Rotundas

Regular Antiprisms can be stretched in an infinte number of infinite sets of Triamond Antiprisms. The faces which are stretch into Triamonds are the band of triangles on the perimeter of the antiprism. As a side effect the two parallel polygons of edge count N most often become irregular. In cases where not every triangle is stretched, the two N edged polygons that are also stretched become no longer an equal number of edges.

The number of ways one N-Antiprism can be stretched is based on the factors of 2N. An 8-Antiprism for instance, has factors of 16 which are 1, 2 ,4 and 8. Then an 8-Antiprism can be stretched such that every triangle becomes a Triamond, or ever second triangle becomes a Triamond, every Fourth, or finally every Eighth.

There are at least three Triamond Antiprisms for every regular N-Antiprism, since 2N can be divisible by at least 1, 2 or N. Because of the multitude of Triamond Antiprisms, a shorthand naming method must be devised. The shorthand for Triamond Antiprism can be TAP N,M where N is the polygon edge count for the regular antiprism, and M is the frequency of triangles around the edge band which are stretch. Then, for instance, the 4 Triamond Antiprisms that can be derived from and 8-Antiprism would be TAP 8,1 , TAP_8,2 , TAP_8,4 , and TAP 8,8.

Triamond Antiprisms start at N=2 instead of N=3 as in regular antiprisms. This occurs because if each face of a tetrahedron is stretched into a Triamond, a Triamond Antiprism (TAP 2,1) results.

Every regular N-Antiprism can be formed into a Triamond Antiprism by stretching every triangular face around the outer band. These prisms always have 2N faces around the perimeter and the major faces have an edge length pattern of 1:2:1:2...

The TAP 2,1 is a section shown in this Truncated Tetrahedron. The TAP 3,1 is a section shown in this Truncated Octahedron. TAP 5,1 is a section shown in this Truncated Icosahedron. The TAP 4,1 does not occur in any regular solid. The TAP 6,1 is the highest of the order which can contain only Triamonds and regular unit polygons but the cupolas covering it are flat.

Triamond Antiprisms of Order 1

Every N-Antiprism can become a Triamond Antiprism by expanding every other triangle to a Triamond. In this set one major face becomes a regular polygon of edge count 2N while the other major face remains a regular polygon of edge count N. The N face will be of edge size 2. In the case of the TAP 2,2 , N equals 2 so one face is reduced to a line and is eliminated. The TAP 2,2 is actually a valid Convex Triamond Solid.

The TAP 2,2 section shown in this Tetrahedron. Actually Tap 2,2 is itself a Stretched Tetrahedron. The TAP 3,2 is a section shown in this Truncated Tetrahedron. The TAP 6,2 is the highest order which can contain only Triamonds and regular unit polygons but the Triamond Hexagonal Frustrum covering it is flat.

Triamond Antiprisms of Order 2

Miscellaneous examples of TAP's of higher orders. One artifact that was found, the TAP 4,4, is in the Stretched J17 Johnson Solid. Another one that was found in the Convex Triamond Solid study is the curious TAP 5,5. It can be found as a section of the Stretched Icosahedron. Three other TAPs are modeled as examples, none of which have yet been found any Convex Triamond Solids.

Miscellaneous Triamond Antiprisms

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

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Triamond Triangular Frustum

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