Augmented Truncated Dodecahedron -- Discussion[edit | edit source]

There are two (regular) ways of augmenting a Truncated Dodecahedron with pentagonal cupolae: cupola pairs connected square-to-square (CS2S) and connected triangle-to-triangle (CT2T).

The Truncated Dodecahedron may also be augmented with pentagonal rotundae in two (regular) ways: rotunda pairs connected pentagon-to-pentagon (RP2P) and connected triangle-to-triangle (RT2T), producing polyhedra that are far from spherical. These models do not appear in this entry, because I haven't tried to build them myself (yet).

Augmented Truncated Dodecahedron (Cupolae Square-to-Square)[edit | edit source]

Augmented Truncated Dodecahedron
(Cupolae Square-to-Square)
Augmented Truncated Dodecahedron S2S V1L .jpg
Truncated Dodecahedron augmented with outward Pentagonal Cupolae, aligned Square-to-Square
Type Polyhedron
Panels 80 × Geomag triangle panel icon.gif
60 × Geomag square panel icon.gif
12 × Geomag pentagon panel icon.gif
Rods 270 × Geomag rod icon.gif
Spheres 120 × Geomag ball icon.gif
Author PolyClare 03:06, 20 January 2009 (UTC)


To the right is a Truncated Dodecahedron augmented with Pentagonal Cupolae. In this case, all the pentagons are aligned so that a sqaure from one cupola is connected to a square from its neighboring cupola.

It is interesting to note that in this model, pairs of connected squares are coplanar, forming rectangles of size 1×2 units. Additionally, blocks of four trangles are also coplanar, forming equilateral triangles of edge length 2 units. The overall result is that this model may be viewed as a stretched (no longer regular) Rhombicosidodecahedron, shown for ease of comparison below:

Rhombicosidodecahedron
Rhombicosidodecahedron.jpg

 

Note that augmenting the Truncated Dodecahedron with even one pentagonal cupola aligned square-to-dodecahedron ends up with cupola triangles that are coplanar with the Truncated Dodecahedron triangles. Therefore there are no Truncated Dodecahedra augmented with pentagonal cupolae oriented this way among the 92 Johnson Solids.


 

Augmented Truncated Dodecahedron (Cupolae Triangle-to-Triangle)[edit | edit source]

Augmented Truncated Dodecahedron
(Cupolae Triangle-to-Triangle)
Augmented Truncated Dodecahedron T2T V1L .jpg
Truncated Dodecahedron augmented with outward Pentagonal Cupolae, aligned Square-to-Square
Type Polyhedron
Panels 80 × Geomag triangle panel icon.gif
60 × Geomag square panel icon.gif
12 × Geomag pentagon panel icon.gif
Rods 270 × Geomag rod icon.gif
Spheres 120 × Geomag ball icon.gif
Author PolyClare 03:06, 20 January 2009 (UTC)


To the right is a Truncated Dodecahedron augmented with Pentagonal Cupolae. In this case, all the pentagons are aligned so that a triangle from one cupola is connected to a square from its neighboring cupola.

It is interesting to note that this model is nonconvex, since pairs of triangles from connected cupolae are NOT coplanar. However, these triangle pairs are very close to being coplanar (see further discussion and photos below). The overall result is that using Geomag one may carefully replace pairs of triangles with Rhombic Panels, resulting in an intriguing near miss described in the entry entitled Pseudosphere From Geomag Box.

Note that augmenting the basic Truncated Dodecahedron with just one pentagonal cupola, aligned triangle-to-dodecahedron ends up with a convex polyhedron. It is the triangle-to-triangle pairs of multiple adjacent pentagonal cupolae that make the Augmented Truncated Dodecahedron CS2S nonconvex. This is why there are only four Truncated Dodecahedron-based polyhedra listed among the 92 Johnson Solids: J68 "augmented truncated dodecahedron" (one cupola), J69 "parabiaugmented truncated dodecahedron" (two cupolae, opposite ends), J70 "metabiaugmented truncated dodecahedron" (two nonadjacent cupolae, separated by one decagon), and J71 "triaugmented truncated dodecahedron" (three nonadjacent cupolae). These are the four ways to attach pentagonal cupolae to a Truncated Dodecahedron without any pentagonal cupola being adjacent to another.


 

Model Gallery and Discussion[edit | edit source]

First, for comparison purposes, displayed below is the Truncated Dodecahedron which has been "diminshed" by pentagonal Cupolae aligned Square-to-Square (CS2S):

Truncated Dodecahedron Diminished by Pentagonal Cupolae, CS2S
Truncated Dodecahedron (Left View) .jpg
Left view
Truncated Dodecahedron (Right View) .jpg
Right View

 

Augmented Truncated Dodecahedron (Cupolae Square-to-Square)[edit | edit source]

Below are two wall-eyed stereo pairs of the Augmented Truncated Dodecahedron (CS2S). You can see that this model is the same as the previous one above, except that the pentagonal cupolae point outward instead of inward (i.e. "augmented by cupolae" instead of "diminished by cupolae").

Truncated Dodecahedron Augmented by Pentagonal Cupolae, CS2S (View 1)
Augmented Truncated Dodecahedron S2S V1L .jpg
Left view
Augmented Truncated Dodecahedron S2S V1R .jpg
Right View

 

Truncated Dodecahedron Augmented by Pentagonal Cupolae, CS2S (View 2)
Augmented Truncated Dodecahedron S2S V2L .jpg
Left view
Augmented Truncated Dodecahedron S2S V2R .jpg
Right View

 

Augmented Truncated Dodecahedron (Cupolae Triangle-to-Triangle)[edit | edit source]

If you detach the pentagonal cupolae from the previous CS2S model above, and twist and reattach each cupola so that they connect at corresponding triangles (instead of at squares), you get the Augmented Truncated Dodecahedron (Cupolae Triangle-to-Triangle) model. Below is a photo of a model with JUST ONE cupola twisted:

One Twisted Pentagonal Cupola
Augmented Truncated Dodecahedron S2T (1 Twist) .jpg

 

Below are two wall-eyed stereo pairs of the Augmented Truncated Dodecahedron (Cupolae Triangle-to-Triangle).

Truncated Dodecahedron Augmented by Pentagonal Cupolae, CT2T (View 1)
Augmented Truncated Dodecahedron T2T V1L .jpg
Left view
Augmented Truncated Dodecahedron T2T V1R .jpg
Right View

 

Truncated Dodecahedron Augmented by Pentagonal Cupolae, CT2T (View 2)
Augmented Truncated Dodecahedron T2T V2L .jpg
Left view
Augmented Truncated Dodecahedron T2T V2R .jpg
Right View

 

Noncoplanarity of Triangle Pairs in the CT2T Model[edit | edit source]

If you look very close and carefully at the preceding photos of the Cupolae-Triangle-to-Triangle model, you may suspect that the triangle pairs are noncoplanar, resulting in a nonconvex model. See closeup below:

Noncoplanar T2T Triangles: CLOSEUP
Augmented Truncated Dodecahedron T2T with Noncoplanar Triangles .jpg

 

Extracting just the corner triangle from the Truncated Dodecahedron, along with its 9 surrounding polygons (3 adjacent squares attached along one edge and 3 pairs of triangles attached at a vertex), you can see that this figure can be flattened out to a planar form:

Corner Polygons: Planar
Augmented Truncated Dodecahedron - Planar Piece .jpg

If any of the squares are flexed at even a slight angle to the central triangle, the outer triangle pairs will be forced to buckle, producing a figure that cannot be convex. Therefore the Triangle-to-Triangle form of the Augmented Truncated Dodecahedron is nonconvex. This is why the Pseudosphere From Geomag Box is a near miss instead of a working convex model.

 

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