Wikia

Geomag Wiki

Augmented Truncated Dodecahedron

Talk1
152pages on
this wiki

Augmented Truncated Dodecahedron -- Discussion Edit

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

Augmented Truncated Dodecahedron
(Cupolae Square-to-Square)
Augmented Truncated Dodecahedron S2S V1L
Truncated Dodecahedron augmented with outward Pentagonal Cupolae, aligned Square-to-Square
TypePolyhedron
Panels 80 × Geomag triangle panel icon
60 × Geomag square panel icon
12 × Geomag pentagon panel icon
Rods 270 × Geomag rod icon
Spheres 120 × Geomag ball icon
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

 

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

Augmented Truncated Dodecahedron
(Cupolae Triangle-to-Triangle)
Augmented Truncated Dodecahedron T2T V1L
Truncated Dodecahedron augmented with outward Pentagonal Cupolae, aligned Square-to-Square
TypePolyhedron
Panels 80 × Geomag triangle panel icon
60 × Geomag square panel icon
12 × Geomag pentagon panel icon
Rods 270 × Geomag rod icon
Spheres 120 × Geomag ball icon
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

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)
Left view
Truncated Dodecahedron (Right View)
Right View

 

Augmented Truncated Dodecahedron (Cupolae Square-to-Square) Edit

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
Left view
Augmented Truncated Dodecahedron S2S V1R
Right View

 

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

 

Augmented Truncated Dodecahedron (Cupolae Triangle-to-Triangle) Edit

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)

 

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
Left view
Augmented Truncated Dodecahedron T2T V1R
Right View

 

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

 

Noncoplanarity of Triangle Pairs in the CT2T Model Edit

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

 

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

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.

 

Advertisement | Your ad here

Around Wikia's network

Random Wiki