Using the Rhombic Dodecahedron and the Rhombic Enneacontahedron to Create a “Near Near-Miss” to the Johnson Solids

This is the rhombic dodecahedron, the dual of the Archimedean cuboctahedron.

Rhombic Dodeca

While the rhombic dodecahedron has 12 faces, there are many other polyhedra made entirely out of rhombi, and most of them have more than twelve faces. An example is the rhombic enneacontahedron, which has two face-types: sixty wide rhombi, and thirty narrow ones. It is one of several possible zonohedrified dodecahedra.

Zonohedrified Dodeca

As the next figure shows, the wide rhombi of the rhombic enneacontahedron have exactly the same shape as the rhombic dodecahedron’s faces, so the two polyhedra can be stuck together (augmented) at those faces. These wide rhombi have diagonals with lengths in a ratio of one to the square root of two.

Zonohedrified Dodeca with RD

The next picture shows what happens if you take one central rhombic enneacontahedron, and augment all sixty of its wide faces with rhombic dodecahedra.

Augmented Zonohedrified Dodeca

Since this polyhedral cluster in non-convex, it can be changed by creating its convex hull, which can the thought of as pulling a rubber sheet tightly around the entire polyhedron. Here’s the convex hull of the augmented polyhedron above.

Convex hull of RD-augmented REC

The program I use to make these rotating images, Stella 4d (which you can try here), has a function called “try to make faces regular.” If applied to the convex hull above, this function leaves the triangles and pentagon regular, and makes the octagons regular as well. However, the rhombi become kites. The rectangles merely change, getting slightly longer, while rotating 90º, but they do remain rectangles.

convex hull of RD-augmented REC after TTMFRegular worked on the octagons

After creating this last polyhedron, I started stellating it. After stellating it eight times, I obtained this polyhedron:

8th

Once more, I applied the “try to make faces regular” function.

Unnamed after TTMFR

This polyhedron has five-valent vertices where the shorter edges of the kites meet. These are also the vertices of pentagonal pyramids which use kite-diagonals as base edges. By using faceting (the inverse operation of stellation), I next removed these pyramids, exposing their regular pentagonal faces.

Faceted Poly

In this polyhedron, all faces are regular, except for the red triangles, which are isosceles. (There are also triangles — the pink ones — which are regular.) Each of these isosceles triangles has ~63.2º base angles, and a ~53.6º vertex angle, with legs just under 11% longer than the base. This is a judgement call, for “near-miss” to the Johnson solids has not been precisely defined, but I see an ~11% edge-length difference as too great for this to be classified as a “near-miss,” even though I would love to claim discovery of another near-miss to the Johnson solids (if it even turns out I am the first one to find this polyhedron, which may not be the case). It is close to being a near-miss, though, so it belongs in the even-less-precisely defined group of polyhedra which are called, quite informally, “near near-misses.”

For the sake of comparison, here is a similar polyhedron (included in Stella 4d‘s enormous, built-in library of polyhedra) which is recognized as a near-miss to the Johnson solids. (I do not know the name of the person who discovered it, or I would include it here — I only know it wasn’t me.) It’s called the “half-truncated truncated icosahedron,” and its longest ledges are just over 7% longer than its shorter edges, with the non-regularity of faces also limited to isosceles triangles. However, this irregularity appears in all of the triangles in the polyhedron below — and in the “near near-miss above,” the irregularity only appeared is some of the triangular faces.

Half-trunc Trunc Icosa

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. The majority of these things are geometrical. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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7 Responses to Using the Rhombic Dodecahedron and the Rhombic Enneacontahedron to Create a “Near Near-Miss” to the Johnson Solids

  1. swo8 says:

    They are really neat!
    Leslie

    Liked by 1 person

  2. Tom R says:

    I call the last one, the dual of the rhombic enneacontahedron, a “rectified truncated icosahedron”,with Coxeter/Johnson name rtI, and Conway name atI.
    See wikipedia:
    https://en.wikipedia.org/wiki/Rectified_truncated_icosahedron

    Liked by 1 person

    • I also like the name you found for the next-to-last one, the “rectified chamfered dodecahedron.” That also answered my “did I find it first or didn’t I?” question. Clearly, I didn’t.

      Like

  3. Billy Ray says:

    I love the 3D images you use. Can you tell me what software you use to create it? I am a newly hired high school Geometry teacher that needs such visual aids when teaching solids.

    Thanks

    Liked by 1 person

  4. I think I have maybe discovered this before you sometime in 2013, I posted to this forum: https://groups.google.com/forum/#!topic/geodesichelp/jtzXKv5hMmg

    I later found a simple way to produce it and posted that here: http://geo-dome.co.uk/article.asp?uname=Regular_subdivision I think this was in sept 2014

    The one at the bottom is a rectified truncated icosahedron, to make this in stella you just truncate (50%) a truncated icosahedron (which is a 33.333% truncation of a icosahedron) I called the new near-near miss a rectified truncated triacontahedron, which is possible in stella but does not produce regular hexagons. I do hope I got there first as I would love to have naming rights.

    Liked by 1 person

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