The yellow figure is a rhombic dodecahedron, and the red pieces form six rhombi which intersect the faces of the yellow figure. There are also hypershort red struts connecting the red rhombi to each other. It’s not exactly a polyhedron, but I had fun making it. I built it using Zome, which you can buy for yourself at http://www.zometool.com.
Here’s a Zome rhombicosidodecahedron, made up of white Zomeballs and short blue struts.
If you replace each of the rhombicosidodecahedron’s thirty squares with golden rectangles in a certain way, by replacing certain short struts with medium struts along the triangle/rectangle edges, you get a Zomeball made of Zome — what some have called a “metazomeball.” It has enlarged triangles, compared to the pentagons.
It’s almost possible to augment each of the Zomeball’s 62 faces with all-blue pyramids. Here’s the attempt.
The Zomeball’s twelve pentagons are augmented with pyramids, using medium struts, so that each of these pyramids’ lateral faces is a golden isosceles triangle. The twenty triangles are also augmented by pyramids, with golden isosceles triangles as lateral faces. This requires the use of long blue struts. As for the augmentation performed on the thirty golden rectangles, that’s a bit more complicated.
Here’s what the golden rectangles are augmented with, with the new pieces all being short blue struts. It isn’t quite a pyramid, nor is it quite a prism. Its lateral faces are two equilateral triangles and two golden isosceles trapezoids. It’s a five-faced polyhedron in need of a good name. Until someone comes up with a better name, then, I’m going to call it a golden pentahedron.
Here’s a close-up of a golden isosceles trapezoid, by itself. It’s made of three short struts and one medium strut.
It’s also possible to do the first change in another way. When the rhombicosidodecahedron has its squares replaced with golden rectangles, rotate the golden rectangles 90 degrees, such that it increases the size of the pentagons, rather than the triangles. This has been called an “antizomeball.”
With the antizomeball, augmentation with pyramids is possible with all 62 faces.
Here’s what the three types of pyramid used in the augmentation of the antizomeball look like, separated from the main structure.
With the antizomeball, the pentagonal pyramids’ bases are made of medium struts, and long struts make the five lateral faces into golden isosceles triangles. The triangular faces of the antizomeball are made of short struts, so medium lateral edges for these pyramids make their lateral faces golden isosceles triangles. The golden rectangles in the antizomeball are different from the ones in the Zomeball, and the changes make it possible to augment the antizomeball’s golden rectangles using four medium lateral edges, forming two golden isosceles triangles and two equilateral triangles as the lateral faces of a true pyramid.
If you’d like to try Zome for yourself, which I strongly recommend, the site to visit is http://www.zometool.com.
The blue figure in the center of this model is the compound of five cubes. If you take a cube, and build pyramids of just the right height on each of that cube’s faces, those pyramids form a rhombic dodecahedron, as seen below.
In the model at the top of this post, yellow rhombic dodecahedra have been built around each cube in the compound of five cubes. The yellow figure in the top is, therefore, the compound of five rhombic dodecahedra.
I made these models out of Zome. If you’d like to try Zome for yourself, the place to go to buy it is http://www.zometool.com.
The polyhedral compound above contains an icosidodecahedron (blue) and a rhombic triacontahedron (red). In this compound, the icosidodecahedron’s edges are bisected, while the rhombic triacontahedron’s edges are split into segments with lengths in the square of the golden ratio (~2.618 to 1).
If you want Zome of your own, the place to buy it is http://www.zometool.com.
The polyhedron above is called the rhombic triacontahedron, one of the Catalan solids. Its thirty faces are each golden rhombi — rhombi with diagonals in the golden ratio.
This yellow polyhedron is called the rhombic enneacontahedron. It has ninety faces — sixty wide rhombi, and thirty narrow rhombi.
This third polyhedron is called the rhombic hexecontahedron, and its faces are sixty golden rhombi. It is the 26th stellation of the rhombic triacontahedron. It can also be viewed as an assemblage of twenty golden parallelopipeds, each meeting at the exact center of the polyhedron. A single golden parallelopiped is shown below, and it resembles a cube that has had too much to drink, causing it to lean over.
These four rhombic polyhedra were all constructed from Zome. If you’d like to have some Zome of your own, the website to visit is http://www.zometool.com.
Zome can be rearranged in trillions of ways. Here are three of them.
I found these yesterday in Facebook’s “10 years ago” memories it likes to give me every day. If you want to get your own Zome, the website to visit is http://www.zometool.com.
Zome is a ball-and-stick modeling system which can be used to make millions of different polyhedra. If you’d like to get some Zome for yourself, just visit http://www.zometool.com.
I’ve been using Zometools, available at http://www.zometool.com, to build interesting geometrical shapes since long before I started this blog. I recently found this: a 2011 photograph of myself, holding a twisting Zome torus. While I don’t remember who was holding the camera, I do remember that the torus is made of adjacent parallelopipeds.
After building this torus, I imagined it as an accretion disk surrounding a neutron star — and now I am imagining it as a neutron star on the verge of gaining enough mass, from the accretion disk, to become a black hole. Such an object would emit intense jets of high-energy radiation in opposite directions, along the rotational axis of this neutron star. These jets of radiation are perpendicular to the plane in which the rotation takes place, and these two opposite directions are made visible in this manner, below, as two dodecahedra pointing out, on opposite sides of the torus — at least if my model is held at just the right angle, relative to the direction the camera is pointing, as shown below, to create an illusion of perpendicularity. The two photographs were taken on the same day.
In reality, of course, these jets of radiation would be much narrower than this photograph suggests, and the accretion disk would be flatter and wider. When one of the radiation jets from such neutron stars just happens to periodically point at us, often at thousands of times per second, we call such rapidly-rotating objects pulsars. Fortunately for us, there are no pulsars near Earth.
It would take an extremely long time for a black hole to form, from a neutron star, in this manner. This is because most of the incoming mass and energy (mostly mass, from the accretion disk) leaves this thermodynamic system as outgoing mass and energy (mostly energy, in the radiation jets), mass and energy being equivalent via the most famous formula in all of science: E = mc².
This is one projection of the four-dimensional hyperdodecahedron, or 120-cell, rendered in Zome. All the part for this come in a single kit, and, if you want it for yourself, you can find it for sale at this website.
I did have student help with the construction of this, for which I am grateful. However, for legal and ethical reasons, I cannot credit the students by name.
Here’s a closer view, through the “core” of all-blue pentagons:
Zome is a great product. I recommend it strongly, and without reservation (and no, they aren’t paying me anything to write this).
RobertLovesPi.net 