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.
There’s a special rhombus which is called the “golden rhombus,” because its diagonals are in the golden ratio. To construct it with compass and straight edge, you first construct a golden rectangle (shown with blue edges and a yellow interior), and then connect the midpoints of its sides to form a rhombus (with edges shown in red).
Several polyhedra can be made which use golden rhombi as their faces. The most well-known of these polyhedra is the rhombic triacontahedron, which has 30 such faces. It is the dual of the icosidodecahedron.
If the rhombic triacontahedron is stellated 26 times, the result is the (non-convex) rhombic hexecontahedron. It has 60 golden rhombi as faces.
Both of these polyhedra can be constructed with Zometools (available at http://www.zometool.com). With white Zomeballs and red Zomestruts, these polyhedra look a lot like this:
The flat image at the top of this post was created using Geometer’s Sketchpad and MS-Paint. The four rotating polyhedral images were created using Stella 4d: Polyhedron Navigator, which you can purchase, or try for free, at http://www.software3d.com/Stella.php.
You can buy your own Zome at 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).
Ordinarily, with Zometools, the compound of five cubes is an all-blue model. However, I wanted to build one in which each cube is a different color, so I made a special request to the Zometool Corporation (their website: http://www.zometool.com) for some off-color parts, to make this possible.
The five colors used in this model are standard blue, a darker shade of blue, red, yellow, and black.
I also received the struts needed to build this model in white, so I will be making a second version of this soon. I didn’t want the Zomeballs used to match any strut color, though, so I will have to wait for the shipment of purple Zomeballs I ordered, today, to arrive, before I can build that model.
Zome is a fantastic tool to use for mathematical investigations, as well as education, and other applications as well. I recommend this product highly, and without reservation.
The President of the Zometool Corporation, Carlos Neumann, gave me a challenge, not long ago: find a solution to the Zome Cryptocube puzzle which uses only B0s, which I call “tiny blue struts.” For the Cryptocube puzzle, though, these “blue” struts actually appear white. Carlos knows me well, and knows I cannot resist a challenge involving Zome. Here is what I came up with, before the removal of the black cube, which is what the Zome Cryptocube puzzle starts with.
In a “pure” Crypocube solution, the red Zomeballs would also be white — not just the “blue” struts. However, when Carlos issued this challenge, I was at home, with all the white Zomeballs I own located at the school where I teach — so I used red Zomeballs, instead, since I had them at home, and did not wish to wait.
Here’s what this Cryptocube solution looks like, without the black cube’s black struts. You can still “see” the black cube, though, for the black Zomeballs which are the eight corners of the black cube are still present. As is happens, this particular Cryptocube solution has pyritohedral symmetry — better known as the symmetry of a standard volleyball.
While the Cryptocube puzzle is not currently available on the Zome website, http://www.zometool.com, it should be there soon — hopefully, in time for this excellent Zome kit to be bought as a Christmas present. Once a child is old enough so that small parts present no choking hazard, that child is old enough to start playing with Zome — and it is my firm belief that such play stimulates the intellectual growth of both children and adults. As far as a maximum age where Zome is an appropriate Christmas gift, the answer to that is simple: there isn’t one.
Also: while I do openly advertise Zome, I do not get paid to do so. I do this unpaid advertising for one reason: I firmly believe that Zome is a fantastic product, especially for those interested in mathematics, or for those who wish to develop an interest in mathematics — especially geometry. Also, Zome is fun!
This is my second solution to the Zome Cryptocube puzzle. In this puzzle, you start with a black cube, build a white, symmetrical, aethetically-pleasing geometrical structure which incorporates it, and then, finally, remove the cube. In addition, I added a rule of my own, this time around: I wanted a solution which is chiral; that is, it exists in left- and right-handed forms.
It took a long time, but I finally found such a chiral solution, one with tetrahedral symmetry. Above, it appears with the original black cube; below, you can see what it looks like without the black cube’s edges.
Last month, in a special Christmas promotion, the Zometool company (www.zometool.com) briefly sold a new kit (which will return later) — a fascinating game, or puzzle, called the “Cryptocube.” Zome usually comes in a variety of colors, with each color having mathematical significance, but the Cryptocube is produced in black and white, which actually (in my opinion) makes it a better puzzle. Here’s how the Crypocube challenge works: you use the black parts to make a simple cube, and then use the smaller white parts to invent a structure which incorporates the cube, is symmetrical, is attractive, and can survive having the twelve black cube-edges removed, leaving only the cube’s eight black vertices in place. I had a lot of fun making my first Cryptocube, and photographed it from several angles.
If this was built using standard Zome colors, the round white figure inside the cube, a rhombic triacontahedron, would be red, and the pieces outside the cube, as well as those joining the rhombic triacontahedron to the cube (from inside the cube), would be yellow.
It isn’t only humans who like Zome, by the way. Jynx the Kitten had to get in on this!
Jynx quickly became distracted from the Cryptocube by another puzzle, though: he wanted to figure out how to pull down the red sheet I had attached to the wall, as a photographic backdrop for the Cryptocube. Jynx takes his feline duties as an agent of entropy quite seriously.
As usually happens, Jynx won (in his never-ending struggle to interfere with whatever I’m doing, in this case by pulling the sheet down) and it took me quite a while to get the red sheet back up, in order to take kitten-free pictures of my Cryptocube solution, after removal of the black cube’s edges.
Here’s the view from another angle.
The Cryptocube will be back, available on the Zometool website, later in 2015. In the meantime, I have advice for anyone not yet familiar with Zome, but who wants to try the Cryptocube when it returns: go ahead and get some Zome now, at the link above, in the standard colors (red, blue, and yellow, plus green in advanced kits), and have fun building things with it over the next few months. The reason to do this, before attempting to solve the Crypocube, is simple: the colors help you learn how the Zome system works, which is important before trying to solve a Zome puzzle without these colors visible. After gaining some familiarity with the differing shapes of the red, blue, yellow, and green pieces, working with them in white becomes much easier.
On a related note, Zome was recommended by Time magazine, using the words “Zometool will make your kids smarter,” as one of the 14 best toys of 2013. I give Zome my own strong, personal recommendation as well, and, as a teacher who uses my own Zome collection in class, for instructional purposes, I can attest that Time‘s 2013 statement about Zome is absolutely correct. Zome is definitely a winner!
For years, I have used Zometools (sold here: http://www.zometool.com) to teach geometry. The constructions for the icosahedron and dodecahedron are easy to teach and learn, due to the use of short reds (R1s) and medium yellows (Y2s) for radii for the two of them, as shown below, with short blue (B1) struts as edges for both polyhedra.
Unexpectedly, a student (name withheld for ethical and legal reasons) combined the two models, making this:
I saw it, and wondered if the two combined Platonic solids could be expanded along the edges, to stellate both polyhedra, with medium blues (B2s), to form the great and small stellated dodecahedron. By trying it, I found out that this would require intersecting blue struts — so a Zomeball needed to be there, at the intersection. Trying, however, only told me that no available combination would fit. After several more attempts, I doubled each edge length, and added some stabilizing tiny reds (R0s), and found a combination that would work, to form a compound of the great and small stellated dodecahedron in which both edge lengths would be equal. In the standard (non-stellated) compound of the icosahedron and dodecahedron, in which the edges are perpendicular, they are unequal in length, and in the golden ratio, which is how that compound differs from the figure shown directly above.
Here’s the stabilized icosahedral core, after the doubling of the edge length:
This enabled stellation of each shape by edge-extension. Each edge had a length twice as long as a B2 added to each side — and it turns out, I discovered, that 2B2 in Zome equals B3 + B0, giving the golden ratio as one of three solutions solution to x² + 1/x = 2x (the others are one, and the golden ratio’s reciprocal). After edge-stellation to each component of the icosahedron/dodecahedron quasi-compound, this is what the end product looked like. This required assembling the model below at home, where all these pictures were taken, for one simple reason: this thing is too wide to fit through the door of my classroom, or into my car.
Here’s a close-up of the central region, as well.