2b:null 2c:["$","title",null,{"children":"Five Tetrahedra and Blue-Green Zome | Scott Vorthmann | Observable"}] 31:T8a1,md` [Zome][zometool] models don't typically last long for me, usually being cannibalized for parts within a couple of months. This model, however, has persisted for at least eight years, for two main reasons. On one hand, it is a very nice object, a compound of five interlinked tetrahedra using the vertices of a dodecahedron, my two favorite polyhedra. Perhaps more significantly, the blue-green struts are nearly useless for anything else. High aesthetic value of the whole combines with low practical value of the parts, ensuring the model's continued survival. The design comes from the book [Zome Geometry][book], by George Hart and Henri Picciotto. George designed the model originally, and he has a [picture of it][george] on his website. As the book points out, "it is just barely possible and holds itself together very snugly." The model is serendipitous: if the struts were a tiny bit shorter, the model would be impossible to build. If they were a tiny bit longer, the model would be unpleasantly loose and wobbly. So why do I say these perfectly-sized struts are "nearly useless"? This takes us into the age-old tension between the purist and the pragmatist. Sometime before 2001, Zometool began manufacturing these blue-green struts (called "gb1" struts) as an extension to the existing Zome system, for the express purpose of allowing the construction of certain [Archimedean solids][arch] that were otherwise inaccurate when constructed with the usual green struts. Here is an example of one of them, the great rhombicuboctahedron, as realized in vZome. *(This is not an image; you can explore the model by rotating, panning, and zooming with your mouse.)* The vZome model file can be [downloaded][model]; I'll explain below how you can use vZome to approximate blue-green struts in this way. [zometool]: https://zometool.com [book]: https://www.amazon.com/Zome-Geometry-Hands-Learning-Models/dp/1559533854 [george]: http://www.georgehart.com/zomebook/additions.html [arch]: http://mathworld.wolfram.com/ArchimedeanSolid.html [model]: https://gist.githubusercontent.com/vorth/51c7216a7d9f57e790de87b4e17cce8c/raw/1201cb8f473ea490156bdf231843f8394d553572/great-rhombicuboctahedron.vZome `32:T1583,md` Standard Zometool green struts are designed to fit the diagonal of a square made from blue struts, so they are longer then the corresponding blue struts, scaled by a factor of √2. Using green struts, three of the Archimedean solids have rectangles where they should have squares, or octagons with two different side lengths. The solution was to create struts that travel in the "green" direction, but have the length of a blue strut. Using these struts, the rectangles now become squares again, and the octagons become regular. Zometool colored these blue-green, to avoid confusion with either blue or green struts, while demonstrating the affinity to both. The desire to model the Archimedean solids correctly is surely a purist's argument. Making that a requirement for selling Zometool kits is equally a pragmatic conclusion, ironically. Those arguments together must have been very compelling -- the cost of creating tooling for injection molding is quite prohibitive. Sadly, when the Zometool factory was [flooded in 2013][flood], the injection molds for the blue-green struts were lost, so they are not made any more. [flood]: https://www.youtube.com/watch?v=1HShXhJMQdA As I heard it, weighing against the Archimedean argument was an opposing purist viewpoint, involving the overall consistency of the Zome system from a mathematical perspective, which we will now explore. One of the beautiful aspects of the Zome system is the remarkable serendipity that seems to pervade it. So often when constructing a free-hand model, connections "just work". One way to explain this is to say that a legitimate (no bending or twisting) Zome model connects a set of balls that can all be assigned coordinates in a very particular subset of ℝ³, using only integers and the golden ratio. This vector space is embedded "densely" in ℝ³ in roughly the same way that the rational numbers, ℚ, are embedded in the real numbers, ℝ. The blue-green struts spoil this ideal world. Adding a single blue-green strut to a Zome model results in a ball whose coordinates are no longer in that same vector space. Instead, it lives in a different vector space, where the coordinates contain terms involving √2 in addition to the golden ratio. In a sense, the blue-green strut bridges between two parallel Zome universes; once in the second universe, the only way back is to follow a path that includes additional blue-green struts in very specific symmetric configurations (stopping at additional universes along the way). Any other connection between the universes is illegitimate: a cheat, an approximation, like pretending that π is equal to 22/7. This admittedly purist analysis has a completely pragmatic side: if you start modeling with blue-green struts mixed in with other Zome struts, unless you follow very stringent rules, you'll quickly find that you've lost that "just works" character of Zome building. Stuff won't connect up, to put it simply. Those three problematic Archimedean solids, when modeled with blue-green struts, *do* follow those stringent rules; every blue-green strut has one or more parallel siblings, and there are three-cycles around corners. The models contain a minimum of six otherwise-disjoint parallel Zome universes, symmetrically arranged. Incidentally, on the subject of symmetry, there is another reason to question the importance of those Archimedean solids, again from a purist perspective. As my friend [Brian Hall][bhall] pointed out to me, the Archimedean condition (that the faces of a polyhedron are all regular) is not very interesting mathematically, nor very useful in teaching mathematics. Pedagogically, polyhedra have more value in illustrating *symmetry groups*. In that context, a pentagonal face on a polyhedron should indicate an element of order five in the symmetry group, and so on. It turns out that exactly those faces that required blue-green struts are the ones that are "lying" to us: the squares do *not* indicate four-fold rotational symmetry in the polyhedron, nor do the regular octagons indicate eight-fold rotational symmetry. The versions using regular green struts are more honest, in this regard. ## Building the vZome Model Switching back to the pragmatic viewpoint, I imagine that you might like to use vZome to model the Archimedean solids. Many of them can be modeled in Zometool (and thus vZome) directly, so I won't talk about them. Two of them, the snub dodecahedron and the snub cube, cannot be modeled even in vZome without some approximations that I have not worked out. The last three, the same ones that prompted the creation of blue-green struts, can be modeled in two different ways. The easiest way is to model them using √2 support provided in vZome. When you create a new document using the menus, you have the option of selecting several different "fields", and one of them is √2. In this field, since all struts can be scaled by √2, green and blue struts can naturally have the same length. I'm not going to go into specifics; if you want to get an idea of how to build the model shown below, [download the vZome file][root2] and use undo/redo to see how it was constructed. *(This is not an image; you can explore the model by rotating, panning, and zooming with your mouse.)* [bhall]: https://www3.nd.edu/~bhall/book/lie.htm [root2]: https://gist.githubusercontent.com/vorth/51c7216a7d9f57e790de87b4e17cce8c/raw/e25b0d2dfd25722cc1301f7f476fdd90849ea514/great-rhombi-root2.vZome `33:T86b,md` Note that the ball and strut shapes in this model are non-standard, necessarily; the √2 field cannot implement icosahedral symmetry, nor even render a golden rectangle. If you want to model blue-green struts faithful to the actual Zometool parts, you'll need to stay in the "golden" field. To illustrate how to do this, I'll model the compound of five tetrahedra that opened this article. First, you'll need the vertices of a dodecahedron, but at just the right size. Since we want the green struts to be shorter than usual by a factor of 1/√2, we'll use a yellow strut scaled the same way. Select the yellow dot in the direction panel triangle. To construct a custom length strut, you first need to show lengths, by checking the "Show Strut Scales" menu item: ![](http://vzome.com/blog/wp-content/uploads/2018/02/system-menu.png) This will cause the strut size panel to show extra information at the bottom, representing the strut size algebraically: ![](http://vzome.com/blog/wp-content/uploads/2018/02/show-lengths.png) Click on the "custom" button, to bring up the custom length dialog. Enter the values "-9" and "6" in the first two fields, as shown, and click on the "Set" button.

This approximation to 1/√2 is due to Brian Hall, and is accurate to about one part in a thousand. Now you're ready to create your yellow strut. Create it, then select the ball at the end:

Next, apply icosahedral symmetry to create all twenty vertices of the dodecahedron. Now, find a square among these vertices and construct a diagonal across it:

Apply icosahedral symmetry, then set the color of the struts while they are all selected, if you like, and remove the yellow strut and center ball, yielding the completed model. *(This is not an image; you can explore the model by rotating, panning, and zooming with your mouse.)* `34:T575,md` If you examine this model carefully, you'll see a little more clearance between the struts than exists in the physical model, but that is due to the way vZome has modeled the green strut shapes; they are a bit narrower than in real life. ## Conclusion I'll admit that I was never a fan of the blue-green struts, favoring the purist argument regarding the mathematics of Zome. That said, I'm happy to keep my model of the compound of five tetrahedra, and take comfort from the one Zome model in my life that I can consider permanent. Incidentally, George Hart informed me that he also has his original model, now twenty years old! If you are a purist of the Archimedean ilk, dismayed by the disappearance of blue-green Zometool struts, you can take heart in this era of cheap 3D printing. I have modeled [many custom Zome parts][shapeways], and I would happily model the blue-green struts if someone wants them. After all, they have the shape of green struts, which are already modeled. I can even go one better, and model the struts that are required for the final two Archimedean solids, the snub dodecahedron and snub cube, so you can finally build a complete set. Please feel free to [contact me](mailto:scott@vorthmann.org) if you are interested; I never charge anything for model design, or any markup on the Shapeways printing. [shapeways]: http://www.shapeways.com/shops/vzome `35:[] 2f:{"id":"aa555e428dd6ffca","slug":"five-tetrahedra","title":"Five Tetrahedra and Blue-Green Zome","thumbnail":"7a4eb3b0e8d7b7f3f70cee50600249775ca36e50801f162662f5da077640160a","access_level":"public","create_time":"$D2019-05-25T23:32:20.344Z","update_time":"$D2025-01-29T04:30:38.324Z","view_time":null,"first_public_time":"$D2019-06-01T16:31:37.723Z","fork":true,"stars":4,"starred":false,"owner":{"type":"team","id":"bd8d3517f25bb386","login":"vorth","name":"Scott Vorthmann","avatar_url":"https://avatars-next.observableusercontent.com/2f38be1791bd361c7d78e3163c0ba8ff19e87d0419eb72a6e479dacc410deb9a","tier":"starter_2024"},"version":3189,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"c80882137d4e42a5","login":"vorth","name":"Scott Vorthmann","avatar_url":"https://avatars-next.observableusercontent.com/2f38be1791bd361c7d78e3163c0ba8ff19e87d0419eb72a6e479dacc410deb9a","tier":null}],"members":[{"type":"individual","id":"c80882137d4e42a5","login":"vorth","name":"Scott Vorthmann","avatar_url":"https://avatars-next.observableusercontent.com/2f38be1791bd361c7d78e3163c0ba8ff19e87d0419eb72a6e479dacc410deb9a","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":2,"collection":{"id":"c6c505abda9887a4","slug":"vzome","title":"vZome","owner":{"id":"bd8d3517f25bb386","login":"vorth","name":"Scott Vorthmann"}},"type":"notebook","body":{"title":"Five Tetrahedra and Blue-Green Zome","cells":[{"id":2999,"value":"md`\n# Five Tetrahedra and Blue-Green Zome\n\n***The Purist Versus the Pragmatist, some Zometool History, and Other Topics.***\n`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3016,"value":"html`\n

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