Polyhedra

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Among all the things I have in my office you may notice the sextant, celestial globe, portable sundial, astrolabe and mariner's astrolabe on the shelf and the poster of Holbein's Ambassadors on the door.

Making polyhedral models 120-cell Runci-truncated 120-cell Links

Making polyhedral models

I love making polyhedral models with plastic building kits like Zome System, Polydron, JOVO (US distirbutor: Jovo Click 'n Construct) or simply paper.

The simplest way to make polyhedral models is to use paper. You can print out the nets from Paper Models of Polyhedra. (It is best to use nets witht flaps. And it is an interesting question to figure out how many flaps you need.)

One of the purposes of this page is to discuss the relative merits of the different plastic kits.

Appearance wise, there are two main types of plastic models, solid models and frame models. Zome is frame, Jovo and Polydron are solid, but Polydron has a compatible frame version called Frameworks. However, Frameworks is not edge based like Zome, but is face based with hollow faces.

Which is more interesting? Solids models can be used to illustrate how to color the faces. How many colors do you need to color an icosahedron? Frame models, however, can be used for making soap bubbles, and for seeing the structure more transparently. I like both types.

From the point of view of construction, face based models (Polydron, Frameworks and Jovo) are quicker to make than edge based models (Zome) since a polyhedron has more edges than faces, so if you want to put together some models quickly, face based models have an advantage.

Some simple edge based systems have edges that can be bent, so any kind of structure can be built. Zome, however, has thick edges that can only be pointed in certain directions. Originally, Zome only had blue, red and yellow struts, and could not be used for making exact tetrahedra or octahedra. However, with the introduction of the green struts, all the Platonic and Archimedean solids, except for the snub cube and snub dodecahedron can be made with Zome. Unfortunately, the green strut can be hard to work with at first.

Polydron contains triangles, square, pentagons, hexagons and octagons (with holes!). That means that it can be used to make all the Platonic and Archimedean solids except the truncated dodecahedron and the great rhombicosidodecahedron. Frameworks contains the same except for the octagons. The compatibility between Polydron and Frameworks is a nice feature.

Jovo only contains triangles, squares and pentagons. However, they also offer a set called Jovo System II that contains among other things hexagons.

Jovo is smaller, which sometimes makes it easier to work with. However, it also creates restriction. It is impossible to make a pentagonal pyramid with Jovo. The five triangles won't bend down low enough to meet.

Jovo comes with a very handy key that makes it easier to put in the last pieces.

All the three systems are good. However, Zome has one advantage. The book Zome Geometry: Hands on Learning With Zome Models by George W. Hart and Henri Picciotto is a masterpiece, and means that you will never run out of things to do with Zome.

My student, Kavitha d/o Krishnan, and I have made some polyhedral models with Zome System. The big model to the right consists of a dodecahedron, a small stellated dodecahedron, a great dodecahedron and a great stellated dodecahedron. In her right hand she holds a small stellated dodecahedron, and in her left hand she holds a great dodecahedron with a small stellated dodecahedron inside. On the table you will see models of all the five Platonic solids and a small copy of the great stellated dodecahedron.

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Here she is showing two compounds of five tetrahedra. They are mirror images of each other. We made them by first building a dodecahedron. These are in fact stellations of the icosahedron. By merging these, we would get a compound of ten tetrahedra, but it's not possible to make that using Zome System, because there are not enough holes in the ball.

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Kavitha has also made some very nice paper models. On the left you see a left and a right snub cube. On the right you see a beautiful model of the great dodecahedron and truncated icosahedron. Do you realize that it has the same shape as the soccer ball?

120-cell

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We finally dared to go for the 120-cell. This is a stereographic projection of a regular polytope in R4. For a simple description, you can start with Viewing Four-dimensional Objects In Three Dimensions from The Geometry Center, The Fourth Dimension by John Savard or 4D Polytope Projection Models by 3D Printing by George W. Hart. For more details you can read Hyperspace Tutorial by Eric Swab. For a detailed mathematical description, look at The Story of the 120-cell by John Stillwell, taken from the Notices of the American Mathematical Society.

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Partially finished. In the beginning and the end of the building, it's not too difficult to move the model around if you're VERY careful. But there is an intermediate stage where it is very hard, because the supporting surface curves in and is very small. Our solution was to use a chair. I put books on the seat of the chair, and in that way I was able to support the model at four points: seat, 2 arm rests and back.

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We made it!

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Runci-truncated 120-cell

At the 2005 Art and Math Conference in Boulder, Colorado, I was part of a team making a beautiful Zometool model of the runci-truncated 120-cell. It is a 3D shadow of a uniform four-dimensional polytope consisting of 120 truncated dodecahedra, 600 cuboctahedra, 1200 triangular prisms, and 720 decagonal prisms. For more details and pictures you can go to George Hart's webpage.

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Links


Helmer Aslaksen
Department of Mathematics
National University of Singapore
helmer.aslaksen@gmail.com

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