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Stretch Polyhedra
Sometimes you just need to use a little mathematical reasoning to come up with some new and interesting models. Inspired by "stretch" limousines, I thought it would be nice to have some "stretch" polyhedra. There are lots of possibilities, and I plan to add a few more examples here when I get a chance.This gallery will grow whenever I find a particularly interesting variation, or someone suggests another application of it that I haven't considered, so submissions and suggestions are always welcome. If you need any help understanding or building one of these models, feel free to e-mail me.
Alex@JovoToys.com
Since the standard JOVO set includes triangles, squares and pentagons, the figures below are shown using just those shapes (using all six colors of JOVO - red, yellow, green, blue, black and white). If a hexagon is needed, you can substitute six triangles. Some intermediate stages contain irregular shapes, but others can be built using JovoToys. The final results should all be constructable with JovoToys.
Let's start with a Stretch-Icosahedron. For step 1, color a regular icosahedron, so that you have 6 pairs of triangles in one color, and eight single triangles (in the planes of an octahedron) in another color:
The pairs of triangles will be situated with their shared edge facing either up, down, left, right, front or back. Step 2, is to slide a copy of the icosahedron along one pair of these edges, in this case, the front and back edges.
We cannot build this model, because there are some overlapping pieces in the "valley" between the two copies. But if we replace the valley with a triangle, square, triangle, we end up with a single direction stretched icosahedron:
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| Front View | Top View |
Already this is pretty cool, because we can repeat this as many times as we want, making super-stretch polyhedra:
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| Front View | Top View |
But we can also stretch in other directions as well. If we look at the top view, we can see the two original top triangles (in red) that are now separated by a square. If we were to slide a copy of our current model along that direction, and again fill in the valleys with traingles and squares:
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| Slide | Fill Valleys |
In the end, once we have done this one more time on the last remaining orthogonal pair of triangles, we should get this:
Below is a table with the Icosahedral models and their Stretched counterparts. In many cases, there were stretched faces that were not regular, so I substituted, so that I would end up with only regular-faced models (and so I could build them with JovoToys). Click on the pictures to see a VRML model, and if you have Great Stella version 3.0 or higher, you can open the .stel file by clicking the name below the picture.
Jump straight to Pictures of Models, Mathematical Models, Creative Models or Lesson Planning,
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Thanks to Robert Webb for his very cool "Stella" program. I have used it extensively to generate VRML files for my site. It is a great tool for rapid exploration of augmentations and excavations of polyhedra (among other things).
Thanks also to Melinda Green and Don Hatch for their excellent Tyler Web Application. This free web tool lets you do great flat designs with polygons in just about any arrangement.
All pages on this site were written in Microsoft's Notepad, using Microsoft Paint and IrfanView for image editing. For more details, see my Help Page
Link to this page as http://www.JovoToys.com/JovoStretchPolyhedra.html