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 daniel erdely (M, 64)Budapest, HU Immortal since Aug 13, 2008 Uplinks: 0, Generation 3 Spidron Main page Spidron presentation 1 Spidron presentation 2 Spidron article Spidron Space-fillers
My main goal is to present the Spidron project we develop with my friends for years. it is a system of triangles with extraordinary properties in space and time.
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Pythagorean Theorem in the Third Dimension
I guess I have found the only possible 3D deformation of the Euclidean plane which - regarding of the 9 vertices of the basic shapes of the Pythagorean Theorem, like the triangle and the squares - is preserving the validity of the theorem. The triangle in the 2D plane is rotating and shrinking, but its proportions and angles don't change. I tried to make some drawings to demonstrate this interesting swirling deformation below.

This picture shows how to make SpHidron deformation from the classical Spidron deformation

The arms of the Sphidron disc are rotating around themselves. The longer distance from the center increases the measure of the rotation. Here the change of the rotation is continuous while in the case of Spidrons it is discrete.

In between every pair of twirled arms must be another arm which like the twirled ones remains in one plane but it isn't not rotating around itself - like an axis - during this deformation.

The spiral arms one by one remain on their own plane, but these planes on which they lie on can be lifted up and pushed down alternately. This way you can make more and more dense surfaces in the same volume.

To make this deformation possible, you have to change your previous imaginations on the plane. It is a little more complicated, as the elements of the plane are not points with no dimension, but they do have dimensions, what make possible their rotation and sliding. While two of the neighboring "spoints" remain neighbors, the rest of them can be changed. Just like in the case of a simple pearl-string where every bounce has only and maximum two permanent neighbors.

Drawings by D Erdely, computer visualization by Janos Erdos

Mon, Mar 15, 2010  Permanent link