Math in Second Life, page 3

By Xah Lee. Date: .
Alexanders horned sphere
This is called Alexander's Horned Sphere. As you can see, the linking of the circles continues on forever.

This shape is a fantastic discovery to mathematicians in 1924, in particular, of the branch topology. It is interesting because it is: “An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected.”. Here's some basic explanation:

On a piece of paper, draw a simply connected closed curve (for example, a circle). Once you draw a circle, it divides the region into two sections: inside, and outside. A theorem says that this will always be so. (silly theorem, eh?) Furthermore, the two regions are similar in the sense that each is a rather coherent blob.

Now, if you have a sphere (aka ball) in space. This sphere also divides the 3D-space into two regions, the inner one, and the other one. Again, the inside region and outside region are similar in the sense that both are coherent blobs.

However, now consider this shape called “Alexander's Horned Sphere”. It divides the space into the inner region and outter region. However, the outer region is no longer a coherent blob, yet the inner region is still a un-pierced blob.

The above is a rough description on why it is interesting. To understand this exactly, a human animal needs to study math for about a decade. For a technical description, see: Alexander's horned sphere.

dodecahedron star 43
A shape similar to a stellated dodecahedron. This is actually a living animal. When it moves, the pointed tips flow freely like some sea creatures, making it extremely cute.

(This shape is not a stellated dodecahedron. Ask Seifert Surface or Bathsheba in-world if you want to know the detail)

fractal nest cube 30
A iterated function system styled fractal. Note how the pattern nests.
fractal nest tetrahedron
Another fractal. The outer shape is a tetrahedron. Bisect each of its edges, forms a octahedron in its center and 4 smaller tetrahedrons at each of the corners. Do this again on the tetrahedrons, will form this shape.

This technique of forming fractals is based on the Sierpinski triangle.

torus eversion
This sequence shows how to turn a torus inside-out. (A torus is the name for the shape of a ring.)

Unless otherwise indicated, the objects shown in this page are created by Henry Segerman (aka Seifert Surface in Second Life).

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