Math in Second Life

By Xah Lee. Date: . Last updated: .
Geo Dome
A screenshot of a location in Second Life.

This sim built by Henry Segerman http://www.segerman.org/ (aka Seifert Surface).

The metal ball above the bridge is made of 4 nested spheres, rotating on different axes. Behind the sphere is a tower of octahedrons. (A octahedron is a regular solid, having 8 faces, each face is a equilateral triangle. (it's like two pyramids with their bases glued together)) On the right is a geodesic dome made of glass. Inside is a garden with mathematical sculptures.

double spiral
The middle sculpture is a double spiral. This sculpture is based on the Equiangular Spiral. Most spirals seen in nature's growths, for example, plants and seashells, are equiangular spirals. (See: Seashells photo gallery.)
hopf bundle
This colorful spiral sculpture is a model of the Hopf bundle. You will need to have a phd in math to understand this one. For mere mortals, suffice it to say it is twisty.
hopf bundle 2
hopf bundle. Standing in front is a Japan geisha in traditional Japanese attire the kimono.
moebius strip
The girl Eureka of the Japanese animation Eureka Seven. (The creator of this avatar is Yamiki Ayakashi)

The background blueish strip is the Moebius strip. Moebius strip is a example of a surface that has only one side. (If you are a ant crawling on the strip, you'll end up on the other side without crossing any edge.) The rim of this strip, is a Trefoil knot. Tie a overhand knot, then connect the lose ends, and you have it!

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 http://www.segerman.org/ (aka Seifert Surface in Second Life).

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