3-D Yin-Yang

Aaron Isaksen

aaron@hkn.eecs.berkeley.edu

Stereoscopic images

In this assignment, we were to create a 3-D Yin-Yang model out of two similar interlocking parts that are of equal volume and are either identical or symmetric. In this model, the pieces are not identical, but are mirror images of each other. These images were modelled and rendered using the POV-Ray 3.0 raytracer (www.povray.org). Here are some stereoscopic pictures of one of the parts. The pictures are layed out for cross-eyed viewing.

Here is a stereoscopic image of the other part.

Exterior and Cross-section images

In this model, when you put the two pieces together, you can not see a Yin-Yang symbol from the outside (as shown below). I feel this is justified, as someone viewing a 2-D YY from a 2-D world (the plane containing the symbol) would only see a black and a white semicircle. The 2-D viewer would have no idea what is contained "inside" the symbol. Likewise, as we are viewing the 3-D object from a 3-D world, we only see two hemispheres. A 4-D viewer would be able to see the interesting nature of the model without taking slices.

However, if we view the object from the inside, we will see our 2-D Yin-Yang. If we take the above shape, and, without rotating or translating, we make any cross section that contains the center of the sphere, a Yin-Yang like-symbol will appear. Note the two examples on the left are exact Yin-Yang symbols. These are obtained when the cutting plane passes through the points where the dividing curve is touches the outside sphere.

Construction of the model

I noticed that one could create a normal 2-D Yin-Yang from a circle "rolled" around the circumference of a larger circle. As the circle is "rolled", its radius increases (non-linearly) such that it is always tangent to the containing circle and the dividing curve. I wanted to extend this method of creating the object to 3-D, by rolling a sphere instead of a circle. After a 4-hour bout trying to figure out the geometry of the problem, I derived that given the angle theta shown below and 1/2 the radius of the containing sphere (let this be r, which is the radius of the smaller circles that create the dividing curve), the radius of the "rolling" sphere is equal to -r*(Cos(theta)-1)/(3+Cos(theta)).

To create the 3-D version, I begin with a half-sphere. On the flat face of the sphere, I "roll" another half-sphere (the two flat faces are touching each other) which increases in radius using the above equation. Then, starting from the other side of the flat face from where I started my sweep, I begin another sweep, this time carving out from the exposed face.