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Hi there, this is Vicky. I'm here because I'm being nice to Mr. Taranta, my math teacher. And besides, this gives me credit. So, I want to create a 3-D Apollonian Gasket. However, since I can't make a physical model, I need to do this with equations that involve a z-axis. The problem here is, I wasn't taught much about 3-D equations, and I probably will need help with that. Also, I would need to get familiar with graphing sphere and lines on 3-D graphs, figuring out centers of spheres, and translating them on a graph to create the 3-D Apollonian Gasket. Anyone there willing to help?

A link for you: [Angles and Circles]

Hope it helps. -Diana

Hi Vicky, I reviewed the method we were using to find the equations for the 2nd-iteration spheres in your image, and it should work in general. To review what we did:

  1. Find the diagonal of the cube using the pythagorean theorem. Let's call this distance D.(Recall: We found the diagonal, d of a rectangular prism with dimensions a x b x c can be found with the formula a2 + b2 + c2 = d2.)
  2. Subtract the diameter of the sphere from the diagonal of the cube, then divide the result by 2 to get that "left-over" distance between the surface of the sphere and one corner of the cube. Let's call this distance p.
  3. Notice that this left-over space for the large sphere is geometrically similar to the left-over space for the small sphere. That means, if the unknown diameter of our new, small sphere is d, then:
\frac{D}{D+p}=\frac{d}{p} (Make sure you understand why this works!)
  1. Now that we have the diameter of the small sphere, we can find its location; the center of this sphere must have distance from the origin that is equal to the sum of the radii of the large sphere and the small sphere. Let's call this distance q.
  2. Knowing q, and knowing the x,y, and z coordinates for the little sphere will be equal to each other (because it's in the corner of a cube), we can use the pythagorean theorem again to find these coordinates. We'll find a point, (x0, y0, z0), where x0 = y0 = z0.
  3. Now we have everything we need to build the equation for a sphere:
(x-x_0)+(y-y_0)+(z-z_0)=\left( \frac{d}{2}\right) ^2

So, check your work again, or consider starting from the beginning (tedious, I know, but sometimes it's easier than self-checking) and see if you can find what's not working.

In terms of graphing lines, is this absolutely necessary? It seems to me that your efforts are better focused toward getting these spheres than toward plotting what I believe to be reference lines. This kind of graphing would probably take you into vector algebra, which -- while very fun -- won't help you much with the main focus of your project.

-Diana (24:03 2/27/12)

It occurred to me that the relationship between the radii and center points of the circles of an apollonian gasket in a square (2D) should have a predictable relationship with the radii and center points of the corresponding spheres of the gasket in a cube (3D). You already have that information for two of the sizes of spheres; from those you should be able to investigate the nature of this relationship and find the radii and center points of your next iteration. This may require some experimentation, though, since two points are not enough to independently define a non-linear relationship.

-Diana (11:23, 3/5/12)