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3-torus

Here’s an equation for a a 3-torus I made:

(((x-4)^2+y^2+z^2+3^2-1^2)^2-4*3^2*((x-4)^2+y^2))*(((x+4)^2+y^2+z^2+3^2-1^2)^2-4*3^2*((x+4)^2+y^2))*((x^2+(y+6)^2+z^2+3^2-1^2)^2-4*3^2*(x^2+(y+6)^2))-5000000

How did I make it? Consider describing it as a function  f(x,y,z)=0. This is called an implicit function. Since it’s the surface of zero values and zero times anything is zero, we superimpose functions by multiplying them. So if t1, t2, and t3 are tori, t1*t2*t3 is 3 tori. Now to smooth it out we subtract a number which causes the surface to move forward along something called a gradient.

Voila, a 3-trous!

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This entry was posted on December 11, 2009 at 00:53 and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

2 Responses to “3-torus”

  1. 3D printing of Mathematical Objects! « Christopher Olah's Blog Says:
    August 21, 2010 at 04:36 | Reply

    […] implicit plot of the 3-torus (see previous post) […]

  2. Manipulation of Implicit Functions (With an Eye on CAD) « Christopher Olah's Blog Says:
    November 6, 2011 at 00:23 | Reply

    […] multiplication and subtraction — I used them a lot when I was younger (for example, making an equation for a 3-torus). The first trick is to realize that multiplying the functions for your implicit functions together […]

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