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Dedication of the Umbilic Torus  
The dedication ceremony of the Umbilic Torus, by Heleman Ferguson, has been held on October 26. Professor Tony Phillips (Department of Mathematics) introduced the speakers, including President Stanley, Marilyn and Jim Simons, and Heleman and Claire Ferguson. A reception and a lecure by the Ferguson's followed in the Simons Center.

The project was supported by the Simons Foundation. The statue was built in Baltimore over the last two years by a team led by the artist. The construction involved, among other things, the creation of a robot that shaped the sandstone forms for the casting of the silicon bronze structure. Photos about the installation and the dedication ceremony are here, username: public, password: photos.

Of all the twisty shapes that one can imagine, this one is special, because it relates to the solutions of the cubic equation, x3+bx2+cx+d=0. Here is a physicists interpretation of the math behind the satutue. (Thanks to John Morgan, who gave an excellent lecture about the subject. Apologies for the oversimplifications and missing some of the beautiful math. --LM)

If we pick the parameters b, c and d randomly, and we solve the eqaution, chances are that we have either one solution or three solutions (assuming we look at real numbers only). For example x3 - 3x2 + x - 3 = 0 has only one solution, x=3. On the other hand, x3 - 6x2 + 11x - 6 = 0 has three solutions: x=1, x=2 and x=3.

We can imagine that the the parameters b, c and d are the axes of a Cartesian coordinate system. In this three-dimensional space every point has well defined b, c and d, and therefore every point corresponds to a cubic equation. We solve the equation, but we do not care about the actual values of x. All we care about is the number of solutions: either one or three, as we have seen above. Now we color each point in the space of b, c and d according to the number of the solutions. For example, we give "blue" color to the points where there are three solutions, and we color "red" the points that yield one solution.

The Umbilical Torus is related to (topologically equivalent to) the surface that separates the blue and the red regions in the three dimensional space of b, c and d. The surface of the Torus itself correspond to the parameters when two of the three solutions happen to be equal. The "rim" or "edge" that runs around and performs a 120o twist (coming back to itself after three full rotations) correspond to the special case when all three solutions are equal (for example, in x3 - 6x2 + 12x - 8 = 0).

The animated gif images on the right hand side were created with the free softwares K3DSurf and Cropper. Here are links to the parametric equations describing the forms: umbilic1.k3ds, umbilic2.k3ds and umbilic3.k3ds


Umbilical torus generated by the free software K3DSurf. Click on the image to see a larger version.


Other variations of the twisted form that do not have the deeper interpretation of the Umbilical Torus.
2012-10-26

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