Umbilic torus

Created by Helaman Ferguson, the umbilic torus is a single-edged 3-dimensional figure. Ferguson created a 27-inch bronze sculpture, Umbilic Torus, and it is his most widely known piece of art. The torus blurs the distinction between mathematics and art forming them into one work of mathematical art. The lone edge goes three times around the ring before returning to the starting point. A cross section of the surface taken from an umbilic torus corresponds with a hypocycloid. The torus is defined by the following set of parametric equations.

:$x = sin u left\left(7+cosleft\left(\left\{u over 3\right\} - 2v ight\right) + 2cosleft\left(\left\{u over3\right\} + v ight\right) ight\right)$

:$y = cos u left\left(7 + cosleft\left(\left\{u over 3\right\} - 2v ight\right) + 2cosleft\left(\left\{u over 3\right\} + v ight\right) ight\right)$

:$z = sinleft\left(\left\{u over 3\right\} - 2v ight\right) + 2sin left\left(\left\{u over 3\right\} + v ight\right)$

:::$mbox\left\{for \right\}-pi le u le pi,quad -pi le v le pi. ,$

*Torus
*Mathematics and art

References

* Larson, Roland E., et al. "Calculus". Ed. Charles Hartford. 6th ed. Boston: Houghton Mifflin Company, 1998.

* [http://www.helasculpt.com/gallery/umbilictorus4inch Umbilic Torus]

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