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Interlocked Tori | Calabi-Yau | 27 Lines on the Clebsch Cubic | The Wolfram Works
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'Interlocked Tori' by Michael Trott, Wolfram Research, Inc

Interlocked Tori


Michael Trott, Wolfram Research, Inc, 1996
ZPrinter Z406 printed plaster sculpture

Given a set of tori (potentially double torus, triple torus, etc.), there are various symmetric ways to interlock them. This figure shows a configuration of such tori with three symmetry planes.

Michael Trott commented on his creation: "I think it depends a little bit on the viewpoint of what you consider to be art. It can look striking, and it does, in many cases, look very beautiful and very interesting. But it is typically not my intention to make something that is beautiful. It just comes out that way. I think this has pretty deep roots in science itself. It turns out that many things that are elegant from a purely mathematical point of view turn out to be pleasing for the human eye."

This 3d model was generated using Mathematica 6.0 by Senior Kernel Developer Ulises Cervantes-Pimentel, Wolfram Research Inc. More on The Science and Art of Mathematica, from Michael Trott, can be found here.

Visit Wolfram's Mathworld for more information on torus, its definition, and mathematics.

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'Calabi-Yau' by Jeff Bryant and A.J. Hanson, Wolfram Research, Inc

Calabi-Yau


Jeff Bryant, Wolfram Research, Inc
and A.J Hanson, Indiana University, January 4, 2008
ZPrinter Z406 printed plaster sculpture

String theory is a prime candidate for unifying quantum mechanics with Einstein's general theory of relativity (a "Theory of Everything"). In string theory, the fundamental physical objects are vibrating strings that sweep out world sheets in spacetime, with elementary particles represented not as points but as different modes of vibration of these strings on the scale of the Planck length, of the order of 10-35 m.

In one version of a consistent string theory, the strings must live in a 10-dimensional spacetime. Since human physical experience appears to be that of a four-dimensional spacetime (three space dimensions plus time), it is presumed that if 10-dimensional string theory is correct, there must be six additional dimensions that are curled up into complicated, undetectably small shapes at the Planck scale, known as Calabi-Yau spaces. Every point in spacetime would therefore possess six additional dimensions whose topology is described by a Calabi-Yau space, of which there are a large number of possibilities.

This model depicts what many believe is the simplest and most elegant of these possibilities, the quintic polynomial in four-dimensional complex projective space. When two of the four complex variables are fixed, the surface that remains can be displayed using ordinary graphics projected from 4D into 3D. The fact that the surface is derived from a quintic or fifth-power polynomial can be perceived directly by noting the fivefold pie-slice arrangements that occur in the model of this Calabi-Yau shape.

This 3d model was generated using Mathematica 6.0 by Senior Kernel Developer Ulises Cervantes-Pimentel, Wolfram Research Inc.

Visit Wolfram's Mathworld for more information on the definition and mathematics of Calabi-Yau space.

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'27 Lines on the Clebsch Cubic' by Ulises Cervantes-Pimtel

27 Lines on the Clebsch Cubic


Ulises Cervantes-Pimtel, Wolfram Research, Inc, January 2, 2008
ZPrinter Z406 printed plaster sculpture

As part of the current collaboration with 3D printing companies such as ZCorp, Wolfram Research has developed several algorithms that allow Mathematica 6 to generate complex 3D models with a minimal amount of effort. One example is a 3D representation of the classical result from algebraic geometry that on every cubic surface there is a configuration of 27 lines. This theorem was proved in 1849 by Salmon and Cayley.

Several people have used previous versions of Mathematica, in conjunction with other 3D modeling programs, to construct 3D printable objects. A few months ago, Wolfram Research was asked if Mathematica could generate this model in a simpler way. With Mathematica 6, you can write a very simple program that can compute the exact equations for the surface, as well as the exact locations for all the 27 lines. Mathematica uses state-of-the-art meshing algorithms that generate watertight models that are directly suitable for 3D printing without any extra manual manipulation.

This 3d model was generated using Mathematica 6.0 by Senior Kernel Developer Ulises Cervantes-Pimentel, Wolfram Research Inc.

Visit Wolfram's Mathworld for more information on the 27 lines that lie on the general cubic surface.

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'The Wolfram Works,' from Wolfram Research, Inc

The Wolfram Works


Wolfram Research, Inc, 2008
several immersive interactive applications

The Wolfram Works are 3D models created by Wolfram Research, Inc using Wolfram's Mathematica. These pieces illustrate a potential trend in art and life in general: a computer object no longer necessarily needs a physical counterpart to claim significance. These virtual sculptures do not need to be fabricated in the material world to be interesting to look at and in fact offer greater interaction to a viewer than a sculpture under glass can.

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The Beckman Institute for Advanced Science and Technology

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