Mar.15, 2013

University of Melbourne mathematician Dr Henry Segerman says all language is about communication, Maths is just the same. Dr Segerman has found a special way to express mathematics in 3D printed sculpture art.

Henry Segerman earned his master of Mathematics degree at University of Oxford in 2001 and then a Ph.D. in Mathematics at Stanford. Using 3D modeling software Rhinoceros and 3D printing company Shapeways' service, Segerman has made more than 100 sculptures.

3D printing gives Segerman huge amount of freedom in the geometry. See the photo below, he used 3D printing to create these representations of regular four-dimensional polytopes, the analogues of the 3-dimensional regular polyhedra.

The 120-cell and 600-cell are shown, which have 120 dodecahedral facets and 600 tetrahedral facets respectively. The left and right objects are dual to each other, which means that the vertices of one correspond to the 3-dimensional facets of the other, and vice versa. This is illustrated in the center object, which is simply copies of the two other objects occupying the same space, interlinking with each other.

The Dual Half 120- and 600-Cells won the "Best Use of Mathematics" award at the Bridges conference.

An interesting mathematical influence in Segerman's work is "self-reference", that an object can describe itself. One example is 'Sphere autoglyph', A self-referential tessellation of the sphere.

An autoglyph is a word written or represented in such a way that it is described by the word itself. In this 3D printed sculpture, 20 copies of the word "SPHERE" fit together to form the sculpture of a sphere. This was really a challenge for Segerman to work out how to fit the shape of the English word SPHERE together with copies of itself.

"This is both an artistic/typographical problem which asks: 'How can the shapes of the letters be deformed to fit together but still be legible'? And a mathematical problem: 'What are the possible ways of making a symmetrical pattern on a sphere?'." says Segerman.

Another example is "Bunny" Bunny: the word "bunny," is repeated many times over to form the surface of the Stanford Bunny, a computer graphics 3D test model developed by Greg Turk and Marc Levoy in 1994 at Stanford University.

One recently project of Dr. Segerman is a moving sculpture: Triple Gear. In this unusual mechanism three gears mesh together in pairs, and yet they can turn. Segerman says as far as he knows, no one has done this before.

If you take three ordinary gears and put them together so that each gear meshes with the other two, then none of the gears can turn because neighbouring gears must turn in opposite directions. Triple gear avoids this problem by having the three "gears" arranged like linked rings - the gears then rotate along skew axes, and the opposite direction rule no longer applies.

Segerman designed these sculptures using scripting and then producing them using 3D printing. The technology allows him to get very close to mathematically precise geometry, which is often difficult to achieve by other means.






Posted in 3D Printing Applications

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