Jun 25, 2014 6:30 AM

Gallery: This Is What Math Equations Look Like in 3-D

Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
01
34-double-deck-filament
In this model of a mathematical surface, every aspect of every swoop, dip and pinch is encoded in a single equation. That equation has a singularity where the plaster would be drawn infinitely thinly. In a concession to physics, the final gap is bridged by a tiny wire.
Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
02
43-do-the-twist-tighter
Knowledge of curves in the plane can be bootstrapped to build surfaces by first making metal arcs then connecting them with string. This model was constructed by Arnold Emch, the only major American model maker.

Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
03
81-floating-triangle
By changing the parameters in the equation defining the Clebsch cubic (discussed below), you can morph the shape of the surface. Here three throats have pinched off into singular points. This process is called degeneration.
Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
04
97-mero-top-view
This surface is the graph of the real part of a complex function (i.e., the square root of -1 is involved). Each peak, called a pole, comes with two towers and two pits.

Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
05
55-string-snapped-string
After a century of tension, many of this model's strings have frayed and snapped. The Smithsonian's National Museum of American History has a large collection of deteriorating models. None of them are on display.
Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
06
174-levitation-b
A single equation cuts out the two separate pieces of this model. This is similar to the way that the equation x^2^ = 1 has two separate solutions, x=1 and x=-1.

Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
07
13-wow-osculating-circle
A glimmering curve is suspended at the center of this model, where two surfaces made of string intersect.
Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
08
63-plaster-logarithmics
This plaster model shows the potential energy of an electric dipole. The negative charge generates an infinite well and the positive charge an infinite peak. Far away the charges approximately cancel and the surface is almost flat.

Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
09
76-bone-ball-full-view
The bends and bulges in this bone-white ball are defined by a single equation of degree four in x, y, and z. The label, in German, hints at the size of the 1911 Schilling model catalog: This model is number 4 in series 9 of 40.
Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
10
88-deep-pinch-
This surface is described by the equation z = xy/(x^2^ + y^4^). A severe singularity forms at the origin where the equation yields z = 0/0. There a vertical sheet drops from the high parabolic ridge to the low parabolic valley. A perpendicular mid-height ridge runs through the sheet.

Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
11
12-bulbous
These six bulges (two are in back) exhibit a [dihedral symmetry](http://en.wikipedia.org/wiki/Dihedral_group) and a confident attitude. The creases where they meet correspond to singularities in the defining equation.
Photos by UIUC <a href="http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=342">Altgeld collection</a>
12
4-copper-bracelet
The spiral of this helicoid is self-similar: Each turn has identical curvature. To verify this, you can slide the brass band around the shaft.

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