In the 2022-2023 exhibition "Imaginary: Beauty and Power of Mathematics", several images show the beauty of surfaces that are determined by a mathematical equation.
Most of these equations and the resulting images can be found in "IMAGINARY", a publication (2014) of the Mathematisches Forschungsinstitut Oberwolfach.

The surface, determined by the polynomial equation

even has a name: "Vis-à-vis".


In the Oberwolfach catalogue we read: "Vis-à-vis means opposite. The singular tip on the left looks at a curved but smooth hill on the right."

As mentioned in the Exhibition Catalogue, it can be fascinating to observe the effects of small changes in the equation on the resulting image.

The suggestion is to generalize the equation to:
bx²-x³+y²+y⁴+az³-z⁴=0
(a and b are parameters taking values between 0 and 1)

For a=1 and b=1 we obtain the "Vis-à-vis" surface.

For a=0 and b=0 the surface is rather smooth, without tip nor hill.


For a=1 and b=0 the surface is smoothly curved at the right side.


And for a=0 and b=1 even two tips are facing each other.


Finally, taking small steps for a and/or b between 0 and 1 or 1 and 0, and combining the images, we obtain the following animation:



All images were obtained using the free raytracer Povray. To combine a number of them into the animation I used Gif Movie Gear

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February 2023
Herman Serras
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