MATHĒMA

MATHĒMA is a mathematical vase generator that runs in your browser. You give it a shape, add some equations, fiddle with the parameters, and eventually arrive at something you can send to a 3D printer. Somewhere along the way, “I’ll just try one more variation” becomes a fairly optimistic assessment of how the evening is going.

The idea is straightforward enough: mathematical functions can produce rather interesting geometry, and a vase provides a convenient excuse to turn some of it into a physical object. Waves become ribs, interference patterns produce folds, and a little twist along the vertical axis can make an otherwise quite sensible shape look as though it has had an argument with gravity.

MATHĒMA gives you a workbench for exploring these possibilities. You start with a silhouette, choose a cross-section, and build up the surface using layers of mathematical patterns. There are harmonic functions, rose curves, Bessel functions, noise, Voronoi cells, and several varieties of fractal behaviour. You can combine them, adjust their strength, and decide where along the vase they should appear.

There are presets to explore, randomisation for when you’re short of ideas, and variations for when you like what you have but suspect that something slightly different might be better. The 3D preview lets you turn the result around and inspect what you’ve actually made. This is occasionally different from what you thought you were making.

One of the more involved features is Julia boundary lofting. Here, the boundary of a Julia set supplies the cross-section of the vase. Change the complex parameter as you move upwards, and the sections evolve into a three-dimensional form. It can produce lobes, curling outlines, and folds that look surprisingly organic for something derived from repeatedly squaring a complex number.

This is also the experimental part. Julia boundaries contain detail at scales that a finite mesh, and certainly a printer nozzle, cannot reproduce. Some parameter paths produce unsuitable geometry; others take a while to calculate. There is still work to do on making this more forgiving and faster. I’m not in danger of running out of things to improve.

The practical side is aimed at FDM printing, particularly spiral vase mode. MATHĒMA checks the generated geometry and provides printability analysis for overhangs, small features, and build-volume limits. You can inspect problem areas in the preview before exporting an STL, OBJ, or 3MF file. Your slicer still deserves a look before you commit several hours and a quantity of filament to the experiment.

Projects can be exported and reopened for further tinkering, and there’s mathematical provenance documentation if you want to keep track of how a particular object came into existence. This may prove useful after the seventeenth adjustment to a parameter whose original value you were absolutely certain you would remember.

The first version is now available at works.baumann.gr/mathema.

Have a play. Start with a preset, move a few curves around, and see what happens. Whether the resulting object ever encounters a flower is entirely up to you.