"My mathematical art generally starts with existing ideas, extrapolated in new directions and combined with disparate elements in an effort to create new and visually interesting results; the more surprising the better. Quick visualisations are mocked up in C++ software and the final models exported to PostScript (2D) and POVRay (3D) formats for rendering."

A harmonogram is the visualisation of a string of characters as a harmonic series. For this image, the character positions and values within the string {" you", 3, 117} define the frequencies and amplitudes of Fourier descriptors, the modulus value 3 induces the three-fold symmetry and the frequency scaling value 117 increases the curve's complexity. The resulting Fourier series is sampled at 14,004 regular intervals and a sphere plotted at each sample with radius based on local path curvature. This method of undersampling introduces periodic errors that manifest as delicate spiral tendrils, lending a lighter touch to this otherwise menacing figure.

This image shows two harmonograms {"a", 3, 141} + {"a", 7, 2102} composited as a single harmonic series. The Fourier descriptor frequencies and amplitudes are identical in both cases (given by "a") but the modulus values (3 and 7) and frequency scaling values (141 and 2102) are relatively prime, resulting in large sweeping paths modulated by finer perturbations. The Fourier series is sampled at 10,000 regular intervals, with local path width based on local path curvature and colour based on a continuous spectrum over the time domain t = [0..1]. The image suggests a skeletal figure riding an equally skeletal bicycle... with a bit of imagination.

“ Impossible Fern.”

This figure shows a fractal structure called the Pythagorean Tree incorporating an impossible multibar motif (based on the Penrose Triangle). The tree curls to the left as its construction is based on alternating squares and 30 degree Pythagorean triangles, which are substituted by a simple multibar design at each iteration. This particular tree uses a greedy algorithm that branches left twice for every right branch, giving a more homogenous spread of detail for a limited number of iterations (10).

“ Abelian Study+ ”

Spanish architect Antoni Gaudi (1852-1926) based many of his designs on elegant polyhedral models deformed into organic-looking shapes. This image shows an extrapolation of a sculpture study by Gaudi, based on the branching pattern of the evergreen shrub Abelia floribunda. The basic unit of construction is a stella octangula with the top and bottom points removed, repeatedly stacked along a spiral path with decreasing size and 185 degree rotation per iteration. The skew quadrilaterals formed by adjacent triangle pairs that meet across iterations are smoothed to form hypar patches, giving the final model an organic spiky look.

“ Trencadis Tori ”

This figure shows two interlocking tori finished in a style inspired by Spanish architect Antoni Gaudi (1852-1926). The periodic undulations, checkerboard colouring and trencadis-style mosaic finish are reminiscent of the design of the tower overlooking the main entrance at Park Guell (Barcelona).