The Octaplex on Elsevier-ScienceDirect

The Octaplex, symmetry in four-dimensional geometry and art, is now available on Elsevier-ScienceDirect @



Update 10, 2016

I was preparing some artwork  (The 12-30 project, #211 – July 30 ) for an upcoming exhibit and I clicked the emboss filter – almost by accident. And here it popped up – a next to perfect representation of a (topological) 4D perspective of a Klein bottle – from the outside looking in. Thanks @Mathematica, and thanks #Richard Bennigan for the original script.



December 13. – 347

MathMod & Morenaments. Breathers. Breathers are pseudospherical surfaces of negative curves. Changing the U & V parameters lead up to a Kuen figure (plate #343). This occurrence was discovered by Richard Palais using 3D-XplorMath. Here I explore the shape of the surface in more of a cultural and historical context and as a reference to mathematician E. Beltrami who discovered the pseudosphere in 1868.

Background: pmm symmetry. This symmetry group contains reflections whose axes are perpendicular and has no glide-reflection.


December 3. – 337

MathMod & Morenaments. Variation on the Duplin (more likely F. Dupin ) cyclides – named after 19th century French mathematician F. Dupin. Theses surfaces have a low algebraic degree and have been proposed as a solution to a variety of geometric modeling problems.


November 13 – 317.
Geogebra. From the 2D-outline sangaku: given the inradius of the square (cube) find the inradius of the circle (sphere) in the triangle (pyramid) – The answer is 1/2 the size of the largest circle…


November 1 – 305.

Moving from JavaView to GeoGebra. Geogebra is a mathematics software designed for mathematics students that creates beautiful visual outcomes. As an anchor to this month’s exploration, I ‘ll use some Sangaku geometry problems from the library of Alex Bogolmony available at Sangaku were meant to address two-dimensional geometry problems. Treating the same problems in a 3D environment is bound to bring some unexpected surprises…

This first – sangaku – is called – Three Tangent Circles sangaku. Given three circles tangent to each other and to a straight line, find the radius of the middle circle via the radii of the other two…


October 11 – 284.

Tri-noid. Tri-noids are part of the k-noid family. A k-noid is a minimal surface with k catenoid openings. The first k-noid minimal surfaces were described by Jorge and Meeks in 1983.

Background: Hubble telescope – a Perfect Storm of Turbulent Gases in the Omega/Swan Nebula (M17)