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 cut-the-knot-.org. 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)



October 3 – 276

JavaView. Icosahedron. The icosahedron. is a 20-face polyhedron. There are 43,380 distinct nets for the icosahedron. Fifteen golden rectangles span the interior of the icosahedron. These rectangles have 30 edges, and each edge pairs up with its opposite edge to form a golden rectangle. Two icosahedra appear as polyhedral “stars” in M. C. Escher’s 1948 wood engraving “Stars”.

Background – Hubble view of the sunward plunging comet ISON. It was discovered in 2012 by Vitali Nevski and Artyom Novichonok and named after the Russia-based International Scientific Optical Network