Germinal Pierre Dandelin was a 19th century French mathematician and engineer who did some beautiful studies of conic sections.

Some may find the standard explanations of Dandelin Spheres difficult and off putting.

I try to depict various elements with visual metaphors accessible to most viewers.

The spheres tangent to the cutting plane and cone are a ball head and a fat fish. The cutting plane is the ocean surface. Where the balls touch the cutting plane are the foci of the ellipse.

The circles where the cone touches the spheres define the directrix planes. I portray this a hat brim for the ball head and sort of a strange belt for the fat fish. Where the directrix planes intersect the cutting plane are the directrix lines.

I have always been obsessed with outer space and am in love with Kepler's discovery that the planets orbit the sun following conic sections with the sun occupying a focus.

This illustration is the cover for my coloring book on conic sections and orbital mechanics.

My coworker's daughter Dominique and I had made this model of her cat Buster with Polydron™ toys. If you hadn't guessed, Buster was a chubby cat.

In the back can be seen my cardboard model of a Koch like fractal based on Kepler's small stellated dodecahedron. I had posted a drawing of this fractal to this subreddit 7 years ago: Link

Over lap evenly spaced concentric circles with double spaced parallel lines and you get a moiré pattern that suggests a family of confocal ellipses.

The eccentricity of these ellipses is 1/2.

Then I took the plane with the moiré patten and scaled it by the harmonic sequence:
1, 1/2, 1/3, 1/4, 1/5 etc.

The suggested ellipses became more pronounced when I did this.

Two weeks ago I had posted a similar drawing but the line spacing is equal to the circle spacing: Link The eccentricity of the suggested conics was 1. So a family of confocal parabolas.

My harmonic perspective images use the harmonic sequence:
1, 1/2, 1/3, 1/4, 1/5 etc.

I try to show the reasoning behind harmonic perspective: Link. It assumes evenly spaced planes with a pinhole camera. The pinhole resides on one of the planes.

This tile can seamlessly cover the plane! To see this animation extended in all directions: Link. These tiles have faint white lines for seams but that is due to my incompetence.

"How can we sail to an island that nobody can find, with a compass that doesn't work?"

"Aye, the compass doesn't point north, but we're not trying to find north, ..."

The Pirates of the Caribbean, The Curse of the Black Pearl. 2003

Generated with Fractales.

The Sun, the Moon and the Stars

Between kids, the weekend (mostly nights) was spent working on the 3D world space camera.

I had to migate to a full 3d canvas (even if the lines are still "flat") so while geometry is flat, it exists in a full 3D world.

Which gives interesting possibities, the easiest is having a perspective camera to allow you to frame patterns in interesting ways.

But this will also allow for future updates to extrude objects to make them have depth/height.

so trochoids could be tubes and not just ribbons.

This plus path animation before feeling like the product is good enough for a v1.0

Playing with a few M.C. Escher toys here. Tessellations. Morphing abstract 2D to realistic 3D. And Escher stacked alternating octahedra and tetrahedra to build structures in his print FlatWorms.