u/HopDavid

Another harmonic perspective drawing, this time with sine waves. As usual they follow the harmonic sequence: 1, 1/2, 1/3, etc..

It was my friend, Michael Carnright, who got me interested in the harmonic sequence when he was playing the bag pipes. It shows up in music: Link. Then Mike started noticing the sequence in other places.

When you tile this rectangle to cover the plane the waves have a hypnotic effect: Link

This one also tiles to cover the plane: Link

One of Ghee Beom Kim's studies inspired this but I can't find it at the moment.

Like my other harmonic perspective images this one relies on the harmonic sequence: 1, 1/2, 1/3, 1/4, 1/5 and so on.

My orbital mechanics and conic sections coloring book has about five pages devoted to Dandelin Spheres.

This illustration looks at eccentricity in terms of ratio of distance to focus to distance to directrix.

Here's another Dandelin illustration: Link

Leonardo da Vinci liked to portray open faced polyhedra so you could see the internal structure. I call this Leonardo style polyhedra drawings.

It is a page from my Geoscapes coloring book published by Dover.

Isaac Newton was undoubtedly one of the greats. However some myths have grown around Newton's memory over time.

Historian Thony Christie examines Neil Tyson's imagined timeline: Why doesn't he just shut up?

My attempt to Op Art in the style of Marvel comic book artist Jack Kirby. Kirby drew The Fantastic Four,The Silver Surfer and other great comic book lines.

The sky was done by a Deviant Art colleague of mine who called himself (or herself) Photic Driver. Photic Driver gave me permission to incorporate it.

Below the horizon is one of my perspective drawings based on the harmonic sequence, 1, 1/2, 1/3, 1/4, etc.

It's a perspective drawing of a parabola. Cartesian coordinates where it crosses corners are (0,0), (±1,1), (±2,4), (±3,9), (±4,16), etc.

I believe this perspective drawing of a parabola is a perfect circle. Can anyone prove (or disprove) this notion?

Is this drawing appropriate in this subreddit? I kinda sorta think it's geometrical. I sometimes see snakes and other reptiles as stunningly beautiful geometric tapestries.

IIRC Escher had contemplated using turtles in his print Gravity. After all, turtles are walking polyhedra! Making the net for the turtles was challenging. I had several pages in Excel using dot products, cross products, scaling matrices and rotation matrices.

I believe the Riemann projection of a loxodrome is a logarithmic spiral. A fanciful drawing of this notion.

A lot of my drawings are imitation Escher. Escher did a gorgeous drawing of a loxodrome: Link

I believe the Lunar Hill Sphere will be a bay opening to the wider solar system.

From EML2 (Earth Moon Lagrange 2) it takes only 1 km/s for insertion to a trans Mars orbit.

EML2 is 2.5 km/s from the moon. And there are heteroclinic paths linking EML2 and EML1. EML1 opens to lower earth orbits.

Here's Neil making this ridiculous claim: Link

Of all Neil's wrong explanations this one is in the top three most annoying (for me). I'm interested in the possibility of space settlement. So possible ways to mitigate huge delta V budgets has been a passion of mine.

Is it possible pop science is making the populace even dumber?

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.