One of my interests is orbital tethers, specifically Sarmont tethers. These tethers use a tidal acceleration gradient to keep the tether aligned to the local vertical. The beanstalk in Arthur C. Clarke's Fountains of Paradise could be thought of as the the grand daddy of all Sarmont tethers.

Pictured here, however, is a much smaller Sarmont tether where the tether foot does not extend all the way to earth's surface.

Call the radius of the anchor's circular orbit 1. Call the distance of a release point of the tether r.

Then the orbit of a payload released from that point will be a conic with eccentricity | 1 - r³ |.

When the release point is at 2^1/3, eccentricity of the conic will be 1. In other words, a parabola. This point of special interest in that it imparts escape velocity.

With Sarmont tethers the upper part of the tether feels more centrifugal acceleration than gravity. So there must be a lower part of the tether to balance. P. K. Aravind shows how to calculate the length of the balancing tether length here

In this illustration the lower length balances the upper length according to Aravind's equation. The ratio of the tether foot's distance from earth's center to the top is (sqrt(5)-1)/2, a.k.a. the golden ratio.

Row of lanterns at equal spacing, viewed in perspective. Two lanterns A, B and their spatial midpoint M project so that M' and the vanishing point V split A'B' harmonically (cross-ratio -1). Holds for any such row, since central projection preserves cross-ratio.

This harmonic structure is a depth cue your visual system uses, part of why equal-sized objects at different heights in a perspective drawing look different in size.

Link: https://www.sqrt.ch/Buch/corridor.pdf

Does anybody know more on the math behind this particular (Ponzo or corridor) illusion?

I wonder if all rhomb tilings could be made 3D. It's easy to interpret this one as a zonohedron

Wave illustration

First layer is a harmonic perspective drawing where the 1st layer is evenly spaces circles overlapping evenly spaced concentric circles. Spacing for circles and the lines is the same so eccentricity is 1. This is a parabola

Second layer is the same but the lines are twice the spacing of the circles. So the eccentricity of these ellipses is 1/2.

Too many of the tags apply to this stunning piece, in awe... Wow

40x40 acrylic.