Another recreation of Spyros Horemis' art. Originally this was from a 1970 Dover book, back before the days of computer graphics. The man's very meticulous drawings are a wonder.
If I look at this for awhile it starts moving, swirling either clockwise or counter clockwise, sometimes both directions at once. Does anyone else see this motion?
I am trying to make an animation of the motion I see, but it's difficult to capture.
I love the artwork of Spyros Horemis! Unfortunately his beautiful Dover book seems to be out of print. A used one is available last time I looked: Link
Spyros was making these images by hand before the days of computer graphics. His meticulous precision is stunning.
Kepler first pinged my radar when I was a teenager making cardboard models of polyhedra. Kepler discovered new polyhedra when he extended the edges of the Platonic solids.
Extend the edges of the pentagons on the Platonic dodecahedron and you get 12 five pointed stars. I am a little obsessed with the small stellated dodecahedron. Pictured here are four pieces where I play with this shape.
Kepler was also responsible for making the sun centered solar system the main stream consensus. Kepler's model made accurate predictions. Unlike the models offered by Galileo, Copernicus and Ptolemy.
Isaac Newton's world changing insight revealed in Principia was that inverse square gravity implies Kepler's three laws.
And where did the notion of inverse square gravity come from? Kepler had suggested the sun exerts a force which causes the planets to follow elliptical orbits. But he had thought the force falls linearly with the inverse of distance.
But it was Kepler who made the geometrical argument that the strength of light falls off with the inverse square of distance.
One of Kepler's followers, Ismael Boulliau, noted that Kepler's argument could also be applied to radiating lines of force. Thus Kepler is indirectly responsible for the notion of inverse square gravity. See The Man Who Inverted and Squared Gravity.
In my opinion Kepler deserves more credit than either Galileo or Copernicus for changing the way we see our star and planets. And more credit than Isaac Newton for laying the foundations of the math we use for celestial mechanics.
There was a king who was proud of his chesse playing skills. Krishna visited his court disguised as a human.
He challenged the king to a game of chess with a friendly wager. The loser would give the winner one grain of rice on the 1st square of the chess board, two grains on the second square, four grains on next square and doubling the grains on each square.
When the king lost the game Krishna revealed his true identity. And the king realized the enormity of the wager.
Krishna told the king to feed pilgrims passing through rice until the debt is paid. Centuries later the debt is still not paid.
This is The Legend of Paal Payasam
It's a nice story about exponential growth.
An animation of the Phobos Deimos ZRVTO (Zero Relative Velocity Transfer Orbit) as described by Paul Penzo on page 45 of the Tethers In Space Handbook: Link
If a payload is released at the right time from the top of 940 tether ascending from Phobos it will arrive at the foot of a 2960 km tether descending from Deimos.
At apoapsis the payload matches the velocity of the Deimos tether foot.
Thus Phobos can send payloads to Deimos with nearly zero propellant. And vice versa.
Penzo notes both Deimos and Phobos are huge momentum banks. So Phobos and Deimos anchored tethers can catch and throw payloads till the cows come home with very little effect on their orbits.
There is a ZRVTO between any two Sarmont tethers in circular orbits of the same inclination.
I believe Penzo's insight can be extremely useful.
Required tether mass goes up exponentially with delta V. Somewhat like propellant in Tsiolkovsky's rocket equation.
With a chain of Sarmont tethers connected by ZRVTOs, the delta V budget is broken into chunks. Thus you can have tethers made from today's materials without ridiculous taper ratios.
I want strokes with variable width but solid and opaque through out the stroke.
Screen capture shows what I've been getting. I try to change opacity jitter and other settings but they return to zero I soon as I close the brush settings panel.
I kind of like having variable opacity based on pen pressure. I can used that for some effects. But for right now I want my pen to behave like it used to.
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 - r^3 |.
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.
I wonder if all rhomb tilings could be made 3D. It's easy to interpret this one as a zonohedron
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.
Neil telling us Newton was no good any more when he had God on the brain: Link
Historian Thony Christie looks at Tyson's imagined timeline: Link
Physicist Luke Barnes notes Newton returned again and again to the problem of modeling the motion of multi-body systems. As did Euler, Lagrange, d'Alembert and Laplace (of course): Link
Neil Tyson gives Newton a starring role in his cautionary tale against belief in Intelligent Design. It is extremely distrespectful to Newton and every thing that he was. It is also a complete fiction.
Another Harmonic perspective drawing.
This can be seen covering the entire plane at.
https://cunews.info/HexPrismsAlienFish.html
On this page you can watch the fish change from red to blue as she crosses the boundary from one tile to another.
Part of the hypnotic quality of these harmonic animations is the patterns that coalesce and then dissolve and coalesce into new patterns.
I am wondering if populating this space with creatures is distracting and breaks the spell.
as usual the successive layers are scaled the harmonic sequence -- 1, 1/2, 1/3, 1/4, 1/5, etc.
To see this cover the plane go to https://cunews.info/HexPrisms.html
Hexagonal array. Layers are scaled by the harmonic sequence: 1, 1/2, 1/3, 1/4, 1/5, etc.
Layers shrink along the harmonic sequence: 1, 1/2, 1/3, 1/4, 1/5 etc.
1st iteration of a Sierpinski tetrahedron is to remove an octahedron from the middle. I put the punched out octahedra to the right.
Perspective drawing of a structure built from truncated octahedra.
I started with the O'odham Man in the Maze design which has 4 arms. Then I did variations with 2 arms, 3 arms, 5 arms and 6 arms.
These labyrinths are a metaphor for life. You start at the beginning, go through many twists and turns and arrive inevitably at the end.
The O'odham people are indigenous to southern Arizona and northern Sonora, Mexico.
The maze is topologically similar to The Labyrinth of The Minotaur on an ancient Grecian coin Link.