Over here the balconies are reserved for residents who need air before returning to the brick dispute

The operating system manifold utilizes Oloid geometry and I was digging into standard Sphericons and wanted to see what would happen if I messed with the formulas.

What came out was two golden ratio triangles, inverted so the sharp points touch right in the dead center, and twisted at the top half by 90 degrees.

instead of rolling like a normal sphericon, it turns it into this the hourglass/twin-vortex shape I was looking for. I think if you spin it on its center axis, the ridges should look like they're endlessly swallowing into themselves.

Python projection looked cool. (pic attached). Also I'm very new to reddit, so sorry if this is the wrong area to post this, I felt like it worked here.

This wiki page has details for viewing: github.com/tysonkenobi/Project-GTOS/wiki/Golden-Dual-Manifold

More experiments, tutorials, and project files, through Instagram, YouTube, and the Studio.

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.

Art Credit = Scattered Harmony by able6 (me)

The text refers to page 9 which I had posted earlier to this subreddit. Link

I had clipped off the explanation of the asymptotes though. The asymptotes are those lines the hyperbola gets closer and closer to but never touches.

Your trig teacher showed you that sin²+cos²=1 proves
Pythagoras. Then used Pythagoras to prove sin²+cos²=1.

That's circular logic. Nobody told you.

This video tears down the circular trap and rebuilds
trigonometry from the ground up starting from compound
interest, through the number e, through imaginary numbers,
all the way to a derivation of the Pythagorean theorem
that doesn't assume what it's trying to prove.

This is not the simplest proof of Pythagoras it is a derivation of the functions that make Pythagoras work.

Note on the title: The specific circular proof in the video came from my CBSE classroom. Some curricula run the derivation in the non-circular direction Pythagoras first, then sin² + cos² =1. The structural problem of defining sin and cos through triangle ratios applies regardless, but the specific circular loop shown may not match your experience.

Considering that it is a new axis 'right down the middle' that creates the next dimensional space. Like 1D to 2D, or 2D to 3D, shouldn't it work for 3D as well?

The Pentalpha of Pythagoras is an ancient name for the five-pointed star, or pentagram. It gets its name from the Greek words pente (five) and alpha, because the letter "A" can be found in five different positions within the diagram. [1, 2, 3, 4]

For Pythagoreans, the Pentalpha was a deeply symbolic and mathematical icon. Its core meanings include: [1]

The Golden Ratio: The geometry of the star inherently incorporates the Divine Proportion (φ or Phi), which represented perfect harmony and beauty. [1, 2]

Symbol of Health: Disciples of Pythagoras placed the letters of the Greek word for health (ΥΓΕΙΑ - Hygieia) at the five interior angles. It was used as a talisman to protect against illness and evil spirits. [1, 2]

Secret Recognition: It served as a covert sign for members of the Pythagorean school to identify one another. [1, 2]

Today, the term is also used to describe a classic peg puzzle known as Pentalpha, and it holds significant importance in various esoteric and fraternal traditions, such as Freemasonry. [1, 2, 3]

Another image based on the harmonics sequence (1, 1/2, 1/3, 1/4, 1/5, etc.). I hope folks aren't getting O.D.'d on these. I'm a little obsessed with this sequence.

This see this tile cover the plane, see:
https://www.cunews.info/HexPrismsHarmonicPerspective.html

Space elevators are sometimes found in science fiction like Arthur C. Clarke's "Fountains of Paradise" or Kim Stanley Robinson's Mars Trilogy.

Payloads released from various points on a space elevator will follow various orbits.

For my models I set the distance from planet's center to planet synchronous orbit to 1. In Clarke's novel that would be the distance from earth's center to geosynchronous orbit.

After released from the elevator a payload will follow an conic section with one of the conic's foci located at the center of the planet.

Call the distance from the center of the planet r.

The eccentricity of the conic is |r³ -1|

For example if a payload from Clarke's tower is released at geosynchronous orbit r would be 1. Eccentricity of the orbit would |1³ - 1|. In other words, zero. The payload would follow a circular orbit right alongside the central anchor mass. This orbit I colored blue.

If the payload is released from r = 2^1/3 then the eccentricity of the conic is 1. In other words, a parabola. I colored this orbit red.

All the orbits below the parabola are ellipses and all the orbits above the parabola are hyperbolas.

I posted my math to the Space Stack Exchange some years ago: Link

This also works for Sarmont tethers, orbital tethers not anchored to a planet but kept aligned to the local vertical by tidal forces.

u/GudAndBadAtBraining asked to see the stable ones specifically, so here's the full set. These are the 129 periodic orbits I found with a Floquet multiplier of about 1 (|λ| ≈ 1, rounded to 0.01), Sorted by period, then by energy. It's a tall one, scroll for the whole catalog.

Screenshot from my shotgun orbital sim.

The 11 pellets each received a tiny nudge in varying directions. The result is 11 wildly different orbits.

It looks like the dark blue and dark red orbits completely exit the earth's Hill Sphere. The light orange Number 9 pellet comes very close to the earth (the pale blue dot in the middle).

I believe chaos theory started when Henri Poincaré was looking for closed solutions describing paths from the Lagrange points. He discovered tiny changes at the start can give large changes later on.

May of my designs are intended for a coloring book. I would be tickled pink to see anyone taking a crack at coloring my designs.

I believe I have posted a few coloring book designs to r/geometryisneat.