It looks suspiciously like a circle. But the regular 65537-gon is
famous. Gauss proved you can construct a regular
It seems Hermes, ironically named after the Greek god of speed, spent a whole decade writing a 200-page manuscript explaining how to construct a regular 65537-gon with ruler and compass. But this paper of his is just 16 pages long:
If you take Pascal's triangle mod 2 and draw black for 1 and white for 0, you get a pleasing pattern.... closely connected to a fractal called the Sierpinski gasket, where you keep cutting out triangular holes from an equilateral triangle:
The rows of Pascal's triangle mod 2 give numbers in binary:
They're the products of distinct Fermat primes
We only know 5 Fermat primes: for example, the next Fermat number is not
prime:
As a result, we only 32 regular polygons with an odd number of sides
that can be constructed by straight-edge and compass: the polygons
with
For more, see my blog article:
People like to make fun of epicycles because you can describe any orbit using epicycles. Above we see an almost square orbit done using epicycles.
But this weakness is also their strength: a Fourier series is a way of writing any periodic function using epicycles.
My former student Brandon Coya discovered this "commutative law". It
can't be new. It's easy to prove. But I'll name it after him. My
student Joe Moeller quickly noticed that the operation
Over on Twitter, someone noticed that this operation distributes over
multiplication:
See what's going on? The key is that
The Milky Way is a spiral galaxy, so it has spiral arms. We are in the inner edge of the Orion Arm. You can see the Orion Arm from here! This photo of the Orion Arm was taken by Ahmed abd Elkader Mohamed in Sinai, Egypt.
The Orion Arm is between the Sagittarius Arm, closer to to the galactic center, and the Perseus Arm. It may be a mere 'spur' connecting these arms — that's what the picture below calls it. Or it may be a truly separate arm. This seems to be the more popular view today.
It seems the 'density wave' theory of spiral arms is still controversial, but it says stars orbiting the galactic center temporarily bunch up to form arms. If so, a given star doesn't stay in the same arm forever.
In fact, most stars within 1000 parsecs of the Sun belong to 5 or more 'streams': groups with similar velocity. The most well-established are the Coma Berenices, Pleiades, Hyades, Sirius and Hercules streams. Here are the U (radial) and V (angular) components of their velocities.
The paper where I got this graph argues that Oumuamua, the interstellar asteroid that shot past us last year, is part of the Pleiades stream! Its velocity looks right.
This is the paper:
In short: the Milky Way is a complex and lively place, though it operates on majestically slow time scales. The Sun has orbited the Galaxy only 20 times in its life so far!