What are they and how do you compose such?
How do you compose euclidean rhythms manually?
Slime
#2
Typically you either need a dedicated sequencer (VST forms exist, although a lot of them are just modular) or something that handles math, but they’re basically an evenly-spaced number pattern that’s distributed as evenly as possible in relation to the whole.
For example, your 2 numbers could be 5 (for the hits) and 16 (for the steps), and what you’ll essentially get is something that sounds like a very tribal / esoteric rhythm (which hits an evenly-spaced 5 times for one or two bars, depending on your BPM and whatever other factors exist for 16 steps and where you even started counting), as these have weirdly been around in the natural world for an extremely long time, and then if you add in something like 4 (for the hits, or ‘top number’) and 16 (for the steps) alongside the first example, you’ll sound an awful lot like Meshuggah because you’ll have the classic euclidean polyrhythm that everyone seems to naturally create when they discover polyrhythms that all revolve around an even 8/16 beats.
Here’s one of the many times when I used them in random videos – pay no attention to the pitches (they’re just sample and hold values, which in plain English just means ‘random number generator’), but notice that the rhythms tend to jump around one another in a very familiar fashion.
This one’s kind of long-winded, but if you skip around, I made a whole metal track in realtime using the same euclidean sequencer. It basically just sounds like djent – but totally unhuman, all machine :D.
(Skip ahead to maybe like 25:00 and you’ll hear what I’m talking about)
Another way to get your hands on some is to just use a pack – I’ve even made packs that go from 1-16 on all 16 beats (so that’s 16 patches in total) so you could just take my MIDI clip and use it as a sequencer if you want.
The real fun tends to begin when you start using random bottom numbers (rather than just an even 16-steps), and then cross two polyrhythms with an AND, OR, XOR or any other logic gate to basically modulate and dictate when your triggers even fire to begin with. Although I’ll admit that this last example is probably not for people who aren’t just a little autistic or something, lol.
Oh, and what about modulating the bottom number at different intervals? OK, I’ll stop. 
Challenge 7C euclidean rhythyms deadline tbd
Slime
#4
...generating a Euclidean rhythm with 5 pulses distributed across 8 steps using the Euclidean algorithm:
P = 5 (number of pulses)
S = 8 (number of steps)
Using the Euclidean algorithm:
Divide 5 by 8: Q = 0, R = 5
Add 0 pulses to the sequence of steps.
Apply the Euclidean algorithm with S = 8 and R = 5.
Divide 8 by 5: Q = 1, R = 3
Add 1 pulse to the sequence of steps.
Apply the Euclidean algorithm with S = 5 and R = 3.
Divide 5 by 3: Q = 1, R = 2
Add 1 pulse to the sequence of steps.
Apply the Euclidean algorithm with S = 3 and R = 2.
Divide 3 by 2: Q = 1, R = 1
Add 1 pulse to the sequence of steps.
---------------------------------------------
Since there is no remainder (R = 1), the algorithm stops.
The resulting sequence of steps is: [1, 0, 1, 0, 1, 0, 1, 0].
Technically if you can write a script, this doesn’t seem too bad. I’d consider making something like this for Sonic Pi, but I don’t think it spits out MIDI (only controls external MIDI devices IIRC). This would definitely work on a Raspberry Pi or Arduino Leonardo though