Let’s play the following “chaos game”. Consider a regular triangle and take some initial point in it. Now pick one of the three vertices at random, connect it with the current point, and mark this line’s midpoint as the new point. If you repeat this process many times, what will the resulting shape look like? You might expect to find the points simply shattered everywhere, without much structure, but surprisingly, the points will in fact create a Sierpinski fractal.


There are lots of variations on this theme, some of which can be found on this Wikipedia page.

Animation made in Mathematica.

n roots of unity

When one is dealing with complex numbers, it is many a times useful to think of them as transformations. The problem at hand is to find the n roots of unity. i.e


As is common knowledge z = 1 is always a solution.

Multiplication as a transformation

Multiplication in the complex plane is mere rotation and scaling. i.e


Now what does finding the n roots of unity mean?

If you start at 1 and perform n equal rotations( because multiplication is nothing but rotation + scaling ), you should again end up at 1.

We just need to find the complex numbers that do this.i.e


This implies that :


And therefore :


Take a circle, slice it into n equal parts and voila you have your n roots of unity.


Okay, but what does this imply ?

Multiplication by 1 is a 360o / 0o rotation.

When you say that you are multiplying a positive real number(say 1) with 1 , we get  a number(1) that is on the same positive real axis.


Multiplication by (-1) is a 180o rotation.

When you multiply a positive real number (say 1) with -1, then we get a number (-1) that is on the negative real axis 

The act of multiplying 1 by (-1) has resulted in a 180o transformation. And doing it again gets us back to 1.


Multiplication by i is a 90o rotation.

Similarly multiplying by i takes 1 from real axis to the imaginary axis, which is a 90o rotation. 

This applies to -i as well.


so on and so forth,

Have a great day!

OK, but there must be hundreds of thousands, if not millions of angels (of all shapes, ranks and sizes) in Heaven


Where’s the fic in which a couple (or maybe just the one, a young and impressionable little fledgling angel whose wings have barely started molting for the first time) listens to Gabriel briefing everybody on the Official Reason The Apocalypse Is Cancelled and thinks to themselves this is a load of bullcrap

(Bc you know. You just know Gabriel would say all the wrong things. Like, what do you mean ‘unforeseen circumstances’? We’re Heaven, our job is to foresee. Circumstances. And how could it be that an angel as wicked and evil as Aziraphale is still, you know. An angel? And hasn’t fallen? Something does not add up here).

And so the young fledgling starts to wonder

And they’re not alone

Soon, there are clusters of angels everywhere, heads bent together and talking furiously in low voices.

… gave away his sword …

… what? why? …

… thought they might need it …

… I don’t understand …

… love for humans …

… and demons from what I hear …

… but why …

… saving the world …

… more important …

… ineffability …

… good guys? …

… are we? …

… is he? …

… Fucking Gabriel that’s for sure …

And as the whispers spread (especially that last one), the young fledgling decides to delve into the Earth Observation Files to go see for themselves what Aziraphale has been up to in the past six millennia.

And there’s no time in Heaven, but if there was, it would take them a lot of it before they came out again, looking very thoughtful.

They spends some more not-time in a quiet corner (somewhere the harp music isn’t too obnoxious), a deep frown on their angelic face and eyes red-rimmed and shining. There’s a sniffle, occasionally.

And then, finally, they seek out the nearest Whispering Angel Cluster and tells them what they’ve seen. And the conclusion they’ve come to.

(One or two angels in the cluster gasp, unbelieving. They decide to take the matter to Gabriel, despite the fledgling’s protests and ask is it true? About Aziraphale and his demon, is it true that they saved humanity? Is it true that they saved humans? Not just from the Apocalypse, but many many many times before? Is it true?

And again, Gabriel scoffs and laughs and says exactly the wrong thing (’Don’t you worry your pretty little head about that naughty angel, sweet child. He and that wily serpent of his won’t bother us any longer.’) and that’s all the proof the young angels need that Something Is Not Right.

They return to Fledgling #1, wings drooping and head bowed. There are more whispers (… can we? … Gabriel would murder … Hellfire, they said… Should we?) and eventually, after eons of deliberation, a decision is made.

And so it happens that on a bright, sunny Saturday morning in SoHo, the door to a bookshop tingles open and half a dozen remarkably bland looking humans walk in. And after Aziraphale has called Crowley back to heel and has wrested the nice, leather bound and heavy first edition of the Principia Mathematica out of his demon’s hands, and after the angels have been persuaded to come down from the top of the bookshelves…

He asks them why they’re here. The answer leaves him confused and makes Crowley laugh so hard he almost cracks a rib:

Teach us

Teach us compassion

Teach us to be kind

Teach us about humans

Teach us about what we’re meant to protect

Teach us how to protect

Teach us how to be the Good Guys, like, for real

And Aziraphale is confused and touched (mostly confused), but he agrees. Under one condition: they will have to listen to the demon too, because Aziraphale knows he would be nowhere near the angel he is today if it hadn’t been for Crowley.

(Crowley goes oddly quiet at that for a moment, before he realises Aziraphale is basically giving him shared command of a mini Host. Then he starts howling with laughter again).

Then the rest of the fic is basically just Aziraphale, Crowley and their mini Host of fledglings getting into various kinds of (food and non food related) shenagigans. 

(‘Angel, when they said teach us, I’m pretty sure they did not mean teach us about the best place to get Vietnamese Pho!’ ‘Shut up, Crowley. Also, don’t think I didn’t hear how you told Amriel about how you were aboard the RMS Carpathia in 1912. And let me just say, I knew it.’ ‘Shut up, angel.’)

At a certain moment, Gabriel and the others get wind of aforementioned shenanigans. But by then, it’s too late as the Mini Host has communicated all their efforts and findings back to the various other MIni Hosts that are still in Heaven and now everybody is thoroughly convinced Gabriel’s a fucking prick who Should Not Be In Charge. It’s not a second Rebellion, per se. It certainly involves a lot less sulphur and brimstone and screaming. But when all’s said and all’s done, Gabriel finds himself permanently retired, along with Michael, Sandolphon and Uriel, as a New Policy is put in place:

To Be The Good Guys, We Have To Be The Good Guys

Eloquently put,’ Crowley sniggers. Aziraphale tries not to laugh before he tsks and shakes his head. ‘They’re learning, dear. Give them time.’