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Fun fact of the day (Day 42 without football):
If you have a map of the place you are in, mathematically, there will always be some point on your map that is directly above the place in the real world it represents. Always.
If you have a map of the place you are in, mathematically, there will always be some point on your map that is directly above the place in the real world it represents. Always.
It doesn't matter how you hold the map. The map can be rotated, flipped, even twisted, folded or crumpled.
It is guaranteed by Brouwer's fixed point theorem. A fixed point is anything that goes nowhere after a transformation.
It is guaranteed by Brouwer's fixed point theorem. A fixed point is anything that goes nowhere after a transformation.
Brouwer's theorem tells us that: It is impossible to completely mix up a set of points if they are bounded, without holes and transformed continuously, with no cutting or gluing.
Without boundaries, every point can be mapped somewhere new. Around holes, every point can be mapped somewhere new, and if you cut or glue, every point can be mapped somewhere new.
But otherwise, mixing will always fail somewhere.
But otherwise, mixing will always fail somewhere.
Here's a question do you think you can completely mix up coffee in a mug by stirring?
Well, of course you can... NOT.
Stirring coffee is a continuous transformation of the coffee and everything stays within the same space, so Brouwer's fixed-point theorem applies.
Well, of course you can... NOT.
Stirring coffee is a continuous transformation of the coffee and everything stays within the same space, so Brouwer's fixed-point theorem applies.
No matter how well you try to stir, there will always be – after the liquid has settled – at least one point that you stirred right back to where it was before, a fixed point.
To be sure, coffee isn't made out of points, it's made out of molecules. But they're quite tiny and very numerous, so within a degree of error it'll pretty much hold.
Fixed points don't just frustrate your ability to fully mix things, they can also suck ... you
toward them, they can be attractive. Like the number nine.
toward them, they can be attractive. Like the number nine.
Try this: think of a number with more than one digit and then add up its digits. Now subtract that sum from the original number to get a new number. If you do this over and over and over again until you're left with a single-digit number, you will always end up with 9.
Always.
Always.
Also notice that any number with two or more digits minus the sum of its digits becomes divisible by nine right away. What's going on here?
Well, there's nothing mystical about nine-ness in general, instead it's just a consequence of how we write numbers.
Well, there's nothing mystical about nine-ness in general, instead it's just a consequence of how we write numbers.
Numbers can be written in all sorts of
ways, but the most common – Base ten
positional notation – makes the nine trick
work.
In this system, a number like 25 doesn't mean 2 and 5, it means ten 2's and one 5, that's twenty-five.
ways, but the most common – Base ten
positional notation – makes the nine trick
work.
In this system, a number like 25 doesn't mean 2 and 5, it means ten 2's and one 5, that's twenty-five.
So subtracting out just the positional digits of a number removes one member from each group. The digit you have one of completely disappears. The digit you have ten of goes down to nine copies (100 becomes 99 etc), so the whole thing becomes a multiple of nine.
My favourite fixed point related thing is the Borsuk–Ulam theorem. It states that at any given moment there must be at least one pair of points on Earth's surface that are diametrically opposite one another but nonetheless, have the same temperature and atmospheric pressure.
How crazy is that? Even though weather is chaotic and always changing and even though the other side of the world is very very far away, there must always be at least two places on complete opposite ends of the earth where the temperature and the pressure are the same.
This is true, by the way, for any two variables that vary continuously across Earth's surface, and it can be mathematically proved. (This proof is fairly straightforward, and is explained very nicely in the linked video at the end).
Opposite points on a sphere are called Antipodes. There are sites that help you locate such opposite points:
https://www.jasondavies.com/maps/antipodes/ https://www.findlatitudeandlongitude.com/antipode-map/#.V-nBkZMrIUF
https://www.jasondavies.com/maps/antipodes/ https://www.findlatitudeandlongitude.com/antipode-map/#.V-nBkZMrIUF
If you try to look for antipodal points on land, you'll notice that most points on land are antipodal to water. Since the Earth's surface is mainly covered with water, this makes sense. In fact, the Pacific Ocean contains its own antipodes.
All this came from a VSauce video called "Fixed Points" (linked below), which is probably one of my favourite VSauce videos to date. Check it out for some extra stuff and explanations that I couldn't fit into this thread.
Thanks for reading!
Thanks for reading!
