Awesome Math Party Trick (Three Points on a Colored Line)

Show your friends a fancy math trick: a problem of finding three specially positioned points of the same color on a colored line. Have fun!

[dalesco]

2:05 If that's the kind of tricks you like to show at a party, I have no doubt you will indeed party responsibly

[benzoT]

We should define the line over a non dense set, otherwise it would be trivial. There is always a non empty closed subset of the same color, just pick the extremity of such subset and the middle.

But that's for people with a bit of maths background, works fine as it is for a party trick :)

[gogyoo]

1:59 that realisation was so cool, I was stuck on finding 3 points of one particular colour. Thanks!

[NonTwinBrothers]

I tried this and they left my party :(

[kendakgifbancuher]

My initial thought:
Every one-coloured part of the line is either continuum or a single point. Continuum has a property that each point in between of any 2 points is also inside a continuum (including midpoint).
On the other hand, we can't do colour painting with only single points (without continuums), because its impossible to transform uncountable infinity (line) into 2 countable infinities (2 sets of single points)

[landonoffmars]

Set playback speed to 0.75. That's a real party voice.

[jonpinkley]

The last move reminded be of forking in chess.

[dcterr]

Very cool result! I figured it out, by the way, but I'm still intrigued. Does this have anything to do with Ramsey theory?

[MindcraftMax]

There is a really interesting follow up question: for any weights α≠0, β≠0 with α+β≠0, can you always find three distinct points A, B, C on the colored line such that they're of the same colour and the barycentre of the first point A with weight α and second point B with weight β is the third point C, that is to say such that α vec(AC)+β vec(BC) = vec(0).
Obviously, α=β is the special case where C is the isobarycentre of A and B a.ka. the middle of the line segment AB, and it has already been proven in the video.



Spoiler:
Yes, you can. Again by contradiction, suppose there are no such points…
I'll post the solution if need be.

[cycklist]

'Math visualized' looks horrendous to a British person like me. We'd say 'maths visualised'.