Rethinking the real line #SoME3

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We take a geometric approach to rational numbers, to rethink how to organize the real line. Along the way, we visualize Diophantine approximation and continued fractions. And your favourite number, pi.

Much of the mathematics here is based on the following article:

A big thanks to the Summer of Math Exposition competition for the motivation to make this happen, and a big thanks to my audience for forgiving my video-editing non-skills.

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Music used in the video:
Walk Through the Park -- TrackTribe
George Street Shuffle -- Kevin MacLeod
Quarter Mix -- Freedom Trail Studio
Love Struck -- E's Jammy Jams

Proof of Concept

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The protective geometry view of the rationals reminds me of the gaps I'd see while driving past a vineyard.

deityblah

Ok so that bit of projective geometry going from the 2D grid to the 3D representation blew my mind. What a fascinating video!

kaisassnowski

YouTube has a lot of trash on it - and then it has things like this. I think this is a serious contender for the best STEM-related video I've ever seen.

juliannewmanndchannelmusi

Everywhere I go with visual representations for math, I ended up seeing infinitely repeating fractals

ghostagent

You've just transformed the way I think about numbers forever

iofish__

I've watched a lot of SoME3 videos, and this is one of my absolute favourites. I can really feel the "It would be cool if I could animate this idea" mindset present throughout the video, and I love that you decided to share them. I can tell you had fun making this.

henryginn

My dumbass read this as “ranking all real numbers” like there would be a tier list of infinite length

RigoVids

Great video. Very briefly I thought this was going to veer into p-adic numbers.

eugenemeidinger

Holy smoke didn't know there is such deep connection between the reals, projective geometry and complex plane.

yqisq

"A real number that cannot be described in finitely many English words"
*boom Berry's paradox*

katakana

As a mathematician, I am impressed and inspired by the beautiful and understandable visualization of some pretty deep mathematical concepts. Great work.

jurjenbos

Thanks, nice introduction to Stern-Brocot type structures. Among bases, unary is of course the most basic. We can start constrution of number system (and lot else) from a chiral pair of symbols, and relational operators < 'increases' and > 'decreases' do fine. From these we get two basic palindromic seeds, outwards < > and inwards > <. So, let's concatenate some mediants from the outwards seed:

< >
< <> >
< <<> <> <>> >
< <<<> <<> <<><> <> <><>> <>> <>>> >

We get countable objects from the second line, <> 'both increases and decreases' as the denominator element, and < and > as the integral numerotor element. This way the numerical count of the second row is familiar looking 1/0, 0/1, 1/0, and it's easy to check that we get the ordered rationals in their reduced forms.

Row by row construction, which preserves the previous mediants on each new row, gives better visual of the binary tree of blanks, which divide the palindromic strings into words. Along that binary tree, as discussed in the video, "irrationals" can be represented as L and R paths. Notationally parsimonous way is to write L as < and R as >. Square roots have repeating periods, which is nice.

Standard representations of continued fractions don't necessarily coincide exactly with Stern-Brocot paths, as here we have inverse paths as NOT-operations, with first bits on the second row interpreted e.g. as positive and negative. Eg. the φ-paths of Fibonacci fractions look like this:

LL <<><><>
LR <><><>
RL ><><><>
RR >><><><>

LL and RR are inverse NOT-operations, and sow are LR and RL. BTW a nice surprice was that the Fibonacci words associated with LL and RR have character count of Lucas numbers, and likewise LR and RL Fibonacci numbers.

BTW Base 10 is not totally arbitrary (2 hands, 5 fingers in each, 10 fingers together), . Transforming standard continued fraction representations of sqrt(n^2+1) to the path information, the periods look like this:
sqrt(2): <>><
sqrt(5): <<>>>><<
sqrt(10):
So, these path periods contain chiral substrings:
<<<>>>
>>><<<
We are beatiful. :)

santerisatama

I remember the first time I started 'getting' continued fractions so fondly. It really does feel like breaking free from the trap of decimal expantion, which fails to elegantly represent even simple ratios like 1/3 or 1/7 (let alone simple irrationals!!!)

xatnu

I have played with and pondered continued fractions so many times and never seen this connection. I'm 7/22 ashamed and 1.618 delighted by this revelation!

johnpeterson