Rethinking the real line #SoME3

Comments · 315

• KA

  @kaisassnowski · 3 years ago

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

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• DE

  @deityblah · 3 years ago

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

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• HE

  @henryginn7490 · 3 years ago

  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.

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• IO

  @iofish__ · 3 years ago

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

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• GH

  @ghostagent3552 · 2 years ago

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

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• DR

  @DrJulianNewmansChannel · 2 years ago

  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.

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• JU

  @jurjenbos228 · 3 years ago

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

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• SO

  @solarcrystal5494 · 3 years ago

  Most real numbers are not computable, most are just the random L and R sequences

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• KA

  @katakana1 · 3 years ago

  &quot;A real number that cannot be described in finitely many English words&quot;<br><b>boom Berry&apos;s paradox</b>

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• XA

  @xatnu · 3 years ago

  I remember the first time I started &apos;getting&apos; 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!!!)

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• SA

  @santerisatama5409 · 3 years ago

  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 &lt; &apos;increases&apos; and &gt; &apos;decreases&apos; do fine. From these we get two basic palindromic seeds, outwards &lt; &gt; and inwards &gt; &lt;. So, let&apos;s concatenate some mediants from the outwards seed: <br><br>&lt; &gt;<br>&lt; &lt;&gt; &gt;<br>&lt; &lt;&lt;&gt; &lt;&gt; &lt;&gt;&gt; &gt;<br>&lt; &lt;&lt;&lt;&gt; &lt;&lt;&gt; &lt;&lt;&gt;&lt;&gt; &lt;&gt; &lt;&gt;&lt;&gt;&gt; &lt;&gt;&gt; &lt;&gt;&gt;&gt; &gt;<br><br>We get countable objects from the second line, &lt;&gt; &apos;both increases and decreases&apos; as the denominator element, and &lt; and &gt; 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&apos;s easy to check that we get the ordered rationals in their reduced forms. <br><br>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, &quot;irrationals&quot; can be represented as L and R paths. Notationally parsimonous way is to write L as &lt; and R as &gt;. Square roots have repeating periods, which is nice. <br><br>Standard representations of continued fractions don&apos;t necessarily coincide exactly with Stern-Brocot paths, as here we have &nbsp;inverse paths as NOT-operations, with first bits on the second row interpreted e.g. as positive and negative. &nbsp;Eg. the φ-paths of Fibonacci fractions look like this:<br><br>LL &lt;&lt;&gt;&lt;&gt;&lt;&gt;<br>LR &lt;&gt;&lt;&gt;&lt;&gt;<br>RL &gt;&lt;&gt;&lt;&gt;&lt;&gt;<br>RR &gt;&gt;&lt;&gt;&lt;&gt;&lt;&gt; <br><br>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 &nbsp;likewise LR and RL Fibonacci numbers. <br><br>BTW Base 10 is not totally arbitrary (2 hands, 5 fingers in each, 10 fingers together),. &nbsp;Transforming standard continued fraction representations of sqrt(n^2+1) to the path information, the periods look like this: <br>sqrt(2): &lt;&gt;&gt;&lt;<br>sqrt(5): &lt;&lt;&gt;&gt;&gt;&gt;&lt;&lt; <br>sqrt(10): &lt;&lt;&lt;&gt;&gt;&gt;&gt;&gt;&gt;&lt;&lt;&lt;<br>So, these path periods contain chiral substrings: <br>&lt;&lt;&lt;&gt;&gt;&gt;<br>&gt;&gt;&gt;&lt;&lt;&lt;<br>We are beatiful. :)

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• VE

  @venpopov · 11 months ago (edited)

  I keep rewatching this video every few months. It gave me such a deep glimpse into what numbers are aside from how we talk and writeabout them. It should have at least 10x the views