Mathematical Coincidences

Поділитися
Вставка
  • Опубліковано 10 січ 2025

КОМЕНТАРІ • 511

  • @Kuvina
    @Kuvina  11 місяців тому +393

    CORRECTION: At 2:29, the identity should say 8pi*k^2, not 8pi^(k^2).
    ANOTHER CORRECTION: At 1:36 I said the chance of getting 6 of any digit in a row within the first 768 digits is < 0.1%. However, I just ran a simulation (on 10 million 768 digit sequences), and I got 0.68446%. In my opinion a 0.69% chance is still notable enough to be in this video, but it's not quite as rare as I thought. The confusion comes from ambiguity in language. I thought the source meant "the chance that you get at least 1 of these patterns is

    • @thumbgoblin4716
      @thumbgoblin4716 11 місяців тому +5

      twas bangin👍

    • @jan_Eten
      @jan_Eten 11 місяців тому +4

      pona suli a!

    • @GoodrichT6
      @GoodrichT6 11 місяців тому +8

      Kuvina I think you're so cool ily 🥺

    • @redstowen
      @redstowen 11 місяців тому

      Relativity ep. 6 when

    • @JohnDoe-ti2np
      @JohnDoe-ti2np 11 місяців тому +2

      Fun video! It's worth mentioning that the theta function explanation of Gelfond's constant e^pi is due to Aaron Doman. By the way, dividing an octave into 19ths arguably gives you even better approximations to "nice" intervals. The minor third (6/5), major third (5/4), and major sixth (5/3) are better approximated by 2^(5/19), 2^(6/19), and 2^(14/19) than by 2^(3/12), 2^(4/12), and 2^(9/12), and the perfect fourth (4/3) and perfect fifth (3/2) are approximated by 2^(8/19) and 2^(11/19) almost as well as by 2^(5/12) and 2^(7/12). So close approximation isn't the only reason for the choice of a 12 equal temperament scale.

  • @FiniteSimpleFox
    @FiniteSimpleFox 11 місяців тому +2072

    In general the chance of a specific coincidence occurring is very low, however the chance of *some* coincidence occurring is very high

    • @bluepiston9347
      @bluepiston9347 11 місяців тому +97

      Almost no one understands this lol

    • @AdiDK
      @AdiDK 11 місяців тому +13

      exactly

    • @pablojuan4679
      @pablojuan4679 11 місяців тому +71

      Chance of tornado vs chance of disaster

    • @samylemzaoui2298
      @samylemzaoui2298 11 місяців тому +31

      i cant even count the number of times i tried to explain that to someone and failed miserably

    • @matze9713
      @matze9713 11 місяців тому +113

      ​@@samylemzaoui2298the chance that a specific person understands this is very low, but the chance that some person does is high

  • @notexactlysiev
    @notexactlysiev 11 місяців тому +894

    This is only half mathematical, but I like how the ratio of miles to kilometers (1.609344) is close to the golden ratio (1.61803...)
    This means you can approximately convert those units using the Fibonacci sequence. 2 miles is about 3 km, 3 miles is about 5 km, 5 miles is about 8 km, etc.

    • @TabooGroundhog
      @TabooGroundhog 11 місяців тому +218

      Don’t use the beginning tho lol “1 mile is about 1 km”

    • @pmmeurcatpics
      @pmmeurcatpics 11 місяців тому +25

      That's so cool!

    • @jongyon7192p
      @jongyon7192p 11 місяців тому +64

      And it's convenient bcuz you only ever need to go *up* the sequence, cuz nobody except USA would convert metric back into Oppression Units.

    • @gudmansal3468
      @gudmansal3468 11 місяців тому +12

      close enough@@TabooGroundhog

    • @12Rosen
      @12Rosen 11 місяців тому

      @@jongyon7192pwtf is an oppression unit

  • @robo3007
    @robo3007 10 місяців тому +70

    Here's another one. 82,000 is 10100000001010000 in base 2, 11011111001 in base 3, 110001100 in base 4, and 10111000 in base 5. It is predicted to be the largest number to be represented using only 1s and 0s in all four bases and is thought to be a massive coincidence that a number so large even has that property to begin with.

    • @skysurfer1679
      @skysurfer1679 7 місяців тому +2

      Wtf bro💀💀💀

    • @NocturnalTyphlosion
      @NocturnalTyphlosion 2 місяці тому +2

      also it goes up to base 5, the number of digits it has :3

    • @wpheavyww
      @wpheavyww 2 місяці тому

      ​@@NocturnalTyphlosion looking at your pfp i bet you are insufferable irl

  • @msman3249
    @msman3249 11 місяців тому +186

    The first 40 seconds of the video is literally "How to memorize 15 digits of e"

    • @mbdg6810
      @mbdg6810 10 місяців тому +6

      That is pretty cool though.

    • @huzzah_2763
      @huzzah_2763 6 місяців тому +1

      2.7TolstoyTolstoyrighttriangle

    • @ÞeSheep1
      @ÞeSheep1 3 місяці тому

      ⁠​⁠@@huzzah_27632.7TolstoyTolstoyrighttrianglefirstthreeprimesdegreesinacircle (2.718281828459045235360)

  • @Karaokedad80
    @Karaokedad80 11 місяців тому +549

    One that I was waiting to see if you mentioned:
    10! seconds = exactly 6 weeks.

    • @AlexanderWeixelbaumer
      @AlexanderWeixelbaumer 11 місяців тому

      10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 #Six weeks, you can strike * 6 and * 7 to have a day, they multiply to 42
      10! / 42 = 10 * 9 *8 * 5 * 4 * 3 * 2 * 1 #Now you strike 2 * 3 * 4 to have an hour, they multiply to 24
      10! / 42 / 24 = 10 * 9 * 8 * 5 * 1 #And finally you can strike 10 * 9 * 8 * 5 to get a second, since they multiply to 3600
      10! / 42 / 24 / 3600 = 1
      So the "magic" is the 6 in six weeks that is left, because you can use the others to make seconds, days and weeks.

    • @TabooGroundhog
      @TabooGroundhog 11 місяців тому +99

      That’s SUPER cool that it’s not just close but exactly three Fortnites. Which I guess makes sense since we use highly composite numbers for times, but it’s lucky that weeks are 7 days. I’ll check the rest cause I’m curious
      4! = 24 seconds
      5! = 2 minutes
      6! = 12 minutes
      7! = 1 hour and 24 minutes
      8! = 11 hours and 12 minutes
      9! = 100 hours and 48 minutes
      11! = 1 year 13 weeks 6 days

    • @letMeSayThatInIrish
      @letMeSayThatInIrish 11 місяців тому +6

      That's lovely and insane!

    • @petterlarsson7257
      @petterlarsson7257 11 місяців тому +118

      @@TabooGroundhog its spelled fortnight not fortnite

    • @XplosivDS
      @XplosivDS 11 місяців тому +48

      @@petterlarsson7257 We have been tainted by those 90's

  • @robo3007
    @robo3007 10 місяців тому +21

    My favourite mathematical coincidence is that if you look at space between e and pi on the number line and mark a point exactly two thirds across, that point is almost exactly the number 3 (3.000489)

    • @ernestoherreralegorreta137
      @ernestoherreralegorreta137 6 днів тому +1

      This is equivalent to the "e + 2π ≅ 9" coincidence that he listed above. Your wording in math: "e + ⅔(π - e) ≅ 3" simplifies to "⅓ (e + 2π) ≅ 3" then just multiply both sides by 3.

  • @Yvelluap
    @Yvelluap 9 місяців тому +9

    2:20 i envy your ability to convey this much raw happiness in a single drawing

  • @ryandupuis5860
    @ryandupuis5860 11 місяців тому +88

    It should also be noted that these situations only occur in base 10, which is a human-based standard. Other bases may have coincidences like these, either more or fewer, though.

  • @kyay10
    @kyay10 2 місяці тому +5

    You note around 1:20 that the pie coincidence is a product of language, but it's important to also note that a lot of the coincidences in this video are a product of using a base 10 system, and that they thus are "arbitrary"

  • @parthhooda3713
    @parthhooda3713 10 місяців тому +29

    Math class initially: 6:40
    me: *blinks for a nanosec*
    7:40

    • @GlowBerryPumpkin
      @GlowBerryPumpkin 4 місяці тому +4

      You can fit 6 circles around a circle
      "Yeah that makes sense"
      You can fit 12 spheres around a sphere
      "Yeah I can see that..."
      24th dimensional Hyperspheres
      "What the duck"

  • @Skyb0rg
    @Skyb0rg 11 місяців тому +71

    The fact that 2^31+1 is prime is one of the most useful coincidences in cryptography, since large primes are needed for the math aspect and modulo multiplication’s runtime is based on the number of 1s digits in binary which is useful for the calculation aspect.

  • @johnbarnhill386
    @johnbarnhill386 11 місяців тому +40

    It’s worth mentioning that a lot of these coincidences arise because we use base 10. There might be other coincidences in other bases that we don’t know about

    • @baconheadhair6938
      @baconheadhair6938 9 місяців тому +1

      whatif we use base 1

    • @smasher_zed8888
      @smasher_zed8888 8 місяців тому +1

      @@baconheadhair6938 base 1 is kinda just tally marks when you dont do the slash for 5 since the number in the base is just the number when you switch digits, for example in base 2 it goes 00, 01, 10, 11 (counting 1 to 4). for base 1, it would be 1, 11, 111, 1111 (counting 1 to 4)

    • @lailoutherander
      @lailoutherander Місяць тому +1

      ​@@baconheadhair6938 literally no better than just counting

    • @baconheadhair6938
      @baconheadhair6938 Місяць тому

      @@smasher_zed8888 you cant use the number 1 in base 1

    • @smasher_zed8888
      @smasher_zed8888 Місяць тому

      @@baconheadhair6938 oh right it would just be 0 right

  • @yellowmarkers
    @yellowmarkers 11 місяців тому +82

    One thing I like a bit more than the fact that π has a string of six 9s at digit 763 is the fact that 2π has a string of seven 9s at digit 837. It isn't the first instance of four characters in a row in 2π's decimal expansion (since there was a "1111" before it) but it's still the first instance of five, six, and seven characters in a row.

    • @DavidSartor0
      @DavidSartor0 11 місяців тому +1

      Yay.

    • @asheep7797
      @asheep7797 10 місяців тому +4

      …49999998…
      turns into
      …99999996…

  • @n16161
    @n16161 10 місяців тому +43

    My dude just causally explained the mathematical basis for music in the middle of this.

    • @LipticZone
      @LipticZone 10 місяців тому +1

      honestly quite incredible

    • @paintspot
      @paintspot 9 місяців тому +1

      Kuvina's not a dude, btw.
      -Paintspot Infez
      Wasabi!

    • @n16161
      @n16161 8 місяців тому +2

      @@paintspot my girl just casually explained the mathematical basis for music in the middle of this.

    • @edvinkarlsson9368
      @edvinkarlsson9368 8 місяців тому

      @@paintspotwhat is kusina then? How is kusina not a guy

  • @brodydrawsstuff
    @brodydrawsstuff 11 місяців тому +18

    Here’s another fun one:
    50/49 will spell out the powers of 2, each spaced 2 decimal places apart.
    (50/49=1.0204081632…)

    • @CliffSedge-nu5fv
      @CliffSedge-nu5fv 11 місяців тому +6

      ...1632653061... aw, shucks. Had to carry the 1 for 128 since it didn't fit in a 2-digit space.

    • @CliffSedge-nu5fv
      @CliffSedge-nu5fv 11 місяців тому +7

      Can write it as a series of fractions instead: (2/100)^0 + (2/100)^1 + (2/100)^2 + (2/100)^3 + ...
      = sum_k=0 to infinity (2/100)^k = 50/49 as infinite geometric series.

    • @brodydrawsstuff
      @brodydrawsstuff 11 місяців тому +2

      Hell yea

    • @findystonerush9339
      @findystonerush9339 8 місяців тому +4

      1/96 Will spill out the powers of four: 1/96=0.01041666666666666666... The powers of fours overlap which makes a string of infinite six's

    • @areadenial2343
      @areadenial2343 8 місяців тому +2

      This one is fun! It works because all rational numbers can be constructed from an infinite series, in this case powers of 2. The same is true for 500/499, 5000/4999, and so on, producing larger spaces between powers of two. Eventually, all the powers of two overlap with each other to form a repeating decimal. However, 5/4 is the only one of these numbers which has a terminating decimal representation: 1.25. (Of course, it can also be represented by the repeating decimal 1.249999999...)

  • @Dissimulate
    @Dissimulate 10 місяців тому +8

    I see average people being surprised by coincidences. I try to explain to them how with the number of things that they see, these coincidences are almost certain.

  • @Relkond
    @Relkond 10 місяців тому +7

    2 points about the 7th US president - he was elected in 1828, and served 2 terms.
    If you draw a diagonal line across his square picture, you get a triangle with 3 angles: 45, 90, and 45
    rewriting all that: 2. 7 1828 1828 45 90 45

  • @pinkraven4402
    @pinkraven4402 10 місяців тому +4

    As an amateur musician it always fascinated me how actually lucky it is that 12 tone equal temperament (where each note is the previous one multiplied by 2^(1/12) can get you so close to the most important musical intervals such as 3/4 and 2/3. Sure, maybe that's not as surprising, because from all the possible reasonable divisions of an octave, like 13, 14, 15 notes, one of them should be good enough in approximating these crucial intervals, but, idk, it's very pleasing to me

  • @PantheraLeo04
    @PantheraLeo04 11 місяців тому +41

    There's probably a explanation for why this is the case, but I find it interesting that the first 2 hyper-operations are both commutative and associative but all the following hyper-operations have neither property.

  • @PlantNocturnal
    @PlantNocturnal 11 місяців тому +34

    The fun thing about this is that It's genuinely confusing whether a mathematical coincidence is a thing that makes sense. You have situations like these where there isn't a clear explanation and doesn't seem like there should be, but then everything is still logically determined and interrelated, to some extent just determining that some weird correlation is going on is an explanation.

    • @fahrenheit2101
      @fahrenheit2101 10 місяців тому +1

      No, most of the times there's no reason at all to find it suspicious
      Most of these are base 10 specific for example, but there's nothing special about 10 at all.

    • @willzhao5889
      @willzhao5889 Місяць тому

      ​​@@fahrenheit2101 our fingers

    • @fahrenheit2101
      @fahrenheit2101 Місяць тому

      @@willzhao5889 i meant mathematically ofc

    • @willzhao5889
      @willzhao5889 Місяць тому

      @@fahrenheit2101 ye our fingers

    • @willzhao5889
      @willzhao5889 Місяць тому

      @fahrenheit2101 reality is math, is it not? ;)

  • @themathhatter5290
    @themathhatter5290 11 місяців тому +26

    Other mathematical coincidences involving pi:
    sqrt(2)+sqrt(3) ~~ pi
    9/5+sqrt(9/5) ~~ pi
    e^(pi*sqrt(163)) ~~ (640320)^3+744
    Not involving exact mathematical numbers
    (mile*Astronomical unit)/(inch*light year) ~~ 1

    • @aioia3885
      @aioia3885 11 місяців тому +3

      if you ever want to approximate pi with ruler and compass you can draw a circle with r = 1 and then the side length of the inscribed square will be √2 and the side length of the inscribed equilateral triangle will be √3 so if you add them you can approximate pi since √2+√3 ≈ π

    • @mykal4779
      @mykal4779 11 місяців тому +10

      i feel like a cooler way to phrase the last one is "a mile is to a lightyear as an inch is to an AU" or "there are as many miles in a lightyear as there are inches in an AU"

    • @ixion2001kx76
      @ixion2001kx76 11 місяців тому +3

      Holy cow, the last one is good to 2.8 parts per Nonilion (10^30)

    • @ixion2001kx76
      @ixion2001kx76 11 місяців тому +3

      The 9/5ths is good to 15ppm-much better than 355/113

    • @Anonymous_MC
      @Anonymous_MC 11 місяців тому

      why am i seeing the number 640320 everywhere

  • @lumipakkanen3510
    @lumipakkanen3510 11 місяців тому +26

    To truly answer "why 12 notes" you need to consider more than 3/2 and 4/3. One component is to make sure that approximations to 10/9 and 9/8 coincide into a "meantone" which might as well give you 31 notes per octave. Another component is to make sure that three approximate 5/4 major thirds stack up to an octave (in other words tempered 125/64 and 128/64 sound alike) . 12-tone equal temperament is the only equal division of the octave satisfying both of these requirements.

    • @jhgvvetyjj6589
      @jhgvvetyjj6589 11 місяців тому +4

      The three major thirds stacking to an octave isn’t that important to most music, it’s something that 12edo happened to have, but it’s also related to the 1024≈1000 approximation.

    • @lumipakkanen3510
      @lumipakkanen3510 11 місяців тому +1

      @@jhgvvetyjj6589 yep. 19-tone equal tone temperament also has that 10/9 ~ 9/8, with a ton of other flavor on top of it.

    • @jhgvvetyjj6589
      @jhgvvetyjj6589 11 місяців тому +1

      ​@@lumipakkanen3510 12 notes per octave however was a better fit for the Pythagorean tuning, which is based on exact prime factors of 2 and 3. And meantone happened to be the most natural way to incorporate the next prime factor (5) in the 12 tone system, where 5 is approximated as +4 fifths (factors of 3) in the circle of fifths (5≈3⁴÷2⁴). The other possible approximation, 5≈3⁻⁸×2¹⁵ was not as commonly available in a 12 tone keyboard and is not part of a major or minor scale without wolf fifths.

    • @lumipakkanen3510
      @lumipakkanen3510 11 місяців тому

      @@jhgvvetyjj6589 Sure, but in that case just say that 531441/524288 is tempered out in the 2.3 subgroup and you're done. No need to involve prime 5. Personally I might even go as far as to interprete 12-TET in 2.3.19 tempering out 513/512 and 729/722.

    • @jhgvvetyjj6589
      @jhgvvetyjj6589 11 місяців тому

      @@lumipakkanen3510 531441/524288 being tempered out is what was mentioned in the video after all.

  • @axbs4863
    @axbs4863 11 місяців тому +38

    the next 6 digits of e are 235 and 360, being the first 3 prime numbers and the angle of a full rotation lol

    • @taltalim6174
      @taltalim6174 11 місяців тому +6

      it's fascinating how e contains a lot of important math numbers so early, like that gotta be like 1 in a big number and other irrational numbers don't have this "property"

    • @leave-a-comment-at-the-door
      @leave-a-comment-at-the-door 11 місяців тому +7

      @@taltalim6174 wdym other irrational numbers don't have this property? no matter what random sequence of digits you pick, if you stare at it long enough you will find neat patterns and coincidences in it. the number of 'important math numbers' is large enough that you can always find things.

    • @jandor6595
      @jandor6595 2 місяці тому

      Funny enough, the next four digits are 2874 which could be seen as a pattern as 28/7=4. And thus we have 25 digits of e that we may remember simply by keeping in mind a few patterns.

  • @paulamarina04
    @paulamarina04 11 місяців тому +8

    5:26 specifically its why we have 12TET (12-tone equal temperament). other tuning systems had existed long before 12TET, which were more focused on having neat ratios between the frequencies of the notes than in having them be logarithmically equidistant from one another. the cool coincidence is that they were /almost/ logarithmically equidistant from one another, which allowed 12TET to be used as a more consistent tuning system
    cool video!!!

  • @DissonantSynth
    @DissonantSynth 11 місяців тому +22

    Love your channel and videos so much. Super high quality, incredibly interesting, and well explained. Also, there's a modesty / sincerity to your videos, which is very special, because I think you are truly creating and sharing these videos purely for your love and appreciation of math, science, and art.

    • @Kuvina
      @Kuvina  11 місяців тому +6

      Thank you so much! That is a very well thought out comment and I really appreciate it!

    • @Metal_Master_YT
      @Metal_Master_YT 11 місяців тому +2

      @@Kuvina you guys have matching profile pictures!

  • @UltiMaker2
    @UltiMaker2 11 місяців тому +6

    Recently there were new SI prefixes. "ronna" means 10 to the 27th, "ronto" is 10 to the negative 27, "quetta" is 10 to the 30, and "quecto" is 10 to the negative 30. This also applies to 5:59, where "quettabyte" (QB) means 10 to the 30 bytes and "quettibyte" (?) (QiB) means 1024 to the 10th power (about 1.267651e30).

  • @FZs1
    @FZs1 11 місяців тому +51

    My favorite coincidence is that the sines of the most commonly used angles (0°, 30°, 45°, 60°, 90°) follow a pattern:
    sin(0°) = 0 = sqrt(0)/2
    sin(30°) = 1/2 = sqrt(1)/2
    sin(45°) = sqrt(2)/2
    sin(60°) = sqrt(3)/2
    sin(90°) = 1 = sqrt(4)/2
    This doesn't work for any other value though.
    Despite that, this is how I always memorized them in school (the cosines are the same but the other way around, because cos(x) = sin(90°-x), and tangents are just sin/cos).

    • @tobysuren
      @tobysuren 11 місяців тому +3

      my favourite too, for the sole reason that it's actually useful.

    • @jursamaj
      @jursamaj 11 місяців тому +1

      For a given definition of "important"…

    • @dryden_drawing
      @dryden_drawing 11 місяців тому +2

      So glad I scrolled one more down in the comments, I still don't have these memorized and I am starting to really need them

    • @baldability
      @baldability 11 місяців тому +2

      @@jursamajthese are definitely the most important angles up to 90 degrees in trig

    • @Shyguy5104
      @Shyguy5104 11 місяців тому +3

      this is less of a coincidence and more the greeks specifically designed it to be like that for 360 degrees

  • @RaichuKFM
    @RaichuKFM 11 місяців тому +6

    This was a really fun video! It was nice how you brought up some fun coincidences and what is and isn't actually unlikely about them; a lot of stuff falls prey to overstating that because of a naive view about expectations. My favorite part was the almost 20 + pi result, and the look at the sum that led into. Really cool!

  • @alansmithee419
    @alansmithee419 11 місяців тому +5

    While I don't think it meets the definition of a "coincidence" as provided in this video, something I find really cool is that numbers of the form 1/(99...)8 where you have any number of 9s can display the powers of 2 in their decimal expansions. With m 9s you will get m+1 digits of space for each number.
    1/998 gives 0.001 002 004 008 016 032 064 128 256 513 failing at the last digit here because the next number (1024) exceeds the space each number has and adds a one to the previous number, 512.
    Now 1/8=0.125 which may not seem to follow this pattern, but it turns out the infinite series:
    sum(n=0 to inf) 2^n/10^(n+1) = 1/8 (0.1+0.02+0.004+0.0008+0.00016+...)
    generally with m 9s:
    1/(999...)8 = sum(n=0 to inf) 2^n/10^((m+1)*(n+1))
    There may be better ways of displaying the infinite sums here.
    Also 1/(999...)7 gives powers of 3, 1/(999...)6 powers of 4 etc. Pretty cool.

  • @kikivoorburg
    @kikivoorburg 11 місяців тому +117

    I’m a big fan of the Kibi-prefix system. Having ambiguity about numbers is basically always a recipe for some sort of disaster

    • @xymaryai8283
      @xymaryai8283 11 місяців тому +6

      i'm always sad that we don't have a 2^x counting system, but the kibi prefix system makes me happy everytime i see it

    • @mushroomcraft
      @mushroomcraft 11 місяців тому +4

      "kibibyte" sounds so stupid, I hate that the power of 10 units even exist, because the way that I see it, they are just a way for drive manufacturers to sell you less storage.

    • @ZachAttack6089
      @ZachAttack6089 11 місяців тому +3

      ​@@xymaryai8283If only base-16 was the standard for regular math 😔

    • @vishalmishra3046
      @vishalmishra3046 11 місяців тому +1

      @@ZachAttack6089 It would be nice if human chromosomes and DNA ensured 8 + 8 = 16 fingers/thumbs in human hands instead of 10 that led to decimal system of numbers.

    • @ZachAttack6089
      @ZachAttack6089 11 місяців тому +1

      @@vishalmishra3046 Exactly haha. Or 4 on each, like most animals, so that it would be base 8 (which would still work with computers).

  • @bartholomewhalliburton9854
    @bartholomewhalliburton9854 5 днів тому +1

    I'm so glad you brought in continued fractions! Love it!

  • @2003LN6
    @2003LN6 11 місяців тому +7

    kuvina is honestly the one person carrying the entire internet's faith, love, and good now

  • @Mask60YT
    @Mask60YT 10 місяців тому +3

    my favourite mathmatical "coincidence" is that to get the derivative, you just subtract 1 from the power and multiply by the power.

    • @Mask60YT
      @Mask60YT 10 місяців тому

      @user-nu9ol8hv9c its a coincidence because that is not how the derivative formula was found, it was found using another formula that and coincidentally you can just simply do nx^n-1.

  • @Xonatron
    @Xonatron 11 місяців тому +3

    4:47 - Explains 12 notes per octave - Very cool. 2:1 or 100% pitch increase (double) is an octave up. Why the same keys resonate perfectly. The 3:2 or 50% pitch increase is called a power chord and resonate the next best. Then the 4:3 or 33.3% is next best. Captured well with 12 notes!

  • @HipsterShiningArmor
    @HipsterShiningArmor 2 місяці тому +2

    My favourite fake mathematical coincidence is how 2^7=128, and the internal angle of a regular heptagon rounded to the nearest whole number is 128 degrees. This is supposedly evidence that 7 is a magical, divine, or otherwise special number.
    I call it a fake coincidence because ofc the internal angle of a heptagon is not a whole number so rounding it to a whole number to invoke some kind of mystical property is kinda cheating. And also because it’s 900/7, or in decimal form 128.571428…, which actually rounds to 129, so it’s not even factually correct. Still kinda fun though.

  • @petachad8463
    @petachad8463 10 місяців тому +4

    7:30 Woah Woah Woah, the what now? shi went from playing with numbers to hyperdimentional spheres real quick

  • @TheWandererOfDreams
    @TheWandererOfDreams 10 місяців тому +5

    That "bye" at the end was so ZESTY.

  • @theodriggers549
    @theodriggers549 9 місяців тому +3

    For the e memorization thing, next is the first three primes (235) and degrees in a circle (360)

  • @LeoStaley
    @LeoStaley 11 місяців тому +36

    I like the coincidences of the ALMOST kind. Near Miss Johnson Solids are really fascinating. They're ALMOST Johnson solids, but are just SLIGHTLY irregular.

  • @stevosteffano5577
    @stevosteffano5577 11 місяців тому +5

    Some old favourites, and several new ones. Great video!

  • @rafakordaczek3275
    @rafakordaczek3275 10 місяців тому +2

    Im so glad that youtube recommeded me this video. I discovered your series on relativity. Your animations are basic, but they are sufficient in explaining any concept. Keep up the good work, and I will see you when you pass 100k subscribers!

  • @GuyMonochrome
    @GuyMonochrome 10 місяців тому +1

    Musician here! 4:49 While it's true that we tune many Western instruments to powers of 1/12, it did not cause us to have 12 notes. Long story short, we call this "Equal Temperament" tuning, which is actually quite new.
    Other systems were used in the past, such as what Pythagorean used. It had 12 notes, but you could argue that the note F# was tuned wildly differently from Gb, so you could say they were separate notes. Other systems evolved, such as Just Intonation, which adopted the "12" notes from Pythagorean tuning.
    Interested? Search up 12 TET, 17 TET, and 31 TET.

  • @NidhishwarReddy
    @NidhishwarReddy 11 місяців тому +4

    2:32 it was at this moment my brain.exe stopped working and now i am ded

  • @lythd
    @lythd 11 місяців тому +24

    a sidenote about the MB vs MiB etc differences, the original definition was that KB MB GB etc used powers of 1024, however it was changed to be consistent with si prefixes, and the new KiB MiB took its place. for legacy compatibility reasons windows keeps using the old definition even though its no longer correct. linux and mac, as well as a lot of programs, use the newer definition of KB/MB/GB or use KiB/MiB/GiB.
    so it isn't a case of MB being ambiguous, and MiB being strictly defined, its a case of the old definitions still having hold over, and sometimes still being incorrectly taught or used especially with a lack of awareness.

  • @jursamaj
    @jursamaj 11 місяців тому +1

    0:54 Regarding these criteria: try representing numbers in other bases. You'll find these coincidences disappear… while others appear.

  • @nanamacapagal8342
    @nanamacapagal8342 11 місяців тому +2

    Some more coincidences (and explanations):
    The 3² + 4² = 5² and 10² + 11² + 12² = 13² + 14² are part of an infinite family of these sums:
    21² + 22² + 23² + 24² = 25² + 26² + 27²
    Where the largest term on the left is exactly 4 * a triangular number.
    This even works in 1st powers (for 2 * a triangular number)
    1¹ + 2¹ = 3¹
    4¹ + 5¹ + 6¹ = 7¹ + 8¹
    9¹ + 10¹ + 11¹ + 12¹ = 13¹ + 14¹ + 15¹
    As well as 3rd (6 * a triangular number) and 4th (8 * a triangular number) powers, though with slight modification...
    5³ + 6³ + 2(1³) = 7³
    16³ + 17³ + 18³ + 2(1³ + 2³) = 19³ + 20³
    33³ + 34³ + 35³ + 36³ + 2(1³ + 2³ + 3³) = 37³ + 38³ + 39³
    7⁴ + 8⁴ + (8/2)³ = 9⁴
    22⁴ + 23⁴ + 24⁴ + (24/2)³ = 25⁴ + 26⁴
    45⁴ + 46⁴ + 47⁴ + 48⁴ + (48/2)³ = 49⁴ + 50⁴ + 51⁴
    There's a great explanation of these on Mathologer, and the comments may leave some insight about the higher powers.
    sqrt(2) + sqrt(3) ≈ pi.
    This one comes from two different approximations of pi.
    Start with a circle of radius 1. Its circumference should be 2pi.
    If you inscribe a square in the circle, its perimeter should be 4sqrt(2), meaning pi is about 2sqrt(2).
    If you circumscribe a hexagon outside the circle, the circumference should be 4sqrt(3), meaning pi is about 2sqrt(3).
    If 2sqrt(2) is an underestimate, and 2sqrt(3) is an overestimate, then the average should come pretty close, and indeed it is.
    Thus, sqrt(2) + sqrt(3) ≈ pi.

  • @joeyhardin5903
    @joeyhardin5903 11 місяців тому +5

    Regarding the one about 2^(7/12) being close to 3/2, I'm pretty sure that's not a coincidence. I've been researching the maths behind 12 tone equal temperament in music for a while now, and actually this property of 12, of being able to approximate lots of rational numbers when in an exponent, is not unique and actually is related to the properties of the golden ratio

  • @jari-0815
    @jari-0815 10 місяців тому +3

    π^2 is almost the gravitational acceleration with 9.81 m/s^2

  • @wuketuke6601
    @wuketuke6601 11 місяців тому +3

    The end of this video just ruthlessly escelates

  • @eriksteffahn6172
    @eriksteffahn6172 11 місяців тому +4

    Another interesting consequence of 2^10 ≈ 10^3 is that 2 ≈ 10^0.3. With this you get nice approximations for 10^0.1, 10^0.2, ... based on powers of 2 and 5:
    1, 1.25, 1.6, 2, 2.5, 3.2, 4, 5, 6.25, 8, 10
    this can be useful to approximate non-integer powers (in particular roots) without a calculator, for example:
    5000^1.2 ≈ 10^(3.7 * 1.2) = 10^4.44, which is between 25000 and 32000, so 5000^1.2 ≈ 28000 (correct result is 27464).

  • @gary.h.turner
    @gary.h.turner 11 місяців тому +3

    4:41 - Actually, pi has a "continued fraction pattern" too: π = 4 + {1/[1+X]},
    where X is the continued fraction (a0)^2/{2+[(a1)^2/(2+…)]}, and an = 2n+1. This arises from the continued fraction for the inverse tangent and the fact that tan^(-1)[1] = π/4.

  • @MathFromAlphaToOmega
    @MathFromAlphaToOmega 10 місяців тому +1

    I just stumbled across this video, and was shocked to see that theta function identity I mentioned a few months ago on a Mathologer video! One of my other favorite not-quite-coincidences is that e^(pi*sqrt(163)) is nearly an integer. It's related to lots of interesting number theory, like elliptic curves, unique factorization, Euler's prime generating polynomial x^2+x+41, and the Ulam spiral. I'm looking forward to watching more of your math videos!

    • @Kuvina
      @Kuvina  10 місяців тому

      Thank you so much! Number theory is so cool even though I don't know that much about it. Do you know if that's also related to 163/ln(163)?

    • @MathFromAlphaToOmega
      @MathFromAlphaToOmega 10 місяців тому +1

      @@Kuvina That's really interesting - I hadn't seen that before. I looked it up and it seems like that one really is pure coincidence. There are a few other small values of n for which e^(pi*sqrt(n)) is almost an integer, like 43, but 43/ln(43) isn't close to an integer. Maybe there is some kind of algebraic "explanation" for 163/ln(163), but it's unlikely to be related to the other number theory properties of 163.

  • @parthhooda3713
    @parthhooda3713 10 місяців тому +1

    3.14159 is also a coincidence as we have very less chance that random 6 digits are 314159

  • @Metal_Master_YT
    @Metal_Master_YT 11 місяців тому +1

    Why hasn't anyone made a video like this until now? its was a great idea!

  • @amichayr3418
    @amichayr3418 11 місяців тому +3

    This is like math asmr. I love this

  • @joeldick6871
    @joeldick6871 10 місяців тому +2

    A really cool "coincidence" that actually has an explanation is the near-approximation of pi in the Borwein integral. 3brown1blue did a video on it recently.

  • @egg1490
    @egg1490 2 місяці тому +1

    i love it when people acknowledge that tau exists instead if using 2π

  • @ckq
    @ckq 7 місяців тому +2

    Hey I discovered a new coincidence that relates e to pi:
    The solution to x^x*(1-x)^(1-x) = 0.5^0.5 (See A102268 on the OEIS)
    x=0.889972
    is almost nearly
    pi^2/16/ln(2) = 0.889927
    the number pi^2/16/ln(2) didn't come out of nowhere either, it represents the ratio between (arcsin(sqrt(1))-arcsin(sqrt(0.5)))^2 and ln(1)-ln(0.5).
    For context on the y(x) = arcsin(sqrt(x)) function, it is the integral of 1/sqrt(x(1-x)), so it maps the numbers from 0 to 1 on a scale that is proportional to the standard deviation to account for the fact that there's a bigger difference between 0.99 and 1.00 than 0.50 and 0.51

  • @unflexian
    @unflexian 11 місяців тому +5

    hey there's an error in the sound at 5:13, you seem to have forgotten the flats when playing it (played C F B E A instead of C F Bb Eb Ab), making the interval from F to B a tritone instead of a fourth. this does not at all detract from the video quality, best esomath video ive seen since the cursed units video, but just fyi.

    • @Kuvina
      @Kuvina  11 місяців тому +5

      Thank you for letting me know! I think I originally had it in a different key and then I moved it down to start at C and wrongfully assumed there wouldn't be any sharps or flats!

    • @unflexian
      @unflexian 11 місяців тому +4

      @@Kuvina ohh i see, well at least now you have a segway into a video about equal temperament or harmony or something if you desire :)

  • @robo3007
    @robo3007 10 місяців тому +1

    240 has its digits arranged in descending order in base 2, base 3 and in base 4.

  • @TheArtOfBeingANerd
    @TheArtOfBeingANerd 11 місяців тому +9

    I saw 2^(integer)/12 and immediately thought of music theory. Confirmed when I saw it was aprox 3/2

    • @FaranAiki
      @FaranAiki 11 місяців тому +3

      Me too, but reversed. I was like... 3/2 and 4/3? Seems like a perfect fifth or fourth or something and then I realized it was 2^... haha

  • @guigui0246
    @guigui0246 11 місяців тому +4

    I realized that :
    - In 1 dimension you need 1 support point to not fall (you can't need 0 but there is no down)
    - In 2 dimensions you need 2 support points to not fall /\ (like a card castle)
    - In 3 dimensions you need 3 support points to not fall /|\ (like stools have)
    Does that mean in n dimension you need n support points even if gravity only takes act in 1 of them ?

    • @Tom-u8q
      @Tom-u8q 11 місяців тому +1

      I can't find a source for this, but I would guess so because you need n points to define a hyperplane in R^n

    • @Tom-u8q
      @Tom-u8q 11 місяців тому +3

      I have the outline of a proof, don't want to do the whole thing.
      Show a congruence between a vector space of dim n-1 and the hyperplane created by taking weighted averages of n points
      Show that equilibrium under gravity is equivalent to a projection from centre of gravity in direction of attraction intersecting a weighted average of supports
      Show that for a stable equilibrium, the same must be true for all points in some ball around centre of gravity, ie true for a nudge in n-1 dimensions (not affected by direction of gravity)
      Hence a vector space with at least n-1 dimensions in required so n support points are needed
      This shows no fewer than n work but to show n works, show that the (n-1)-simplex can be arbitrarily scaled to cover the projection of any n ball

    • @b.clarenc9517
      @b.clarenc9517 11 місяців тому +2

      You are right. It also leads to the following puzzle you can ask around:
      Why is a 3-leg stool always stable, but a chair never is? Because we live in a 3D world.

    • @leave-a-comment-at-the-door
      @leave-a-comment-at-the-door 11 місяців тому +1

      yes, n fixed points will fully determine a system in n dimensions. if you want to think about why, it's easiest to invoke linear algebra: think of rows as dimensions and colums as your points, so a square matrix with a non-zero determinant will be well-defined.

  • @loftyTHEOWNER
    @loftyTHEOWNER 11 місяців тому +1

    The leech lattice could be really useful to create my input vectors to randomly associate them to an output and train the model in a supervised way...

  • @OwenGalaxy
    @OwenGalaxy 8 місяців тому +7

    I came from that other video that ripped off your video - yours is much better.

  • @hypercoder-gaming
    @hypercoder-gaming 9 місяців тому +1

    Also noteworthy is 13,14,15. If you take just the 3 from 13, it's 31415 which are the first few digits of pi.

  • @MrTomyCJ
    @MrTomyCJ 11 місяців тому +2

    0:57 I'd say the mathematical explanation is precisely that it's just a coincidence. Coincidences don't need "explanation" in some sense.
    Also notice some of these are mathematical coincidences but only in base 10. This serves to show that there's nothing "fundamental" about them.
    But they're not useless! as the video gives some examples on how sometimes paying them attention can result in practical use cases.

  • @revimfadli4666
    @revimfadli4666 11 місяців тому +4

    Is this the math equivalent of "names alike" memes?

  • @kayleighlehrman9566
    @kayleighlehrman9566 9 місяців тому +1

    The real coincidence was the maths we learned along the way

  • @rociochave1066
    @rociochave1066 8 місяців тому +3

    I just realized...
    e has the exact digits of pi just scrambled.........

  • @mikemac8159
    @mikemac8159 8 місяців тому +1

    this video has both comforted me and put my brain into a number-obsessed mode
    thank you very much :3

  • @QP9237
    @QP9237 11 місяців тому +1

    Saw Gelfond's constant (my favorite number) in the thumbnail and knew I had to watch the video 😂

  • @Nikolas_Davis
    @Nikolas_Davis 11 місяців тому +2

    People are notoriously bad at intuiting how (un)likely something is. I'd be very suspicious of my own intuition in this case, especially since the patterns we're looking for have *not* been specified in advance. "An interesting coincidence" covers so much ground, that you're virtually guaranteed to run into one looking at almost any sequence of random digits.

  • @bennett2201
    @bennett2201 11 місяців тому +1

    wonderful video kuvina!! i loved the chords you made for 12th roots of 2!!

  • @kephalopod3054
    @kephalopod3054 6 місяців тому +2

    The first 360 digits, after the decimal point, of pi end with 3, 6, 0.

  • @glowstonelovepad9294
    @glowstonelovepad9294 10 місяців тому +1

    If you divide or multiply pi by 2, there will be 7 9s in a row there because the 6 9s have 4 before them and 8 after them.

  • @diegovasquez840
    @diegovasquez840 11 місяців тому +1

    My favorite is from physics, where pi^2 is very close to the gravity constant for earth in meters per second

  • @lilyyy411
    @lilyyy411 11 місяців тому +3

    wake up babe! new video from the nerdy enby is here!

  • @_Gam3r
    @_Gam3r 11 місяців тому +2

    fun fact: the sequence 24242424 occurs in the 242,424th digit of pi

    • @gabrielgabi543
      @gabrielgabi543 3 місяці тому

      24 leech lattice and base 10 go brrr

    • @ErikLeppen
      @ErikLeppen 29 днів тому

      Wow, that's nice! I looked it up and it's actually 242424242, where the first 2 is the 242421st digit.

  • @ErikLeppen
    @ErikLeppen 29 днів тому +1

    What I find funny is that people point out the Feynman point in pi (the "999999" in its decimal expansion), not knowing that tau actually can do an even better trick. There's actually a "9999999" at the exact same spot in tau's decimal expansion, which is easily demonstrated because tau = 2 * pi and 2 * 49999998 = 99999996.

  • @ouroya
    @ouroya 11 місяців тому +4

    the reason we have 12 notes in an octave is much more historical than mathematical, though it is intuitive to choose an octave (×2) rather than a tritave (×3) or anything higher. the western 12-tone equal temperament tuning has only been in use since around the mid-1580s at the very earliest. there are a lot more tunings out there based on things other than the twelfth root of two for 12-tone octave subdivision that have been around a lot longer, all with different benefits and drawbacks, though 12TET became standardized as it allowed things in any key to sound equal with the same tuning, whereas most other tuning systems result in needing to retune to the specific key of a piece or different keys having different qualities.

  • @ÞeSheep1
    @ÞeSheep1 9 місяців тому +4

    1:22 hehe 6 9’s

    • @gabrielgabi543
      @gabrielgabi543 3 місяці тому

      Haha so funny

    • @ÞeSheep1
      @ÞeSheep1 3 місяці тому +1

      @@gabrielgabi543 also it’s 0.69% of happening (hehe 69)

  • @mkwilson1394
    @mkwilson1394 11 місяців тому +1

    I really enjoyed this, and I didn't expect a stack of 4ths to show up, so even better!

  • @lawrencejelsma8118
    @lawrencejelsma8118 11 місяців тому +1

    It is accidental. By the way ... The approximation of exact real values such as e^π = 20 + π is of the form e^x = (1)(x + 20) where the Lambert W function solves for x. It might approximate to π (checking in Wolfram Alpha online calculator) with sufficient error minumal in significant digits truncated and rounded, etc.

  • @glowstonelovepad9294
    @glowstonelovepad9294 10 місяців тому +2

    Pi plus pi squared is very close to 13 and 1/90.

  • @puzzleticky8427
    @puzzleticky8427 11 місяців тому

    This is what mathematicians like about maths

  • @DATA-ig9qj
    @DATA-ig9qj 8 місяців тому +8

    i came from the ripped video hi guys

  • @morgan0
    @morgan0 10 місяців тому +1

    i don’t remember any specifics rn but i remember being fascinated by repeated multiples of some N in the digits of a number divided by that N, i think mainly 7 but also possibly others, maybe it has to be coprime to the base

  • @norude
    @norude 11 місяців тому +3

    1828 is the birth year of a famous Russian poet Alexander Pushkin.

  • @norude
    @norude 11 місяців тому +2

    Wow, I thought it would be some person, far from math explaining, how 13 is the devils number because of some coincidence, but it was really interesting, especially the last cannon-ball part

  • @Drawoon
    @Drawoon 11 місяців тому +6

    A lot of these coincidences also rely on a base 10 number system. If you examine things through another number system, you would probably find new coincidences

  • @TheBalthassar
    @TheBalthassar 9 місяців тому +1

    Here's a weird coincidence, I only just now watched this video, after completely missing it when it released. And both this video and the new one care about 70.

  • @notevennelson
    @notevennelson 10 місяців тому +2

    the sum of all the digits of a number acquired for a multiple of 9 is 9:
    9 x 2 = 18 (1 + 8 = 9)
    9 x 16 = 144 (1 + 4 + 4 = 9)
    if you wanna know if a number is divisible by 3, just add all its digits, and if they equal 3, 6 or nine, then it's divisible by 3:
    is 74223 divisible by 3?
    yes, because 7 + 4 + 2 + 2 + 3 = 11 + 7 = 18 -> 1 + 8 = 9.

  • @paulcastelein1405
    @paulcastelein1405 10 місяців тому +1

    I belive that the kilobyte idea comes from the binary system. Since computers use base 2 some people decided to use base 2 for their bases, and 2^10=1024, however some other people decided they'd rather use the base 10 system as it is the one we typically use and they changed the units accordingly, this makes it different to coordinate

  • @Swagpion
    @Swagpion 10 місяців тому

    Here is a Mathimatical coinsidence I noticed. For a circle of radius 2, the area and circumference have the same number with a different exponent. And a sphere of radius 3 has the surface area and volume have the same number, just different exponents.
    Weird how it works out for both. Im not sure if this extends to higher dimension, but maybe it does.

  • @lagomoof
    @lagomoof 11 місяців тому +2

    Look at pi and e in binary out to about 23 places. Look for the parts where they both have six 1s in a row. Reverse one around the midpoint of those six bits. 19 of the bits line up perfectly with the other number. The odds of this are about one in half a million. (Also, since tau = 2pi, its bits are the same, so tau can stand in for pi here.)

  • @ckq
    @ckq 7 місяців тому +1

    2:45 is a good explanation, i dont think it needs more explanation

  • @humbledb4jesus
    @humbledb4jesus 11 місяців тому +1

    whenever unusual structure and design comes from seemingly random events or actions, i assume there is a fractal buried in it somewhere...
    and, like the population growth formula, [N(t) = 2N(t - 1) ], it will fits right in the mandelbrot structure

  • @JoMama-b3k
    @JoMama-b3k 11 місяців тому +2

    Imagine if every digit of e had a pattern

  • @caspermadlener4191
    @caspermadlener4191 11 місяців тому +1

    The Leech lattice indeed has a cool construction using its Weyl vector, but the even unimodular lattice in 8 dimensions does not have this construction (although the construction for this lattice is way simpler). It is a bit of a coincidence, also 1 is a cannonball number as well.