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
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.
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.
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.
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
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.
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)
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.
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 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)
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.
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...)
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.
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.
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.
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.
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.
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.
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"
@@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.
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
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
"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 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.
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).
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.
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 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.
@@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.
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
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"
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.
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).
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.
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!
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.
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!!!
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!
@@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.
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.
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!
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 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.
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
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.
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).
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.
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 ?
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
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.
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.
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
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
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
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.
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.
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!
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.
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
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.
someone prob already mentioned this but xkcd 1047: approximations is very relevant here. I'll include it here, removing the jokes: Relation: Accurate to within: One light-year(m) 99^8 one part in 40 Earth Surface(m2) 69^8 one part in 130 Oceans volume(m3) 9^19 one part in 70 Seconds in a year 75^4 one part in 400 π*10^7 one part in 220 Age of the universe (seconds) 15^15 one part in 70 Plancks constant 1/(30^π^e) one part in 110 Fundamental charge 3/(14 * π^π^π) one part in 500 Electron rest energy (joules) e/7^16 one part in 1000 Light-year(miles) 2^(42.42) one part in 1000 sin(60°) = √3/2 e/π one part in 1000 √3 2e/π one part in 1000 γ(Eulers gamma constant) 1/√3 one part in 4000 Feet in a meter 5/(e√π) one part in 4000 (note: taking the e-th root of pi is very cursed) √5 2/e + 3/2 one part in 7000 Avogadros number 69^π^√5 one part in 25,000 Gravitational constant G 1 / e^(π - 1)^(π + 1) one part in 25,000 R (gas constant) (e+1)*√5 one part in 50,000 Proton-electron mass ratio 6*π^5 one part in 50,000 Σ 1/n^n from 1 to ∞ ln(3)^e one part in 100,000 √2 3/5 + π/(7-π) one part in 220,000 Liters in a gallon 3 + π/4 one part in 500,000 g 6 + ln(45) one part in 750,000 Proton-electron mass ratio (e^8 - 10) / ϕ one part in 5,000,000 Ruby laser wavelength 1 / (1200^2) [within actual variation] Mean Earth Radius 5^8*6*e [within actual variation] γ(Eulers gamma constant) e/34 + e/5 one part in 2,000,000 √5 (13 + 4π) / (24 - 4π) one part in 13,000,000 (he had a joke about 1/137 but I was curious if I could find anything. sadly nothing interesting, but here's the attempt): Fine structure constant 1/137 one part in 3,800 1/137.036 one part in 150,000,000 (actual inverse is 137.035999084...) 1000/((5^2+π)^2-1)^√π one part in 120,000,000 averaging those two gets you one part in 1,400,000,000 attempt 2: 1/ (100* (4√5−1÷8) ) one part in 100,000
Mathologer www.youtube.com/@Mathologer also shows some number sequence returning into the same digits/ciphers like your video's first example, i think to remember it was something about Ramanujan material or perhaps in th-cam.com/video/XR2u7izwZ04/w-d-xo.html . So how can two sums follow the same value, divert and get the same values in the same positions again? The world where humans roam or slurp must be downgraded so they might relate?
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.
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.
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.)
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
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.
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
Correlation does not imply causation. Although these are nice patterns,😊 however you can choose any random irrational number and some form of ‘‘low probability’’ pattern will become apparent to you.
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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
twas bangin👍
pona suli a!
Kuvina I think you're so cool ily 🥺
Relativity ep. 6 when
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.
In general the chance of a specific coincidence occurring is very low, however the chance of *some* coincidence occurring is very high
Almost no one understands this lol
exactly
Chance of tornado vs chance of disaster
i cant even count the number of times i tried to explain that to someone and failed miserably
@@samylemzaoui2298the chance that a specific person understands this is very low, but the chance that some person does is high
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.
Don’t use the beginning tho lol “1 mile is about 1 km”
That's so cool!
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.
close enough@@TabooGroundhog
@@jongyon7192pwtf is an oppression unit
The first 40 seconds of the video is literally "How to memorize 15 digits of e"
That is pretty cool though.
One that I was waiting to see if you mentioned:
10! seconds = exactly 6 weeks.
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.
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
That's lovely and insane!
@@TabooGroundhog its spelled fortnight not fortnite
@@petterlarsson7257 We have been tainted by those 90's
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.
Wtf bro💀💀💀
My dude just causally explained the mathematical basis for music in the middle of this.
honestly quite incredible
Kuvina's not a dude, btw.
-Paintspot Infez
Wasabi!
@@paintspot my girl just casually explained the mathematical basis for music in the middle of this.
@@paintspotwhat is kusina then? How is kusina not a guy
Math class initially: 6:40
me: *blinks for a nanosec*
7:40
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)
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.
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
whatif we use base 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)
Here’s another fun one:
50/49 will spell out the powers of 2, each spaced 2 decimal places apart.
(50/49=1.0204081632…)
...1632653061... aw, shucks. Had to carry the 1 for 128 since it didn't fit in a 2-digit space.
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.
Hell yea
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
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...)
2:20 i envy your ability to convey this much raw happiness in a single drawing
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.
Surely you mean 2^31-1
@@themathhatter5290yeah
Did you mean 2^16+1 (65537) which is the extremely common RSA public key exponent or did you mean 2^31 - 1 ?
2^31 - 1 is not a cryptographically useful prime.
2^31 + 1 is divisible by 3
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.
7:30 Woah Woah Woah, the what now? shi went from playing with numbers to hyperdimentional spheres real quick
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.
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.
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.
Yay.
…49999998…
turns into
…99999996…
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.
*commutative
@@jursamaj thanks
the next 6 digits of e are 235 and 360, being the first 3 prime numbers and the angle of a full rotation lol
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"
@@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.
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
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
I’m a big fan of the Kibi-prefix system. Having ambiguity about numbers is basically always a recipe for some sort of disaster
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
"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.
@@xymaryai8283If only base-16 was the standard for regular math 😔
@@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.
@@vishalmishra3046 Exactly haha. Or 4 on each, like most animals, so that it would be base 8 (which would still work with computers).
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).
my favourite too, for the sole reason that it's actually useful.
For a given definition of "important"…
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
@@jursamajthese are definitely the most important angles up to 90 degrees in trig
this is less of a coincidence and more the greeks specifically designed it to be like that for 360 degrees
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.
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.
@@jhgvvetyjj6589 yep. 19-tone equal tone temperament also has that 10/9 ~ 9/8, with a ton of other flavor on top of it.
@@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.
@@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.
@@lumipakkanen3510 531441/524288 being tempered out is what was mentioned in the video after all.
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
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
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"
Holy cow, the last one is good to 2.8 parts per Nonilion (10^30)
The 9/5ths is good to 15ppm-much better than 355/113
why am i seeing the number 640320 everywhere
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.
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).
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.
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!
kuvina is honestly the one person carrying the entire internet's faith, love, and good now
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.
Thank you so much! That is a very well thought out comment and I really appreciate it!
@@Kuvina you guys have matching profile pictures!
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!!!
Some old favourites, and several new ones. Great video!
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!
this video has both comforted me and put my brain into a number-obsessed mode
thank you very much :3
my favourite mathmatical "coincidence" is that to get the derivative, you just subtract 1 from the power and multiply by the power.
how is that a coincidence how else would you find the gradient of tangent
@@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.
That "bye" at the end was so ZESTY.
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.
I really enjoyed this, and I didn't expect a stack of 4ths to show up, so even better!
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!
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!
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)?
@@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.
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...
wonderful video kuvina!! i loved the chords you made for 12th roots of 2!!
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
2:32 it was at this moment my brain.exe stopped working and now i am ded
12, 1 *3, 14, 15* , 16, 17
Why hasn't anyone made a video like this until now? its was a great idea!
The end of this video just ruthlessly escelates
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.
For the e memorization thing, next is the first three primes (235) and degrees in a circle (360)
This is like math asmr. I love this
1:22 hehe 6 9’s
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).
I saw 2^(integer)/12 and immediately thought of music theory. Confirmed when I saw it was aprox 3/2
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
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.
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.
Love the videos! keep them coming!
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 ?
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
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
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.
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.
3.14159 is also a coincidence as we have very less chance that random 6 digits are 314159
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
0:54 Regarding these criteria: try representing numbers in other bases. You'll find these coincidences disappear… while others appear.
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
The real coincidence was the maths we learned along the way
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
I personally like how e/π is very close to (√3)/2, the value of cos(30°).
Is the string piece in the intro the first half of Iceland from the "Musical Map of Europe" video? I thought I recognized those notes.
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.
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.
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!
@@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 :)
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.
Always love a Kuvina video 😊
π^2 is almost the gravitational acceleration with 9.81 m/s^2
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
Another great video!
I'm sure that you could find tons more coincidences if some of these equations were written in a base other than decimal.
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.
MIND. BLOWN
My favorite is from physics, where pi^2 is very close to the gravity constant for earth in meters per second
someone prob already mentioned this but xkcd 1047: approximations is very relevant here.
I'll include it here, removing the jokes:
Relation: Accurate to within:
One light-year(m) 99^8 one part in 40
Earth Surface(m2) 69^8 one part in 130
Oceans volume(m3) 9^19 one part in 70
Seconds in a year 75^4 one part in 400
π*10^7 one part in 220
Age of the universe (seconds) 15^15 one part in 70
Plancks constant 1/(30^π^e) one part in 110
Fundamental charge 3/(14 * π^π^π) one part in 500
Electron rest energy (joules) e/7^16 one part in 1000
Light-year(miles) 2^(42.42) one part in 1000
sin(60°) = √3/2 e/π one part in 1000
√3 2e/π one part in 1000
γ(Eulers gamma constant) 1/√3 one part in 4000
Feet in a meter 5/(e√π) one part in 4000 (note: taking the e-th root of pi is very cursed)
√5 2/e + 3/2 one part in 7000
Avogadros number 69^π^√5 one part in 25,000
Gravitational constant G 1 / e^(π - 1)^(π + 1) one part in 25,000
R (gas constant) (e+1)*√5 one part in 50,000
Proton-electron mass ratio 6*π^5 one part in 50,000
Σ 1/n^n from 1 to ∞ ln(3)^e one part in 100,000
√2 3/5 + π/(7-π) one part in 220,000
Liters in a gallon 3 + π/4 one part in 500,000
g 6 + ln(45) one part in 750,000
Proton-electron mass ratio (e^8 - 10) / ϕ one part in 5,000,000
Ruby laser wavelength 1 / (1200^2) [within actual variation]
Mean Earth Radius 5^8*6*e [within actual variation]
γ(Eulers gamma constant) e/34 + e/5 one part in 2,000,000
√5 (13 + 4π) / (24 - 4π) one part in 13,000,000
(he had a joke about 1/137 but I was curious if I could find anything. sadly nothing interesting, but here's the attempt):
Fine structure constant 1/137 one part in 3,800
1/137.036 one part in 150,000,000 (actual inverse is 137.035999084...)
1000/((5^2+π)^2-1)^√π one part in 120,000,000
averaging those two gets you one part in 1,400,000,000
attempt 2: 1/ (100* (4√5−1÷8) ) one part in 100,000
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.
my fav is that when you have a point A=(x_1,y_1) that when you take the graph x/y=x_1/y_1 that just is a straight line that goes trough (0,0) and A
240 has its digits arranged in descending order in base 2, base 3 and in base 4.
Mathologer www.youtube.com/@Mathologer also shows some number sequence returning into the same digits/ciphers like your video's first example, i think to remember it was something about Ramanujan material or perhaps in th-cam.com/video/XR2u7izwZ04/w-d-xo.html . So how can two sums follow the same value, divert and get the same values in the same positions again? The world where humans roam or slurp must be downgraded so they might relate?
Saw Gelfond's constant (my favorite number) in the thumbnail and knew I had to watch the video 😂
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.
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.
It's like a massage for my brain ☺
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.)
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
Not really a coincidence but the square root of 2 added with the square root of 3 is almost pi.
i have been trying to tell a the 22/7 is closer one to everyone for ages
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.
I came from that other video that ripped off your video - yours is much better.
This video made my day
wake up babe! new video from the nerdy enby is here!
babe wake up new kuvina video dropped
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
very cool video!!!
awesome video!
Correlation does not imply causation. Although these are nice patterns,😊 however you can choose any random irrational number and some form of ‘‘low probability’’ pattern will become apparent to you.