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
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"
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.
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.
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)
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)
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.
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.
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.
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.
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.
"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.
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.
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...)
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.
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).
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
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.
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.
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!!!
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).
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
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"
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!
@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.
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
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!
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!
e = 2.718281828459045235360... 1828 -> repeated twice 45, 90, 45 -> angles of the isosceles right angle triangle 2, 3, 5 -> first 3 primes 360 -> degrees in a full rotation
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
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.
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.
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.
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.
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
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.
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!
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.
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
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
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.
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.
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.
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.)
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
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.
Here's a big one: 2^100,000 is about 9.99002093x10^30102 The number starts with 999 then 00. Also, 2^200,000 is ≈ 9.98x10^60205 2^300,000 is about 9.97x10^90308 all start with 99.
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.
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.
Here are the odds of that btw Expected number of 9s in first 125 = 12.5 Using Poisson Distribution with mean 12.5, that means the odds of less than 4 9s is 0.16%, but since you chose 9 specifically, we need to multiply the odds by 10 since you could have picked any of the 10 digits, so 1.6%, then if we investigate why you chose specifically the first 125 digits, it would go up more.
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.
