I just wanted to thank this channel for existing. A little over a year ago I started watching numberphile videos which helped me discover my interest in math. I am now back in school because of it and just completed calculus 1 with an A+ and am currently taking Calculus 2! Update: got an A in calc 2, now onto calc 3! Thanks so much Numberphile!
Computerphile (and Numberphile) are part of the reason I went back to school in computer science! (Where I did kind of an equivalent to calc-3, advanced linear algebra and probability and statistics classes). Best decision I made in my life! Kudos to you
No they are not. The only time you got a better chance with binary is from 1-8. For 7 you have a chance of 1/10 in decimal, 1/8 in binary. For 8 you have a chance of 1/10 in decimal, 1/16th in binary. For 512 you have a chance of 1/100 in decimal, and a chance of 1/1024 in binary. For 8192 you have a chance of 1/1000 in decimal and a chance of 1/16384 in binary.
@@1R0QU012 I do programming, but my main field is mathematics. I don't understand your response though, do you mean that computer scientists call binary digits "decimals" even though they are not decimal digits? Or do you mean something else?
Years ago I'd have said "recreational math" was an oxymoron. Not anymore. One thing great about Numberphile / Computerphile is that they have great speakers who are passionate and know how to express their passion. That's also due to their hosts & editing skills. As viewer we don't see just the cool maths, we also choose to absorb all that passion, and that makes a whole difference compared to the boredom of a grade school math class where students have no idea _why_ they are learning that stuff to begin with.
40 wont necessarily show up more than once though..its not guaranteed right unless you can prove that it is...can you please correct this or clarify what you meant.
Hi, Tom, thanks for the very interesting video! I was wondering what would happen if you skip the self-locating string and choose the next/secondary matching string to avoid local looping when searching for the global loop? For example, 211－93－14－1－3－9－5－4－2－6－7－13－110－174－155－314－2120－5360－24671－...
@@TomRocksMaths Yes..., it looks like that. So I kept tracking after 24671－119546－ 193002－240820－274454－153700－..., then 153700 doesn't exist in the first million digits of pi. How come the numbers in "169's circle" are so special that they can form a loop!?
Maths guy: My brain: spike in his ear spike in his ear spike in his ear spike in his ear spike in his ear spike in his ear spike in his ear spike in his ear
I am a software developer by trade. It never ceases to amaze me how mathematicians come up with these problems that seem simple but are actually far beyond the bounds of mere computation; you could never have a "brute-force" solution to a problem like this.
I literally learned the first 100 digits of Pi by making it my password for a couple of weeks, and now I can't get them out of my system anymore, btw great channel. Keep it up!!!
As well as self-locating strings, there's another way to show that some numbers don't sit within the loop they settle into, which accommodates loops larger than 1 - If we take the 169 example, the next digit is a 3. If 169 first appears at the 40th place in pi, then 1693 ALSO first appears at the 40th place in pi. So 1693 has to settle into a loop that doesn't pass through 1693 again.
does this mean that it's very likeable that there exist some numbers whose loop actually did not end? Would that numbers be called "primes" by some means?
Tom Crawford: "You don't really need more than 7 decimal places". Emma Haruka Iwao: "What have I done with my life" Rajan Mahadevan: "I know right" Matt Parker: "Amatuer!"
Considering that NASA uses pi to 16 decimal places in software that stabilizes spacecraft trajectories, the national institute for standards and technology uses pi to 32 decimal places when calculating the fundamental constants of the universe, and pi to 100 decimal places, if memory serves, would be sufficient to calculate the circumference of the observable universe to a precision of less than the width of a hydrogen atom (if the geometry of the universe allows that such a calculation would make sense and we had enough precision in our measurement of the diameter to make such a calculation meaningful, of course), I think it’s safe to say that most of us unwashed lumpenproletarians are in no danger from using a value of pi with seven decimal places in our day-to-day lives!
@@zeldaandTwink Let's use a simple encryption algorithm: EncryptedMessage = character(x) XOR decimal(x)ofPi To decrypt 1 MB of Data you would need to know pi up to 1 million of decimal places. Not just 31.
@@d34d10ck That is using Pi as a seed, which is not using it as the mathematical constant. When using it it as the ratio of a circle's circumference to diameter, the largest physical circle we know of would be the circumference of the universe. Pi to 31 places is the correct ratio within the width of a hydrogen atom, so you never need to go to more digits for PHYSICAL purposes. Any smaller circle, pi to 31 digits is even more accurate as the ratio, very quickly getting below the planck length in terms of error (i.e., pointless for modern physics). That is what was meant by practical applications, not using it as a random large seed, which always benefits from having more digits.
If the digits of pi were completely random you would expect there to be infinitely many self locating strings, but each one should be bigger than its predecesor by about 12.915 (10^(10/9)) times. This aligns roughly with whats shown. Ignoring the massive gap at the start, the average ratio of succesive terms is about 12.184.
I'm reminded of a much earlier video they did on the recreational "happy numbers" and "melancoil", which also had loops and various self-locating numbers, specifically at 1.
[12:37] «It's not the peak that's amazing, it's the awesome landscape and the methods and techniques you're going to learn and use to climb a mountain when you try.»
thanks for the video, all of them really, i do quite enjoy them all, but i would also like to see more of Tom on here too, i especially enjoyed the Navier-Stokes and Reynolds number episodes, glad to see him back with this.
I was thinking base 16 since there's a method to determine the nth digit of pi in base 16 without having to calculate all the digits before it. Google BBP Formula to see the method.
Really enjoyed this video--AND understood it to the end! All of his previous ones were way over my head, so I couldn't finish them. This one was fun. π rocks!
this reminds me of aliquot sequences. all known sequences either terminate at 0 or end in a perfect, amicable, or sociable number. it is an open question whether there are any aliquot sequences that increase without bound
@@hyperhippyhippohopper 242424 is actually very nice, i hate the fact that they not only skipped 3, but they even had the audacity to say "What if we consider 1 to be the 0th digit of pi?"
This reminds me of the Collatz conjecture, which on its own does not have any real significance but the techniques and strategies used to attack it will probably be important tools in solving other problems.
1:25 I think it would be more natural and general for the 1 to be in the 11th position and the 2 to be in the twelth. Since 12 trillionths would have the 1 and 2 in those positions. (and 10^-12 is a trillionth) That's _kind of_ like having a 12 in that position. Except we don't have a digit for 12 and the 10 carries over to 1 in the position to the left.
@@alonamaloh Because you always go to the earliest appearance of the number. Which means that if you happen to end up at a self-locating string, it will have to be the first appearance of that string.
I think this is so awesome! I went to grab a pencil and do my own loop, but then I realized that there no way to easily count where those numbers are in pi 🙃
I made a quick excel to try this, and found it is incorrect. Using the first 32759 digits of pi (the most excel can hold in 1 cell, seems an odd number, 2^15-9, anyways...) the following order/list is produced: 19, 39, 45, 62, 22, 137, 861, 269, 1395, 6482, 228, 2529, 18335; and then it doesn't find '18335' in the first ~32k digits.
I disliked the 32b limitation of excel cells, so I did a bit more exceling and made some formulas to read multiple cells and find the correct text. Long story short, I didn't find anything interesting, even tho at first I thought I found a cool 188 step loop (I had had a bug in one of my formulas). Using a total of 1 million digits of pi, the sequence terminates as follows (without finding the next step, it presumably will find one with enough digits, per this video): 19, 39, 45, 62, 22, 137, 861, 269, 1395, 6482, 228, 2529, 18335, 68539, 22166, 169545, 96010, 67419, 272547, 414384, 148332 (can't find this in the first 1 million digits)
OK, final thoughts, cuz this is pointless unless I wanna python this. I tried numbers 0 to 100 with the first 1 million digits and found just 2 loops, your loop and one that starts at 40 with a cycle length of 20: 40, 70, 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169, 40 This 20 length loop also can have a relatively long lead-up phase starting with 61, which has a 7-step lead-up phase: 61, 219, 716, 39, 43, 23, 16, 40 I also found a number of values that terminate at repeating 1's: 1, 14, 21, 45, 73, 93 The longest path to 1 starts at 45 and is 10 steps to 1, and has the following steps: 45, 60, 127, 297, 737, 299, 2643, 21, 93, 14, 1 In that path, you can see that the other digits 14, 21 and 93 pop up (and 45 obviously), but not 73, which has the following path: 73, 299, 2643, 21, 93, 14, 1 This is at least a little interesting that both 73 and 737 both go to 299, which then leads to 1's. The next couple digits starting at position 299 are 737(2458), so we know a couple more large numbers that go to 1.
There's another possibility for a chain: ending at a smaller loop. For example, if you found one of the strings that were part of a known loop in a position not in that loop, tracing the path back up the chain would be infinitely long, but would end with a loop of known (finite) length.
Self-location can’t be infinite as the increasing enumeration order means it becomes an order of magnitude less likely that one appears. You can use statistics to prove this. An argument can be said about the loops, i.e. That there are no infinite loops, because you can always find the next index, and you always select the lowest valid index, meaning there will for any multiple number be a lower index to be selected, which cuts out the possibility of infinites.
But wouldn’t there still be countably infinite self-locating numbers? Just because they become less common doesn’t mean they stop completely. Would love to see an explanation for that
Tom Navier-Stokes Crawford! That was an epic series. This video is also fascinating; those loops in pi are like the swirling of a viscous fluid... Now, if only we had a set of equations to describe the motion of viscous fluids...
Jeez, so many comments about the topic not being serious enough or people clutching their pearls about the bloke having a few piercings. Lighten up, guys!
10:35 This statement is not correct: self-locating strings could already appear earlier. For example, 44899 could also appear at a smaller position, meaning you would not loop at position 44899
@#Miqdaad Indori It can only stop when 44899 is found at position 44899, and there is no 44899 in an earlier position. (This is not the case as 44899 also appears at position 13714)
@David Gerick Yes, but you would still get 44899, which will take you yet again yo 44899 and so on... You would only het 44899, so in a sense you don't loop between numbers
I've memorized 51 digits of PI. Honestly if I were to replace PI everywhere in my renderers with the number 3.0, then you would barely notice the difference.
I wrote a program to try this out myself and apart from the 1-cycle on the 1 and the 20-cycle with 169->40->..., there is also a 3-cycle with 19->37->46, but apart from that my program has not found any cyles up to 100,000,000 (though I am not certain yet that my program does not contain any mistakes). Not all self-locating strings necessarily form 1-cycles, since they may also appear earlier, in fact only 1 does. Starting the numbering at zero, I also found two more self-locating strings (which also happen to form 1-cycles): 71,683,711 and 78,714,901
Since we are considering the number pi (not the number pi-minus-three), we have the initial three at position zero. So there is a 4-cycle with 0->32->15->3. There is also another 1-cycle at 711939213.
Step 1: Choose a self-locating string. Step 2: Concatenate the self-locating string with the digit immediately subsequent to the string Step 3: Travel to the location of the new string Step 4: Concatenate numbers starting from the first digit of the new location until a string is reached which has not previously appeared in the irrational number Step 5: Record the string Step 6: Repeat steps 3-5 Example: 1->14->93->21->264->603->etc... Every number listed on the new series is a number eliminated as a candidate for infinite loops. You can also freely concatenate additional digits in steps 2 and 4 to obtain new branches and add to the list because the first portion of those numbers only appeared in that step. For example, the 211 used in the video isn't actually the lowest number which leads to location 93. That honor belongs to 21, and 211 is concatenated from there.
I love the fact that this tattooed, pierced guy is a teacher in a prestigious university. Kudos to the people who recognized his brilliance without being distracted by his appearance, I am sure that does not happen nearly enough
pi is 3 if the circle is drawn on a sphere and so diameter of said circle is a slightly elevated arc. how would one approach finding such sphere's diameter?
I played with this and found that if you had considered the first digit to be the 3 (the true first digit), then the first match is 5 at the 5th and, later on, the 242,424 position is 242424.
This video just feeds the mysticality of pi and doesn't even attempt to grow the fascination of numbers in general. Of course pi has many interesting features, but its most interesting features have nothing to do with its decimal expansion. The interesting features of its decimal expansion are shared with many other numbers (I can't prove it, but nonetheless I am confident that almost all numbers, in almost all bases, have the properties pointed out here.)
When Tom said the rules are arbitrary I began to think of what happens when you allow the removal of the necessity of the string being the first occurence, and that there might be some branched loops. I dunno what the branches would do aside from resembling an algorithm, but could be in the direction of deriving proof of infinite occurences of this looping phenomenon. The way the index is described could allow some wiggle room with the right tools I believe.
I just wanted to thank this channel for existing. A little over a year ago I started watching numberphile videos which helped me discover my interest in math. I am now back in school because of it and just completed calculus 1 with an A+ and am currently taking Calculus 2!
Update: got an A in calc 2, now onto calc 3! Thanks so much Numberphile!
All the best bud
how is it?
That's great! Best of luck for your second course.
Computerphile (and Numberphile) are part of the reason I went back to school in computer science! (Where I did kind of an equivalent to calc-3, advanced linear algebra and probability and statistics classes). Best decision I made in my life! Kudos to you
I'm sure the people running this channel live to see comments like yours. Thanks for sharing!
What I want to know is, if the sentence: "We don't know if it's true but we know one thing." is the most common sentence in Numberphile videos.
I know! The same reason he has a wenoos which is unused
Should be the most common sentence in SCIENCE, actually. This is how these guys talk. They actually try to admit what they don’t know. Imagine that.
However, you need it after ‘however’
Most common sentence in all of mathematics
Why does he think 40 occurs a lot? It could just as easily not.
1 is the 1st digit of pi
3: :(
We could count 1 as the 0th digit of pi
3: :C
I laughed too long at this
I thought so too. At least we'd get a 3rd way of counting with 3 as 1st digit, making 1 the 2nd.
This. All of this. When he said "there's another way to count" I'm expecting to number 3 as the 1st position, but no, poor 3 always gets left out
in context, he means the first fractional digit of pi. 3 of course is the whole number portion of pi.
Those that make you 0 are scum.
But... those that make your friends 0 are worse than scum.
Hey the navier stokes guy.
Supriyo Chowdhury Had the same thought :D
Aka the Pokeballs guy
Cute boy with big brain
the navier stonks guy*
3:24 Or in other words "left as an exercise for the reader"
@Goran Newsum
I have read it somewhere 🙃
*3:14
false.
from Navier-Stokes fluid dynamics to number theory, this guy is quite versatile I'd say!!!
Yeah he stripped equation but UA-cam doesn't allow further explanation of Navier-stokes equation
And also waiting for female version
@@अण्वायुवरीवर्त wdym UA-cam doesn't allow further explanation?
I didn't know Machine Gun Kelly does math as a side hustle. It's great to see there is at least an intellectual in the music industry.
Well he kinda did a small brain move with Eminem
My exact same thought haha
I think Kurt Hugo Schneider went to yale university and studied physics or something
Brian May
HAHAHA love it
13598 "It's not far away from the 16000"
Car salesman detected!
Numberphile: there are loops and self-locating numbers in the decimals of π!
Engineers: what decimals??
Samuele
Nobody:
Engineers: sin(0.1) = 0.1
π = sin(π) = 0
isnt pi like 4
@@whatisthis2809 yes
@@whatisthis2809 If you're buying lace to attach into the rim of a circular table cloth, pi is 3,4, just to be safe. 4 costs too much.
You think the strings are common.
*THEY'RE NOT*
Anamika Sinha they are common. There’s an infinite number of self locating strings
Anthony Ross Doesn't mean they're common, if one in a million is a string, it isn't common but there are a lot of them
s dude I guess it’s just a debate of what common means things that are common in some areas might not be common in other parts
3:14
@@S44LT that's where density comes in
The fact that 169 looped back to itself is the wildest shiz to me. I guess I'm easily impressed.
Curious what this all looks like when pi and the positions are written in base 2.
Have you seen binary decimals?
They're a trip.
No they are not. The only time you got a better chance with binary is from 1-8.
For 7 you have a chance of 1/10 in decimal, 1/8 in binary.
For 8 you have a chance of 1/10 in decimal, 1/16th in binary.
For 512 you have a chance of 1/100 in decimal, and a chance of 1/1024 in binary.
For 8192 you have a chance of 1/1000 in decimal and a chance of 1/16384 in binary.
@@1R0QU012 they are not decimals, because "decimal" refers to the digits of the base ten expansion.
@@ahmedouerfelli4709 lol you've obviously don't work with computers.
@@1R0QU012 I do programming, but my main field is mathematics. I don't understand your response though, do you mean that computer scientists call binary digits "decimals" even though they are not decimal digits? Or do you mean something else?
Next: Strings and Loops within e?
Its 2.718281828459045235360 to 21 d.p
The 338th-340th digits of e are 338. The next self-locating string is at 2543, then 91668.
@@floydmaseda If you start counting at the 2, the first self-locating string is at position 8.
Strings and loops within arbitrary irrational numbers?
@@MichaelWBauer Nonono, _transcendental_ numbers!
Years ago I'd have said "recreational math" was an oxymoron. Not anymore. One thing great about Numberphile / Computerphile is that they have great speakers who are passionate and know how to express their passion. That's also due to their hosts & editing skills. As viewer we don't see just the cool maths, we also choose to absorb all that passion, and that makes a whole difference compared to the boredom of a grade school math class where students have no idea _why_ they are learning that stuff to begin with.
it seems the comments have entered an infinite loop in the far reaches of the decimal expansion of pi, never to be seen again...
40 wont necessarily show up more than once though..its not guaranteed right unless you can prove that it is...can you please correct this or clarify what you meant.
Hi, Tom, thanks for the very interesting video!
I was wondering what would happen if you skip the self-locating string and choose the next/secondary matching string to avoid local looping when searching for the global loop?
For example, 211－93－14－1－3－9－5－4－2－6－7－13－110－174－155－314－2120－5360－24671－...
@@hui-yuanchen8454 Nice idea - i imagine you would create an infinite string disappearing further and further into pi...
@@TomRocksMaths Yes..., it looks like that.
So I kept tracking after 24671－119546－
193002－240820－274454－153700－..., then 153700 doesn't exist in the first million digits of pi.
How come the numbers in "169's circle" are so special that they can form a loop!?
@@leif1075 40 shows up 6 times in the first 1000 digits of pi. Now it's proven!
Maths guy:
My brain: spike in his ear spike in his ear spike in his ear spike in his ear spike in his ear spike in his ear spike in his ear spike in his ear
They might be spacers but I have spike earrings that are just an ~illuuuusioonnnn~
I was checking out his pokeball.
I am a software developer by trade. It never ceases to amaze me how mathematicians come up with these problems that seem simple but are actually far beyond the bounds of mere computation; you could never have a "brute-force" solution to a problem like this.
I loved his explanation for why one would bother with this problem. It gives me hope with my own research.
The digits tattooed on his right arm are part of the decimal expansion of e.
If I saw this man on the street my last expectation would be that he is a recreational mathematician.
Not just a recreational; he's a professional mathematician.
Mas628 he’s a professor at Cambridge
@@HabeKeinMitleid he's a professor at oxford actually😡
@@AG-zo5es correct!!
@@recklessroges i rest my case
I literally learned the first 100 digits of Pi by making it my password for a couple of weeks, and now I can't get them out of my system anymore, btw great channel. Keep it up!!!
What an amazing world where we have people thinking these deep thoughts. Million thumps up.
Never could have thought you could fill up 14 minutes, talking about pi ;-) Now that's some contagious enthousiasm Tom ! Nice topic !
I am not a maths person....but you explained this so well that I watched it all
Thanks Lauren, that's awesome.
Ahhhh Tom so dashing and smart! Tom, i think you're the first alt/rocker looking person I've seen be into math.
I love how classic numberphile this is!
This is looking more and more like numerology.
71 with Blue for table number 21
car reference to the Pi movie?
So, something oogy-boogy.
car it looks more like a meaningless attempt at being interesting. This has zero math implication
@@codycast Actually I can see a possibility of this technique being use to prove if pi or e is normal.
As well as self-locating strings, there's another way to show that some numbers don't sit within the loop they settle into, which accommodates loops larger than 1 - If we take the 169 example, the next digit is a 3. If 169 first appears at the 40th place in pi, then 1693 ALSO first appears at the 40th place in pi. So 1693 has to settle into a loop that doesn't pass through 1693 again.
does this mean that it's very likeable that there exist some numbers whose loop actually did not end?
Would that numbers be called "primes" by some means?
Best newcomer at the numberphile awards 2020: Tom Crawford
Kyle
The detail of that pokeball tattoo is amazing.
Thanks!!
Tom Crawford: "You don't really need more than 7 decimal places".
Emma Haruka Iwao: "What have I done with my life"
Rajan Mahadevan: "I know right"
Matt Parker: "Amatuer!"
Considering that NASA uses pi to 16 decimal places in software that stabilizes spacecraft trajectories, the national institute for standards and technology uses pi to 32 decimal places when calculating the fundamental constants of the universe, and pi to 100 decimal places, if memory serves, would be sufficient to calculate the circumference of the observable universe to a precision of less than the width of a hydrogen atom (if the geometry of the universe allows that such a calculation would make sense and we had enough precision in our measurement of the diameter to make such a calculation meaningful, of course), I think it’s safe to say that most of us unwashed lumpenproletarians are in no danger from using a value of pi with seven decimal places in our day-to-day lives!
PI IS EXACTLY THREE
gurrrn Nah, by rounding down as 3 is less than 5, Pi is therefore 0.
@@VenomOnPC In base pi, pi = 1, and 1 is a transcendental number. Don't mess.
@@DrKaii Where did you get your doctorate? In base π 1 is merely 1, and no funny business happens until you reach 10 which, of course, is π.
Commenting to check what's going on with missing comments. Great video btw
"You don't really need more than 7 decimal places"
- Continues to write down the first 100 digits of pi.
On brown paper as well. Should have gone with a subtle off-white colouring.
You survived the 80s?
We don't need more. We can write though.
@@zeldaandTwink Let's use a simple encryption algorithm: EncryptedMessage = character(x) XOR decimal(x)ofPi
To decrypt 1 MB of Data you would need to know pi up to 1 million of decimal places. Not just 31.
@@d34d10ck That is using Pi as a seed, which is not using it as the mathematical constant. When using it it as the ratio of a circle's circumference to diameter, the largest physical circle we know of would be the circumference of the universe. Pi to 31 places is the correct ratio within the width of a hydrogen atom, so you never need to go to more digits for PHYSICAL purposes. Any smaller circle, pi to 31 digits is even more accurate as the ratio, very quickly getting below the planck length in terms of error (i.e., pointless for modern physics). That is what was meant by practical applications, not using it as a random large seed, which always benefits from having more digits.
If the digits of pi were completely random you would expect there to be infinitely many self locating strings, but each one should be bigger than its predecesor by about 12.915 (10^(10/9)) times. This aligns roughly with whats shown. Ignoring the massive gap at the start, the average ratio of succesive terms is about 12.184.
Reminds me of letting youtube autoplay videos. I've noticed loops and onramps to major loops.
I'm reminded of a much earlier video they did on the recreational "happy numbers" and "melancoil", which also had loops and various self-locating numbers, specifically at 1.
Wow this went deeper than I thought
This reminds me of Melancoil numbers ... great to see Numberphile getting back to its 'roots'!
Can't wait for the next Vsauce video to be about self-locating strings.
Coming out tomorrow
@@Wyattporter, sadly, it didn't happen. :(
Coming out tomorrow
@@netstatgrep, how do you know that?
Coming out tomorrow
I was also wondering why we weren't using base 0 index and was pleased to watch the following
Hmm... Pi to the 0th place... well, it wouldn't have a decimal.
i would give this guy 10/10 on fashion and looks
[12:37] «It's not the peak that's amazing, it's the awesome landscape and the methods and techniques you're going to learn and use to climb a mountain when you try.»
6:40 one hundred and si… nice.
thanks for the video, all of them really, i do quite enjoy them all, but i would also like to see more of Tom on here too, i especially enjoyed the Navier-Stokes and Reynolds number episodes, glad to see him back with this.
i love their energy
Very interesting and thought-provoking.
This is suspiciously equivalent to the collatz conjecture
Top notch camera skills.
This reminded me of the Collatz Conjecture.
First thing that came to mind once they started talking about loops and that they can get "stuck"
"recreational maths" I love this guy
Curious as to how these strings function under different number systems, such as those of base 8 or 12
Prime bases better...
@@SierraDN No? You want many divisors to reduce the amount of infinitely repeating expansions, like 0.333333... etc.
I was thinking base 16 since there's a method to determine the nth digit of pi in base 16 without having to calculate all the digits before it. Google BBP Formula to see the method.
Pi base sixty is finite
@@christopherellis2663 Huh? Is that a math joke (like Grothendieck's Prime), or am I missing something?
Really enjoyed this video--AND understood it to the end! All of his previous ones were way over my head, so I couldn't finish them. This one was fun. π rocks!
Love it. I like how you said "decimal", occasionally, too. Thanks.
"P.s." Gee I wish I could... (recollect pi easily today now ...)
This is so cool I wonder if the relationship between those numbers in loops can be generalized
Now, as a completely new rule, start counting from the infinite side towards 0th position. :]
I like this new format of LinusTechTips
I thought that too. He has the same contagious energy too! 😂
I wonder what the function will look like if you graph Looplength(x), y being the number of iterations before terminating or getting stuck
Pretty random I'd guess
was actually thinking the same :D Maybe I should do a script to plot that for me...
@@erikbrendel3217 Notify us when done :)
Erik Brendel oh yes that‘d be cool
this reminds me of aliquot sequences. all known sequences either terminate at 0 or end in a perfect, amicable, or sociable number. it is an open question whether there are any aliquot sequences that increase without bound
Poor 3, often left out. What if you count 3 as the first position?
that's mental!
I had the same idea, it’s prob been tested though and there’s probably a reason it’s only decimals they use
I did the numbers. The self locating strings with 3 as the first digit are as follows:
5
,
242424
,
271070
,
9292071
,
29133316,
70421305
@@hyperhippyhippohopper 242424.
@@hyperhippyhippohopper 242424 is actually very nice, i hate the fact that they not only skipped 3, but they even had the audacity to say "What if we consider 1 to be the 0th digit of pi?"
In love with numbers. Thanks Numberphile!
The most attractive male mathematician in the world. Calling it right now.
This reminds me of the Collatz conjecture, which on its own does not have any real significance but the techniques and strategies used to attack it will probably be important tools in solving other problems.
1:25 I think it would be more natural and general for the 1 to be in the 11th position and the 2 to be in the twelth.
Since 12 trillionths would have the 1 and 2 in those positions. (and 10^-12 is a trillionth)
That's _kind of_ like having a 12 in that position. Except we don't have a digit for 12 and the 10 carries over to 1 in the position to the left.
"You don't really need more than seven decimal places"
Matt Parker will remember that.
The statement at 10:36 doesn't seem right: Self-locating strings may or may not map to themselves, because the string could appear earlier.
You answered to your own mistake in that sentence.
@@kirkanos771 Sorry, I don't understand. Care to explain what my mistake is?
Actually, I just checked, and when he said "...or you get to 16,470 and then loop around", he's incorrect: 16,470 maps to 1,602 and you keep going.
@@alonamaloh Because you always go to the earliest appearance of the number. Which means that if you happen to end up at a self-locating string, it will have to be the first appearance of that string.
But you're correct in that not every self-locating string will act as a deadend to those sequences.
I think this is so awesome! I went to grab a pencil and do my own loop, but then I realized that there no way to easily count where those numbers are in pi 🙃
you may need a computer program to do so, it would be a faster check
There is a shorter circuit: 19, 37, 46 and again 19
EDIT: I failed, see post 3 lol.
I made a quick excel to try this, and found it is incorrect. Using the first 32759 digits of pi (the most excel can hold in 1 cell, seems an odd number, 2^15-9, anyways...) the following order/list is produced:
19, 39, 45, 62, 22, 137, 861, 269, 1395, 6482, 228, 2529, 18335; and then it doesn't find '18335' in the first ~32k digits.
EDIT: I failed, see post lol.
I disliked the 32b limitation of excel cells, so I did a bit more exceling and made some formulas to read multiple cells and find the correct text. Long story short, I didn't find anything interesting, even tho at first I thought I found a cool 188 step loop (I had had a bug in one of my formulas). Using a total of 1 million digits of pi, the sequence terminates as follows (without finding the next step, it presumably will find one with enough digits, per this video):
19, 39, 45, 62, 22, 137, 861, 269, 1395, 6482, 228, 2529, 18335, 68539, 22166, 169545, 96010, 67419, 272547, 414384, 148332 (can't find this in the first 1 million digits)
OK, final thoughts, cuz this is pointless unless I wanna python this.
I tried numbers 0 to 100 with the first 1 million digits and found just 2 loops, your loop and one that starts at 40 with a cycle length of 20:
40, 70, 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169, 40
This 20 length loop also can have a relatively long lead-up phase starting with 61, which has a 7-step lead-up phase:
61, 219, 716, 39, 43, 23, 16, 40
I also found a number of values that terminate at repeating 1's:
1, 14, 21, 45, 73, 93
The longest path to 1 starts at 45 and is 10 steps to 1, and has the following steps:
45, 60, 127, 297, 737, 299, 2643, 21, 93, 14, 1
In that path, you can see that the other digits 14, 21 and 93 pop up (and 45 obviously), but not 73, which has the following path:
73, 299, 2643, 21, 93, 14, 1
This is at least a little interesting that both 73 and 737 both go to 299, which then leads to 1's. The next couple digits starting at position 299 are 737(2458), so we know a couple more large numbers that go to 1.
There's another possibility for a chain: ending at a smaller loop. For example, if you found one of the strings that were part of a known loop in a position not in that loop, tracing the path back up the chain would be infinitely long, but would end with a loop of known (finite) length.
The leading 3 is so neglected. It would be interesting knowing what happens if indexing starts there.
Self-location can’t be infinite as the increasing enumeration order means it becomes an order of magnitude less likely that one appears. You can use statistics to prove this.
An argument can be said about the loops, i.e. That there are no infinite loops, because you can always find the next index, and you always select the lowest valid index, meaning there will for any multiple number be a lower index to be selected, which cuts out the possibility of infinites.
But wouldn’t there still be countably infinite self-locating numbers? Just because they become less common doesn’t mean they stop completely. Would love to see an explanation for that
The task of writing down all digits of pi is left as an exercise to the viewer.
I really really want that paper sheet! Pi actually is a big part of my life and id really really love to have that on my wall!
ngl this fella is my man-crush.
Tom Navier-Stokes Crawford! That was an epic series.
This video is also fascinating; those loops in pi are like the swirling of a viscous fluid... Now, if only we had a set of equations to describe the motion of viscous fluids...
Jeez, so many comments about the topic not being serious enough or people clutching their pearls about the bloke having a few piercings. Lighten up, guys!
This kind of reminds me of the ”social numbers”, in the context of ”Aliquot”-sequences 🤔.
10:35 This statement is not correct: self-locating strings could already appear earlier. For example, 44899 could also appear at a smaller position, meaning you would not loop at position 44899
Yes ikr! I commented the same few seconds back
@#Miqdaad Indori It can only stop when 44899 is found at position 44899, and there is no 44899 in an earlier position. (This is not the case as 44899 also appears at position 13714)
Indeed, 44899 occurs at position 13714 first.
@David Gerick
Yes, but you would still get 44899, which will take you yet again yo 44899 and so on...
You would only het 44899, so in a sense you don't loop between numbers
@@qubatistic4788 The sequence for 44899 is: 44899 -> 13714 -> 120330 -> 2293915 -> 43742 -> 126470 -> ...
I've memorized 51 digits of PI. Honestly if I were to replace PI everywhere in my renderers with the number 3.0, then you would barely notice the difference.
Impressive, somehow I have memorized Tau better than Pi
@@zacharyhandy9606 Quite nice actually, perhaps some day we're going to switch to Tau.
That moment when you actually see your phone number in pi...
there are search pages that can find any 7 digit string
I wrote a program to try this out myself and apart from the 1-cycle on the 1 and the 20-cycle with 169->40->..., there is also a 3-cycle with 19->37->46, but apart from that my program has not found any cyles up to 100,000,000 (though I am not certain yet that my program does not contain any mistakes). Not all self-locating strings necessarily form 1-cycles, since they may also appear earlier, in fact only 1 does.
Starting the numbering at zero, I also found two more self-locating strings (which also happen to form 1-cycles): 71,683,711 and 78,714,901
Since we are considering the number pi (not the number pi-minus-three), we have the initial three at position zero. So there is a 4-cycle with 0->32->15->3. There is also another 1-cycle at 711939213.
@4:40
Matlab user: *suffers*
Haha yes
Tom Crawford: "You don't really need more than 7 decimal places".
Somewhere on the Earth SImon Pampena shed a single tear.
Remember first 8 digits of pi......
"May I have a large cup of coffee."
Thanks me later 🙏
"cup" can't be right. Pi is 3.14159265 not 3.1415326
@@soumilshah1007 May I have a large container of coffee
It would be an interesting exercise to play with this in different bases.
the loops remind me of an orbit in chaos/dynamical systems...
Alisa Beaubien congrats on your comment
Step 1: Choose a self-locating string.
Step 2: Concatenate the self-locating string with the digit immediately subsequent to the string
Step 3: Travel to the location of the new string
Step 4: Concatenate numbers starting from the first digit of the new location until a string is reached which has not previously appeared in the irrational number
Step 5: Record the string
Step 6: Repeat steps 3-5
Example: 1->14->93->21->264->603->etc...
Every number listed on the new series is a number eliminated as a candidate for infinite loops.
You can also freely concatenate additional digits in steps 2 and 4 to obtain new branches and add to the list because the first portion of those numbers only appeared in that step. For example, the 211 used in the video isn't actually the lowest number which leads to location 93. That honor belongs to 21, and 211 is concatenated from there.
Pi, e and phi are the universe, the time and the energy.
Gonzalo Garcia What?
I love the fact that this tattooed, pierced guy is a teacher in a prestigious university. Kudos to the people who recognized his brilliance without being distracted by his appearance, I am sure that does not happen nearly enough
Everyone ignores the 3...
They should instead say pi - 3
And why not make the 3 the 1st position? All self-locators and loops will be completely different.
Just put 3 at the 0th location.
/ivide by 10
@@ChrisWalshZX i found by hand that 5 is a self locating string but everything else i think is massive
pi is 3 if the circle is drawn on a sphere and so diameter of said circle is a slightly elevated arc. how would one approach finding such sphere's diameter?
I played with this and found that if you had considered the first digit to be the 3 (the true first digit), then the first match is 5 at the 5th and, later on, the 242,424 position is 242424.
😂 Too bad it's not 424242.
Dude looks and sounds like he's about to yell at me for my math being half baked and still raw.
@@awindwaker4130, I thought he was a golfer. He stores his tees in his ear lobes.
he does look like ramsy lol
He looks like a mash-up of Gordon Ramsay and Linus from Linus TechTips. 😄
This video just feeds the mysticality of pi and doesn't even attempt to grow the fascination of numbers in general. Of course pi has many interesting features, but its most interesting features have nothing to do with its decimal expansion. The interesting features of its decimal expansion are shared with many other numbers (I can't prove it, but nonetheless I am confident that almost all numbers, in almost all bases, have the properties pointed out here.)
0:30 "2rd"
Officially calling it “Gangstar mathematics”
In honour of this guy
the path could be in a ρ shape imo
You've heard of talking with your hands, but this guy practically dances while speaking.
Frankie’s Theorem: There is a trivial sequence for finding these numbers.
Proof.
Exercise 1
QED
0 strings are lower than 1 strings.
Enjoyed it, thank you.
Are we assuming Pi is incompressible in this video?
When Tom said the rules are arbitrary I began to think of what happens when you allow the removal of the necessity of the string being the first occurence, and that there might be some branched loops. I dunno what the branches would do aside from resembling an algorithm, but could be in the direction of deriving proof of infinite occurences of this looping phenomenon. The way the index is described could allow some wiggle room with the right tools I believe.