The mystery of 0.577 - Numberphile

The harmonic series and the elusive Euler–Mascheroni constant.
More links & stuff in full description below ↓↓↓

Featuring Dr Tony Padilla.

NUMBERPHILE

Videos by Brady Haran

[firstnamelastname]

I love when a Numberphile video has a mindblow moment. It grows BEHIND it too!

[treeinthewood]

the example with the elastic band can be misunderstood easily, so Brady's argument isn't that wrong actually: if you just say "every second, we add another meter to its circumference", you can always add the additional meter in front of the ant, and of course it will never reach the end that way. instead, you have to emphasize that the band is stretched, i.e. it is uniformly expanded so that every part of it grows the same percentage. that means the distance behind the ant and and the distance in front of the ant grow proportionately to their relative size, and as the distance behind the ant becomes larger and larger in relation to the distance in front of it (it will since the ant travels), more and more of the additional meter is in fact added behind the ant, not in front of it.

[ChrisBandyJazz]

If you add 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8, you are just adding 1/2 over and over again, so we can clearly see that it diverges.

And if you add 1+2+3+4..., you are just adding 1 over and over again, so we can clearly see that to -1/12

[lucamaci]

LET'S BRING THIS TO THE TOP
This video was more closely related to the armonic series than 0.577.
You can't just say "0.577 appears all over physics" and "it knows about primes too" and not expect me to demand a more in depth separate video with .577 as its star.

[harry_page]

I worked out the ant band time:
The circumference of the band is given by C = t+1 where t is time elapsed.
Distance travelled by the ant is s.
The speed of the ant seems to be 0.01 m/s, but also has a component given by the band stretching behind it, which gives it further displacement. The rate at which this happens is the proportion of the band that the ant has already travelled across at a given time: s/C = s/(t+1)
So we get the differential equation ds/dt = s/(t+1) + 0.01
(ds/dt)/(t+1) - s/(t+1)^2 = 0.01/(t+1)
d/dt(s/(t+1)) = 0.01/(t+1)
s/(t+1) = 0.01ln(t+1) + c
s = 0 when t = 0 so c = 0
s = 0.01(t+1)ln(t+1)


The point at which the ant makes it back to the start is when s = C = t+1:


t+1 = 0.01(t+1)ln(t+1)
1 = 0.01ln(t+1)
t = e^100 - 1

[ameto]

A numberphile video on numbers, they are getting really rare these days

[Derpster]

All I know is 1+2=12 and even that might be wrong.

[eideticex]

Certainly an interesting one. Have seen .577~ pop up from time to time in the maths I work with in computer graphics and physics simulation. Always thought it was just the result of some personal bias, a product of how I do things. Never realized it actually had a more profound meaning.

[ericperu]

Love the excitement he exhibits when he talks about this stuff. Way over my head but fascinating nonetheless.

[suncu]

I can see motivational poster with that ant story coming up

[alexbabits]

This has got to be one of my favorite Numberphile videos. Just completely thought provoking.

[cuentadeyoutube]

I love how they know the whole -1/12 affair works, at the very least, as a delicious trolling act. (but of course the video itself was so insightful it went over many people's head)

[rickmorrow]

Your passion for mathematics is infectious. Gauss, Newton, Leibnez, Pascal, Euclid, Pythagoras and Archimedes are all subscribing to your YouTube videos.

[the_feature_selector]

So it's the % of the meter increase (relative to ant) that gets smaller as ant goes till eventually the increase relative to the ant is smaller than his cm traveled per increase. This was such a cool problem!! I did using excel spreadsheet as ant going 1 cm and circle increasing by 2cm (starts 2cm big) and when circle reaches 22cm the ant has gone fully around the circle.

[uuu]

"Yeap, -1/12, totally not controversial"
Someone has been working on his sarcasm skills

[wuuglyv]

2:50 "I knew you were going to say that". That's because solving the Reimmer zeta function for a divergent series does not give you an equivalence for n=infinity. It is an "associated" value, not the "answer" of the series.

[lukeusperez]

It's fun to see Brady's reactions in the window reflection

[jimi]

A fun fact is that it shows up in the 'block stacking problem' (or the Leaning tower of Lire). The idea is that you stack blocks or bricks on top of each other on an edge of a table and make the stack of blocks lean over the edge as much as possible without it falling over. Then you want to know how many blocks you need in order to make the tower lean over the edge, for example 4 times the legth of one block. You can calculate the exact number of blocks you need by rounding (to the closest integer) the value of this formula: e^(2*o-y) where "o" is the number of brick lengths the tower leans over the edge and "y" is the Euler-Mascheroni constant.

[eantropix]

It saddens me to see such an amazing channel, with less than 2 million subscribers. Where are all the Numberphiles out there?

[curtiswilson]

Truly thought provoking! This is numberphile at its best, Brady!