Why is pi here? And why is it squared? A geometric answer to the Basel problem
Summary
TLDREl video explora el problema de Basilea, una suma infinita de inversos de números cuadrados que converge al cuadrado de pi dividido por 6, una relación descubierta por Euler. Se utiliza una metáfora de faros en una línea numérica para representar la suma y se introduce el teorema de la inversa de Pitágoras para transformar un faro en dos, manteniendo la luminosidad percibida. A través de esta técnica y la creciente complejidad geométrica, se demuestra que la suma de los inversos de los cuadrados de los números enteros positivos es igual a pi cuadrado dividido por 6, revelando una conexión sorprendente entre la geometría y la serie matemática.
Takeaways
- 🔵 El problema de Basilea, planteado hace 90 años, fue resuelto por Euler, quien descubrió que la suma de los inversos de los números cuadrados se acerca a π² dividido por 6.
- 🌐 La relación entre la suma y π es sorprendente, ya que usualmente no se ve a π elevado al cuadrado en contextos no relacionados con círculos.
- 🏙️ La demostración alternativa presentada en el video utiliza una representación física de la suma a través de la intensidad de luz de faros (linternas) colocados en los enteros positivos.
- 💡 La intensidad percibida de los faros se reduce según la ley inversa del cuadrado de la distancia, lo que es aplicable no solo a la luz sino a otras formas de energía que se propagan desde un punto.
- 📐 La transformación de un faro en dos utilizando el teorema de Pitágoras invertido es clave para manipular la configuración de los faros sin cambiar la intensidad total percibida.
- 🌟 La suma de la intensidad de los faros se puede reorganizar en un patrón que se asemeja a la suma a lo largo del borde de un círculo infinitamente grande, lo que lleva a la conexión con π.
- 🔄 El proceso iterativo de duplicar el tamaño del círculo y transformar cada faro en dos nuevos, mantiene constante la intensidad percibida y la distribución uniforme de los faros.
- 🔢 La suma de los inversos de los cuadrados de los números enteros positivos impares conduce a una serie infinita que se relaciona con π²/4, una aproximación a la solución del problema de Basilea.
- 🚫 La suma original del problema de Basilea incluye todos los números naturales positivos, no solo los impares, lo que requiere ajustes adicionales para alcanzar la solución final.
- 🎥 El video fue escrito y animado por Ben Hambricht, un nuevo miembro del equipo de 3Blue1Brown, con apoyo de los patrocinadores en Patreon.
Q & A
¿Qué serie infinita se resuelve en el problema de Basilea?
-La serie infinita que se resuelve en el problema de Basilea es la suma de los inversos de los números cuadrados, es decir, 1 más 1/4 más 1/9 más 1/16 y así sucesivamente.
¿Cuál fue la contribución de Euler al problema de Basilea?
-Euler resolvió el problema de Basilea después de 90 años de ser planteado, encontrando que la suma de la serie infinita se acerca a π² dividido por 6.
¿Por qué se llama el problema de Basilea 'basilea problem'?
-El problema de Basilea se llama así en honor a Euler, cuyo pueblo natal era Basilea.
¿Qué es la ley del inverso cuadrado y cómo se relaciona con la luz?
-La ley del inverso cuadrado es un fenómeno tridimensional que se da cuando una cantidad se propaga uniformemente desde un punto fuente, como la luz, el sonido o la energía de una señal de radio. La luz disminuye su brillo proporcionalmente a la inversa del cuadrado de la distancia, lo que significa que si la distancia se duplica, el área impactada por la luz se cuadra, y por lo tanto, la cantidad de luz recibida disminuirá a la cuarta parte.
¿Qué es el teorema de Pitágoras inverso mencionado en el guion?
-El teorema de Pitágoras inverso es una relación que se da en el contexto de la geometría esférica y se refiere a la igualdad entre la suma de las recíprocas de las distancias cuadradas de dos puntos y la recíproca de la distancia cuadrada del tercer punto, que es 1/a² + 1/b² = 1/h².
¿Cómo se utiliza la idea de los faros en el guion para representar la serie infinita del problema de Basilea?
-En el guion, los faros son utilizados como una representación física de los términos de la serie infinita del problema de Basilea, donde el brillo aparente de cada faro corresponde a los términos de la serie, y la suma total del brillo es igual a la solución del problema.
¿Qué es la 'ángulo sólido' y cómo se relaciona con la luz?
-El ángulo sólido es la proporción de una esfera que cubre una figura vista desde un punto dado. En el guion, se relaciona con la luz al considerar la cantidad de luz que impacta en una pantalla o sensor, lo que se relaciona con el ángulo que la luz cubre en ambas direcciones perpendiculares a la fuente de luz.
¿Cómo se justifica la igualdad de brillo entre un faro y dos faros en el guion?
-La igualdad de brillo entre un faro y dos faros se justifica utilizando la idea de que la luz de un faro puede 'reformarse' en dos faros nuevos de tal manera que el ángulo de los rayos de luz que impactan en una pantalla es el mismo, lo que mantiene la cantidad de luz recibida igual.
¿Cómo se relaciona la solución del problema de Basilea con la geometría circular?
-La solución del problema de Basilea se relaciona con la geometría circular al considerar la suma a lo largo de un número lineal infinito, lo que se asemeja a sumar a lo largo del borde de una circunferencia infinitamente grande, donde la suma de los inversos de los cuadrados de los números corresponde a la suma a lo largo de la circunferencia.
¿Cómo se transforma la suma sobre los números enteros positivos en la suma sobre los números enteros positivos impares para resolver el problema de Basilea?
-Para transformar la suma sobre los números enteros positivos en la suma sobre los números enteros positivos impares, se multiplica la suma por 4/3, ya que la suma de los recíprocos de los números pares es una cuarta parte de la suma total, y la suma de los recíprocos de los números impares es tres cuartas partes de la suma total.
Outlines
🔍 Análisis del Problema de Basilea
El primer párrafo introduce un desafío matemático que involucra la suma de inversos de números cuadrados, conocido como el problema de Basilea. Este problema permaneció sin resolver durante 90 años hasta que Euler descubrió que la suma se acerca a pi cuadrado dividido por 6. La sección plantea preguntas sobre la presencia de pi y su cuadrado en la respuesta, y sugiere que la solución que se presentará es diferente a la de Euler. Además, se establece una conexión entre la aparición de pi y la relación con las circunferencias, sugiriendo que cualquier ecuación que incluya a pi tiene un lazo con la geometría.
🌐 Representación Física del Problema
El segundo párrafo transforma el problema matemático en una representación física usando faros (linternas) en los enteros positivos de una línea numérica. Cada faro emite una cantidad de luz que se mide en términos de 'brillo'. La relación de brillo entre los faros se basa en la ley de la inversa del cuadrado, donde el brillo disminuye con la distancia al cuadrado. Esta representación física permite abordar el problema de Basilea de una manera visual y geométrica, y se plantea la idea de reorganizar los faros sin cambiar el brillo total para el observador.
🏰 Transformación de Faros y la Teorema de Pitágoras Inverso
El tercer párrafo explora cómo transformar un faro en dos otros sin cambiar el brillo percibido por el observador, utilizando el Teorema de Pitágoras Inverso. Se describe un método para dividir un faro en dos, manteniendo el mismo brillo total, y se utiliza esta técnica para construir una serie infinita de faros a lo largo de un lago circular. La sección también introduce la idea de que la suma de la inversa de los cuadrados de los números impares positivos, que se deriva de esta configuración, se relaciona con pi cuadrado, estableciendo un vínculo con la geometría circular.
🔄 Suma de la Serie y Conclusión del Problema de Basilea
El cuarto y último párrafo concluye el video explicando cómo la suma de la serie de fracciones que involucran solo números impares positivos se relaciona con pi cuadrado. Se ajusta la suma para incluir tanto números impares como pares, y se utiliza el concepto de distancia y brillo para transformar la serie original en una que abarca todos los enteros positivos. Finalmente, se multiplica la suma por un factor para obtener la solución al problema de Basilea, mostrando cómo una serie matemática inicialmente aparentemente no relacionada con la geometría, termina siendo intrínsecamente conectada con ella.
Mindmap
Keywords
💡Basel problem
💡pi (π)
💡inversos de los números cuadrados
💡inversa del teorema de Pitágoras
💡brújula aparente
💡serie infinita
💡geometría esférica
💡ley del inverso cuadrado
💡círculo y radio
💡transformación geométrica
Highlights
El problema de Basel, una suma infinita de inversos de números cuadrados, quedó sin resolver durante 90 años hasta que Euler descubrió que su suma es pi cuadrado dividido por 6.
La aparición de pi en la solución plantea preguntas sobre su conexión con las formas geométricas, especialmente los círculos.
La representación física del problema de Basel mediante faros en una línea de números positivos, donde la brillantez disminuye según la ley del inverso del cuadrado de la distancia.
La brillantez aparente de los faros se relaciona con el ángulo sólido que cubren en la geometría esférica, proporcional al área de una pantalla que intercepta los rayos de luz.
La ley del inverso del cuadrado describe cómo la disipación de la luz (o cualquier otra forma de energía) se propaga desde una fuente puntual en el espacio tridimensional.
La transformación de un faro en dos utilizando el teorema de Pitágoras inverso, manteniendo la brillantez percibida por el observador.
La demostración del teorema de Pitágoras inverso utilizando la intuición de la luz y las pantallas, y su aplicación en la geometría esférica.
La construcción de un arreglo infinito de faros a lo largo de un lago circular, donde la brillantez total se mantiene constante a medida que se duplica el tamaño del círculo y se transforman los faros.
La utilización del teorema de los ángulos inscritos para entender la distribución uniforme de los faros en el lago circular y su relevancia en la solución del problema de Basel.
La transición del arreglo de faros en el lago circular a una línea horizontal infinita, lo que permite sumar la serie de inversos de cuadrados de números enteros positivos y negativos.
La suma de la serie de inversos de cuadrados de números impares, que resulta en pi cuadrado dividido por 8, y su relación con la solución del problema de Basel.
La diferencia entre la suma de los números impares y la suma de todos los enteros positivos, y cómo se relaciona con la disminución de la brillantez a medida que se duplica la distancia de los faros.
La multiplicación de la suma por 4/3 para obtener la solución final del problema de Basel, que es pi cuadrado dividido por 6.
La explicación de cómo la suma de fracciones simples, que a primera vista no tienen nada que ver con la geometría, resulta estar relacionada con pi y las formas geométricas.
La conexión entre la suma a lo largo de la línea numérica y la suma a lo largo del borde de una circunferencia infinitamente grande, sugiriendo una relación con la geometría circular.
La contribución de Ben Hambricht al video, un nuevo miembro del equipo de 3Blue1Brown, posible gracias al apoyo de los patrocinadores en Patreon.
Transcripts
Take 1 plus 1 fourth plus 1 ninth plus 1 sixteenth and so on
where you're adding the inverses of the next square number What
does this sum approach as you keep adding on more and more terms?
Now this is a challenge that remained unsolved for 90 years
after it was initially posed until finally it was Euler who
found the answer Super surprisingly to be pi squared divided by 6.
I mean isn't that crazy?
What is pi doing here?
And why is it squared?
We don't usually see it squared in honor of Euler whose hometown was basil This infinite
sum is often referred to as the basil problem But the proof that I'd like to show you
is very different from the one that Euler had I've said in a previous video that whenever
you see pi show up There will be some connection to circles and there are those who like
to say that pi is not fundamentally about circles and Insisting on connecting equations
like these ones with a geometric intuition stems from a stubborn insistence on only
understanding pi in the context where we first discovered it and That's all well and
good But whatever your own perspective holds as fundamental the fact is pi is very much
tied to circles So if you see it show up there will be a path somewhere in the massive
interconnected web of mathematics Leading you back to circles and geometry The question
is just how long and convoluted that path might be and in the case of the basil problem
It's a lot shorter than you might first think and it all starts with light Here's the
basic idea Imagine standing at the origin of a positive number line and putting a little
lighthouse on all of the positive integers one two three four and so on that first
lighthouse has some Apparent brightness from your point of view some amount of energy
that your eye is receiving from the light per unit time and Let's just call that a
brightness of one For reasons I'll explain shortly the apparent brightness of the second
lighthouse is 1 fourth as much as the first and the apparent brightness of the third is
1 9th as much as the first and then 1 16th and so on and you can probably see why this
is useful for the basil problem It gives us a physical representation of what's being
asked Since the brightness received from the whole infinite line of lighthouses is going
to be 1 plus 1 4th plus 1 9th Plus the 16th and so on So the result that we are aiming
to show is that this total brightness is equal to pi squared divided by 6 times the
brightness of that first lighthouse And at first that might seem useless I mean,
we're just re-asking the same original question But the progress comes from a new
question that this framing raises are there ways that we can rearrange these lighthouses
That don't change the total brightness for the observer And if so,
can you show this to be equivalent to a setup that's somehow easier to compute To start
let's be clear about what we mean when we reference apparent brightness to an observer
Imagine a little screen which maybe represents the retina of your eye or a digital camera
sensor or something like that You could ask what proportion of the rays coming out of
the source hit that screen or phrase differently What is the angle between the ray
hitting the bottom of that screen and the ray hitting the top?
Or rather since we should be thinking of these lights as being in three dimensions.
It might be more accurate to ask What is the angle the
light covers in both directions perpendicular to the source?
In spherical geometry you sometimes talk about the solid angle of a shape Which is the
proportion of a sphere it covers as viewed from a given point You see the first of two
places this story we're thinking of screens is going to be useful is in understanding
the inverse square law Which is a distinctly three-dimensional phenomenon think of all
of the rays of light hitting a screen one unit away from the source as You double the
distance those rays will now cover an area with twice the width and twice the height So
it would take four copies of that original screen to receive the same rays at that
distance And so each individual one receives 1 fourth as much light This is the sense in
which I mean a light would appear 1 fourth as bright two times the distance away Likewise
when you're three times farther away You would need nine copies of that original screen
to receive the same rays so each individual screen only receives 1 9th as much light and
This pattern continues because the area hit by a light increases by the square of the
distance the brightness of that light decreases by the inverse square of that distance
and As I'm sure many of you know this inverse square law is not at all special to light
It pops up whenever you have some kind of quantity that spreads out evenly from a point
source whether that's sound or heat or a radio signal things like that and Remember it's
because of this inverse square law that an infinite array of evenly spaced lighthouses
physically implements the Basel problem But again what we need if we're going to make
any progress here is to understand how we can manipulate setups with light sources like
this without changing the total brightness for the observer and The key building block
is an especially nice way to transform a single lighthouse into two Think of an observer
at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane
Now draw a line from that lighthouse to the observer and then another line perpendicular
to that one at the lighthouse Now place two lighthouses where this new line intersects
the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and
lighthouse B on the upper side It turns out and you'll see why this is true in just a
minute the brightness that the observer Experiences from that first lighthouse is equal
to the combined brightness experienced from lighthouses A and B together And I should
say by the way that the standing assumption throughout this video is that all lighthouses
are equivalent They're using the same light bulb emanating the same power all of that So
in other words assigning variables to things here if we call the distance from the
observer to lighthouse a little a and The distance from the observer to lighthouse B
little B and the distance to the first lighthouse H We have the relation 1 over a squared
plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse
Pythagorean theorem which some of you may recognize from math ologer's most recent and
I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool
relation don't you think and if you're a mathematician at heart you might be asking right
now how you prove it and There are some straightforward ways where you express the
triangles area in two separate ways and apply the usual Pythagorean theorem But there is
another quite pretty method that I'd like to briefly outline here that falls much more
nicely into our storyline because again It uses intuitions of light and screens Imagine
scaling down the whole right triangle into a tinier version and think of this miniature
Hypotenuse as a screen receiving light from the first lighthouse If you reshape that
screen to be the combination of the two legs of the miniature triangle like this Well,
it still receives the same amount of light, right?
I mean the rays of light hitting one of those two legs are precisely the same as the rays
that hit the hypotenuse Then the key is that the amount of light from the first
lighthouse that hits this left side the limited angle of rays that end up hitting that
screen is Exactly the same as the amount of light over here coming from lighthouse a
which hits that side it'll be the same angle of rays and Symmetrically the amount of
light from the first house hitting the bottom portion of our screen is The same as the
amount of light hitting that portion from lighthouse B Why you might ask well,
it's a matter of similar triangles This animation already gives you a strong hint for how
it works And we've also left a link in the description to a simple GeoGebra applet for
those of you who want to think this through in a slightly more interactive environment
and in playing with that One important fact here that you'll be able to see is that the
similar triangles only apply in the limiting case for a very tiny screen The inverse
Pythagorean theorem Alright buckle up now because here's where things get good We've got
this inverse Pythagorean theorem, right?
And that's going to let us transform a single lighthouse into two others without
changing the brightness experienced by the observer With that in hand and no small
amount of cleverness we can use this to build up the infinite array that we need
Picture yourself at the edge of a circular lake directly opposite a lighthouse We're
going to want it to be the case that the distance between you and the lighthouse Along
the border of the lake is one so we'll say the lake has a circumference of two now
the apparent brightness is one divided by the diameter squared and In this case the
diameter is that circumference 2 divided by pi so the apparent brightness works out
to be pi squared divided by 4 Now for our first transformation draw a new circle twice
as big so circumference 4 and Draw a tangent line to the top of the small circle then
replace the original lighthouse with two new ones where this tangent line intersects
the larger circle an Important fact from geometry that we'll be using over and over
here Is that if you take the diameter of a circle and form a triangle with any point
on the circle?
The angle at that new point will always be 90 degrees the significance of that in our
diagram here is that it means the inverse Pythagorean theorem applies and the brightness
from those two new lighthouses equals the brightness from the first one namely pi squared
divided by 4 as The next step draw a new circle twice as big as the last with a
circumference 8 Now for each lighthouse take a line from that lighthouse through the
top of the smaller circle Which is the center of the larger circle and consider the two
points where that intersects with the larger circle Again,
since this line is a diameter of that large circle Then the lines from those two new
points to the observer are going to form a right angle Likewise by looking at this right
triangle here whose hypotenuse is the diameter of the smaller circle You can see that
the line from the observer to that original lighthouse is at a right angle With a new
long line that we drew Good news, right?
because that means we can apply the inverse Pythagorean theorem and that means
that the apparent brightness from the original lighthouse is the same as the
combined brightness from the two newer ones and Of course,
you can do that same thing over on the other side drawing a line through the
top of the smaller circle and getting two new lighthouses on the larger circle
and Even nicer these four lighthouses are all going to be evenly spaced around
the lake Why?
Well, the lines from those lighthouses to the center are at 90 degree angles with each
other So since things are symmetric left to right that means that the distances along
the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going,
but I want to walk through this for just one more step You draw a circle twice as big so
circumference of 16 now and for each lighthouse You draw a line from that lighthouse
through the top of the smaller circle Which is the center of the bigger circle and then
create two new lighthouses where that line intersects with the larger circle Just as
before because the long line is a diameter of the big circle those two new lighthouses
make a right angle with the observer, right and Just as before the line from the observer
to the original lighthouse is Perpendicular to the long line and those are the two facts
that justify us in using the inverse Pythagorean theorem But what might not be as clear
is that when you do this for all of the lighthouses to get eight new ones on the Big lake
those eight new lighthouses are going to be evenly spaced This is the final bit of
geometry proofiness before the final thrust To see this remember that if you draw lines
from two adjacent lighthouses on the small lake to the center They make a 90 degree angle
If instead you draw lines to a point anywhere on the circumference of the circle that's
not between them the very useful inscribed angle theorem from geometry tells us that this
will be Exactly half of the angle that they make with the center in this case 45 degrees
But when we position that new point at the top of the lake These are the two lines which
define the position of the new lighthouses on the larger lake What that means then is
that when you draw lines from those eight new lighthouses into the center They divide
the circle evenly into 45 degree angle pieces and that means the eight lighthouses are
evenly spaced around the circumference with the distance of two between each one of them
and Now just imagine this thing playing on at every step doubling the size of each circle
and Transforming each lighthouse into two new ones along a line drawn through the center
of the larger circle at every step the apparent brightness to the observer remains the
same pi squared over 4 and at every step the lighthouse has remained evenly spaced with
a distance 2 between each one of them on the circumference and In the limit what we're
getting here is a flat horizontal line with an infinite number of lighthouses evenly
spaced in both directions and Because the apparent brightness was pi squared over 4 the
entire way that will also be true in this limiting case And This gives us a pretty
awesome infinite series the sum of the inverse squares 1 over n squared Where n covers
all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in
the leftward direction Adding all of those up is going to give us pi squared over 4
That's amazing and it's the core of what I want to show you and Just take a step back
and think about how unreal this seems The sum of simple fractions that at first sight
have nothing to do with geometry nothing to do with circles at all Apparently gives us
this result that's related to pi Except now you can actually see what it has to do with
geometry the number line is kind of like a limit of ever-growing circles and As you sum
across that number line making sure to sum all the way to infinity on either side It's
sort of like you're adding up along the boundary of an infinitely large circle and a very
loose But very fun way of speaking But wait, you might say this is not the sum that you
promised us at the start of the video And well, you're right.
We do have a little bit of thinking left First things first,
let's just restrict the sum to only being the positive odd numbers which gets us pi
squared divided by 8 Now the only difference between this and the sum that we're looking
for that goes over all the positive integers odd and even is That it's missing the sum
of the reciprocals of even numbers what I'm coloring in red up here Now you can think of
that missing series as a scaled copy of the total series that we want Where each
lighthouse moves to being twice as far away from the origin one gets shifted to two two
gets shifted to four three gets shifted to six and so on and Because that involves
doubling the distance for every lighthouse.
it means that the apparent brightness would be decreased by a factor of four and That's
also relatively straightforward algebra going from the sum over all the integers to the
sum over the even integers Involves multiplying by 1 4th and what that means is that
going from all the integers to the odd ones Would be multiplying by 3 4ths since the
evens plus the odds have to give us the whole thing So if we just flip that around that
means going from the sum over the odd numbers to the sum over all positive integers
requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds
badda boom badda bing We've got ourselves a solution to the basil problem Now this video
that you just watched was primarily written and animated by one of the new three blue
one brown team members Ben Hambricht an addition made possible.
Thanks to you guys through patreon You
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