Beautiful Trigonometry - Numberphile

Summary

El guión describe una animación en la que dos manchas amarillas parecen rotar alrededor una de la otra, pero en realidad, cada punto se mueve en línea recta. Esta percepción visual se debe a la interacción de movimientos lineales y circulares, lo que lleva a una comprensión más profunda de las funciones trigonométricas como el seno y la coseno. El video también explora cómo se pueden crear ilusiones ópticas y efectos visuais usando líneas y puntos en movimiento, subrayando la belleza y utilidad de las matemáticas en la creación de conceptos visuales interesantes.

Takeaways

• 🔵 Los puntos amarillos en la animación se mueven en líneas rectas, no en círculos.
• 👁️ Las personas ven diferentes interpretaciones: algunos ven un movimiento orbital, mientras otros ven líneas rectas.
• 📊 La animación se puede recrear utilizando programación en Visual Basic o GeoGebra.
• 🔄 El movimiento circular percebido es el resultado de la proyección de movimientos lineales en un círculo.
• 📈 Los valores senos y cosenos son las coordenadas y y x de un punto en movimiento circular.
• 🌀 La función seno se puede visualizar como una onda senoidal, mientras que la función coseno se asemeja a una onda senoidal invertida.
• 🔺 La tangente es la relación entre la función seno y la función coseno, y se define como la pendiente de una línea tangente a una curva.
• 📊 Las funciones trigonométricas sec, cosec y cotangente son las recíprocas de las funciones seno, coseno y tangente, respectivamente.
• 🌐 En una representación tridimensional, los movimientos lineales proyectados en diferentes direcciones crean un efecto de espiral.
• 🎥 La comprensión de estos conceptos matemáticos puede mejorar a través de visualizaciones y animaciones.
• 🤓 La apreciación por las matemáticas puede surgir tanto por su belleza como por su utilidad práctica.

Q & A

¿Qué sucede con los puntos amarillos en la animación que se menciona en el guión?

Los dos puntos amarillos en la animación dan la impresión de rotar alrededor del uno al otro, aunque en realidad se mueven en líneas rectas.

¿Por qué algunas personas ven una trayectoria circular en la animación?

Algunas personas ven una trayectoria circular porque las coordenadas de los puntos amarillos son proyecciones de un punto azul que se mueve en círculo, lo que crea una ilusión de movimiento circular a pesar de que cada punto se mueve en línea recta.

¿Cómo se relacionan las funciones seno y coseno con el movimiento circular?

Las funciones seno y coseno son la coordenada y en el eje vertical y horizontal, respectivamente, de un punto que se mueve en un círculo. Son fundamentales en trigonometría porque describen el movimiento cíclico y repetitivo en una dimensión.

¿Qué es la función tangente y cómo se relaciona con el círculo?

La función tangente es la relación entre la longitudinal del radio y la longitud tangente en el punto de contacto con el círculo. Se define como la pendiente de la tangente en ese punto y está relacionada con el círculo porque es una línea que toca el círculo en un solo punto.

¿Por qué se llama trigonometría a la rama de las matemáticas que estudia las relaciones en triángulos rectángulos?

Se llama trigonometría porque originalmente se utilizaba para medir y estudiar triángulos rectángulos, aunque su aplicación más amplia es la descripción de movimientos circulares y cíclicos a través de funciones trigonométricas.

¿Qué son las funciones circulares y cómo se relacionan con las trigonométricas?

Las funciones circulares son funciones matemáticas que describen movimientos cíclicos o repetitivos, como el seno y el coseno. Se llaman circulares porque su forma y comportamiento están relacionados con el movimiento de un punto en un círculo.

¿Cómo se pueden visualizar las funciones trigonométricas en una representación tridimensional?

En una representación tridimensional, las funciones trigonométricas se pueden visualizar como un punto que se mueve en un círculo en el plano XY, proyectando su posición en un eje Z que sale en perpendicularidad. Esto crea una espiral tridimensional que refleja el movimiento circular en dos dimensiones.

¿Qué es un trammel de Arquímedes y cómo se relaciona con las funciones trigonométricas?

Un trammel de Arquímedes es un dispositivo mecánico utilizado para trazar círculos en el plano. Se relaciona con las funciones trigonométricas porque permite dibujar un círculo a partir de dos puntos que se mueven en líneas rectas, demostrando cómo el movimiento lineal puede resultar en una forma circular.

¿Cómo se pueden crear efectos visuales interesantes con el movimiento de puntos en líneas rectas?

Pueden crearse efectos visuales interesantes al alinear y proyectar el movimiento de puntos en líneas rectas de manera que creen ilusiones ópticas de movimiento circular o de formas como círculos o espirales. Esto se logra mediante la programación y la geometría, utilizando funciones trigonométricas para controlar el movimiento de los puntos.

¿Qué es un simple harmonic motion y cómo se relaciona con el movimiento de los puntos en la animación?

El simple harmonic motion es un tipo de movimiento oscilatorio que se repite de manera regular, como el de un péndulo. En la animación, los puntos se mueven en un simple harmonic motion de tal manera que se evadan mutuamente y creen un efecto visual de puntos que se mueven en círculos, aunque en realidad se mueven en líneas rectas.

¿Qué es la ilusión de un círculo en movimiento creada en la animación?

La ilusión de un círculo en movimiento es un efecto visual creado en la animación donde los puntos se mueven en líneas rectas de una manera que, al alinearse correctamente, producen la impresión de un círculo que se mueve internamente. Esto es un ejemplo de cómo el movimiento lineal puede resultar en percepciones visuales complejas.

¿Cómo se pueden utilizar las funciones trigonométricas para resolver problemas prácticos?

Las funciones trigonométricas pueden utilizarse para resolver problemas prácticos que involucran movimientos cíclicos o variaciones periódicas, como en la ingeniería, la física, la música o la animación. Permiten modelar y predecir comportamientos que siguen patrones similares al movimiento de un punto en un círculo.

Outlines

00:00

🔄 Movimientos en línea recta y percepciones visuales

Este párrafo discute un experimento visual que muestra dos manchas amarillas moviendose en lo que parece ser un movimiento circular, pero en realidad son líneas rectas. La curiosidad de los espectadores sobre el movimiento real de los puntos lleva al creador a explicar la percepción visual y cómo las líneas rectas pueden resultar en una ilusión de movimiento circular. Se menciona la importancia de la programación y la geometría en la construcción de este experimento, y cómo los espectadores pueden ver diferentes interpretaciones basadas en sus perspectivas individuales.

05:03

📐 La relación entre el movimiento circular y las funciones trigonométricas

Este párrafo profundiza en la relación entre el movimiento circular y las funciones trigonométricas, como el seno y el coseno. El creador explica cómo estos conceptos matemáticos pueden ser visualizados a través del movimiento de los puntos en el experimento. A través de la construcción de un círculo en 3D y la visualización de las funciones seno y coseno, se demuestra cómo estos son aspectos del mismo movimiento, pero desde diferentes perspectivas. Además, se introduce el concepto de la función tangente y su conexión con el movimiento circular y las coordenadas x e y.

10:03

🌐 Aplicaciones y extensiones de los conceptos trigonométricos

Este párrafo explora las aplicaciones y extensiones de los conceptos trigonométricos, incluyendo las funciones tangente y las funciones circulares adicionales como sec, cosec y cot. El creador muestra cómo todas estas funciones están relacionadas con el movimiento circular y cómo pueden ser visualizadas en un diagrama. Además, se menciona la utilidad de estos conceptos matemáticos más allá de la ilusión visual, como en la resolución de problemas prácticos. Finalmente, se hace un llamado a la acción para que los espectadores prueben los desafíos y cursos en Brilliant.org, un sitio que ofrece recursos para profundizar en matemáticas y lógica.

Keywords

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Highlights

The animation demonstrates how two yellow blobs moving in straight lines can create the illusion of circular motion.

People perceive the motion differently, with some seeing orbiting and others seeing straight lines.

The creator built this animation to understand the nature of the motion, which led to learning programming in Visual Basic and later in GeoGebra.

The blue dot controls the yellow blobs, with its vertical and horizontal coordinates dictating their paths.

The motion is driven by a circular path, which is why the straight lines create a circular illusion.

The animation is an example of a trammel of Archimedes, a mechanical construction for drawing circles using straight lines.

Sine and cosine waves are introduced as the y and x coordinates of a point moving in a circle, respectively.

Trigonometry, despite its name, is more about circles than triangles, as sine and cosine are circular functions.

The tangent function is the ratio of the y-coordinate to the x-coordinate of a point on the circle, and it is the gradient of the radius.

The other trigonometric functions (sec, cosec, and cot) are the reciprocals of sine, cosine, and tangent, respectively.

All trigonometric functions are related to circles and are called circular functions, not just about right-angled triangles.

The optical illusion of a circle rolling inside the animation is created by dots moving in simple harmonic motion in a specific alignment.

The creator's fascination with the animation stems from the beauty of mathematics and its ability to produce captivating illusions.

The animation serves as an example of how mathematical concepts can be both aesthetically pleasing and practically useful.

The discussion on the animation leads to a deeper understanding of trigonometry and its geometrical basis.

The transcript highlights the educational value of engaging with mathematical concepts through visual and interactive means.

The animation showcases the power of mathematical relationships in creating complex patterns and illusions from simple components.

Transcripts

Nothing is moving in a circle.

Each one of these dots is moving in a straight line

- Nice

Talk about burying the lead. That's the best bit. Is that the best bit?

I want to show you an animation that I like and I want to give too much away

I would like to know your opinion of this animation, so

There are two yellow blobs, I'll give you that for free I'm curious what you see happening.

It's kind of like they're rotating around each other. - Go on say more.

So if I talk about this dot, yeah. - It's like it's drawing a circle

You know, it's it's looping around the other one or they are the ones looping around you. - Yeah, it goes either way, doesn't it?

What I like about this animations, I've shown lots of people is animation, but they see different things.

So some people see this orbiting thing when one is orbit in the other, but the other one keeps moving.

Is that a fair description of what you saw? - Yeah.

Other people see straight lines. - Oh, I definitely see straight lines.

Okay

So there's definitely a straight liney motion

and is obviously there and there but at the same time other people have insisted that there's some sort of circular orbit-y

flavor which is a first of all I like because

Straight lines and circles feel opposites in some sense and though there's a mathematical sense where there might be the same too

But both of them are seen by people in a naive description of what's happening

I ended up building an animation on this when I was at school

because I was curious about what this motion actually wasn't that's how

I learned to program in Visual Basic

I have re-built this one in geogebra

I want to show you how I built it and also why I built it. You were right to talk about

Orbiting things and also right about the straight lines

If I put these lines on it's really obvious that one of them is going up and down

one is going left to right...but they always dodge each other and I didn't make them move like this by putting their straight lines on.

What I did was I made the circle around the outside exist.

It's still not obvious to me how they works, but I made this point also exist

So this blue point moving around is how I built this file. That blue dot controls the yellow.

In fact the one going up and down is precisely just the vertical coordinate of the blue dot

It's just tracking wherever the blue dot is in a vertical axis

It's like the projection of it onto a vertical thing and the horizontal one is just the horizontal projection

The reason why we see sort of circular motion going on is because it's driven by a circular motion.

It's possible that you saw something else which was the connector between these dots is a natural thing to sort of see because it's actually

because it is always the same length is equidistant and that's a fluke and some people in their head are kind of

Averaging the two dots and they're seeing the midpoint of the two dots.

Literally the average position that you dress and that is moving in a circle, which is really nice

There's a bunch of mechanical constructions called a trammel of Archimedes where you can draw a circle

From two things moving in straight lines and it's a mechanical way of drawing a perfect circle

You just constrain these two dots to move in straight lines

I didn't say that. - A lot of people don't see it but they they feel there's something intuitively circular going on

So let me go back to the reason why I built the thing. The yellow points are the coordinates of the blue point

Let's just focus on the vertical one

What do you think would happen if I track the vertical position of that over time? Essentially draw a graph.

What will that graph be? Next time we start on the right side

I'm gonna track the y coordinate and you'll just see a graph which maybe is familiar.

Any...any thoughts, Brady?

It's like a sine curve or something. - It is precisely a sine curve. In fact, it's not just any sine curve.

It is...is THE sine curve. So I was a teacher for a long time

I was teaching people trigonometry. Sine and cosine turn up and

Every single time I ever had a class learning that they will... "Sir, what is sine? What is it..." and

There are lots of answers. People think of it as a ratio, opposite and hypotenuse.

That might be familiar words. Other people think of as a function. It's got an input and an output.

These things are true

But what I like about a third answer, I'm gonna give you now, is that it captures

Both of them. Sine is the y-coordinate of a point moving in a circle. Interestingly nothing to do with triangles

Despite the name trigonometry coming from Trigon metry: measuring triangles

I think was the worst named topic in mathematics. Sine is a circular function. In fact

It's one of the circular functions because there's another coordinate we haven't tracked yet.

So let's just do that one the horizontal

One moving left and right there. If I track that going upwards you're gonna see the beginning of it

It's not gonna be a huge surprise when you realize the trace is out the beginning of a cosine wave

And if you flip that back down you see the cosine wave which is precisely the same as a sine wave that's shifted along.

It's just like the circles going around a bit

So sine and cosine as functions or as mathematical objects are

precisely the y coordinate and x coordinate of a point moving in a circle

That's why they're important because almost everything that goes in a cycle or repeats ever ie most things are described by circles and therefore described

by trigonometry. - Is a cosine wave very different to a sine wave then?

It's exactly the same. Literally sine wave

Is there in a cosine wave is the same way shifted is the sine of the other angle this there's no obvious angles

But triangles have angles and we use trigonometry with angle

so the angle comes from, if I draw the radius on, from the center of this thing

the angle gives me a way of measuring where the blue

Dot is at any point. So angle of zero there.

It's the angle the radius makes and that's where you do sine of an angle and it gives you coordinate

and so the

Cosine is actually sine of the other angle, is the angle between the vertical and this radius instead of the horizontal and that's why sine and cosine

Are basically the same. I love the fact that seeing this move makes me understand trigonometry

There's not a triangle in sight until you put that radius back on and then you can see there's a right angle triangle

That's kind of sweeping around inside here. And that's why trigonometry is to do with right angle triangles

It's because x and y coordinates are right angles and the radius makes the hypotenuse

But that means sine and cosine are like two aspects of the same motion. There's just a different perspective

And when someone pointed this out. Could you draw it in three dimension? That sounds like and

I got this on geogebra at the inspiration of a student I was teaching at the time.

So let me show you a three-dimensional version of this

So there's my circle I had originally that's the view we had before. That's the same motion

I just had. five here from a three dimensional mode. I've got this axis going off into the distance

We're gonna track the position of a and time is now gonna go sort of back in into that direction

And we're gonna project the point along that axis. I'm gonna start it now and you see that point going off into space

It's kind of hard to see without

moving your head around in a

three-dimensional world so I can put the path on and you can see, maybe not surprisingly, you get sort of spiral coming out

and I reckon

Most people will predict that but that means the sine and cosine are precisely somehow in that spiral because they're just aspects of that spiral

If you look from the side

There's the sine wave. Which means if you want to see the cosine wave you should probably look from the top or the bottom

There's the cosine wave. You can see the same way if it starts in a different place and if you go back to the front again

They're all

just a projection of a circle and so sine and cosine are useful because they are the sort of

Compressions of circular motion in within one dimension and I thought this 3d

Diagonal that really helped me understand where this comes from. It does beg the question

There's a third trig function that most people know about and I haven't shown you that yet

Tan! - Tan. Do you know what tan is short for, Brady?

Isn't it tangent? - Yeah, what's a tangent?

I mean, I'm in full teacher mode now, but I think you know what tangent is

Yeah, it's a line like kind of like grazing a circle. - Yeah, so tangent it from Latin for touching

So tangent is something just touches the curve doesn't have to be a circle.

The word comes from precisely that set up

This is a slightly more advanced file

I can show you all of the things first will reminder a sine function and there's the graph we're building and cosine is the other

Direction and you can see the graphs arriving tracing their way out. Tangent comes from drawing a tangent

but if I drew a tangent

Vertically to the place where we start all this motion and I track where the radius line would hit the tangent

You'll get a situation that looks like this. There's the tangent line and the radius line

Will always cut the tangent line somewhere and then this green length is precisely the tangent function.

It is also the gradient of the radius because tangent is defined to be sine of a cosine

It was kind of a huge relief to me personally. The word tangent wasn't a coincidence with the other definition of tangent

Which is a line that just touches the circle these three graphs are all related to a circle. They're all really geometrical things

they're not really to do with triangles except by accident because of the coordinate axes and

There are three more functions you learn at A level at school which are one over these functions. So

sec, cosec and cot

are one over these three and they're also on this diagram. If I turn on cosec, cosec otherwise known as one over sine

Just like sine, which is a vertical lengthwise diagram, cosec is where the tangent

cuts the vertical axis and you can see a sine gets bigger cosec gets smaller and then the same at one point and then it

Goes back the other way

and so this

U-shaped graph here is the cosec function otherwise known as one over sine and the same thing happens with sec is where the tangent intersects

the x axis and cot, one over tan, ends up being

The other tangent across the top here is where the radius intersects that all of the trig functions that we learn at school

Are actually altered with circles which is why they are called the circular functions

And I kind of wish we call trigonometry the circular functions from the word go

Even if the use we most make of it is for finding missing sides in a right-angled triangle

There's one more thing that I wanted to you

I had this diagram happening before where these two dots are moving, but you now know how I did it

I put those two lines on there, kind of the projection of that outer point, the projection vertically and horizontally

Now once I realize I can do that. I don't have to restrict it to two lines. I could put three lines on

You get a really nice effect of these three points moving

And they always dodge each other. They're moving in what they might call simple harmonic motion, but then they're nicely sort of lined up

So they don't ever hit each other. Well, I just really like this. So I'm going to crank up the number of lines

Here we go

Now what I love about this is that everybody can see the yellow circle and...

Mind blown! - The dots are in a circle definitely but nothing, nothing is moving in a circle each one of these dots is moving in a

Straight line. So if I grab one with my finger this one

Is just moving backwards forwards in a sinusoidal motion as for they call this with this wave-like motion

But all of them because they're lined up in a certain way create

There's a lovely illusion of a circle rolling around the inside of thing. To program this you just needed some knowledge of how to make

A sine wave just shift around each time and it's just a really lovely optical illusion I like. - Nice!

Talk about burying the lede. That's the best bit. - Is that the best bit? - Yeah.

That's the bit that's got no use

But then I'm a sucker for things that've got no use. The thing is that I love Maths 'cause it's beautiful

It turns out it's also useful but mathematicians usually do stuff because it's nice and then they're like, "Oh yeah, it's also useful

But I would have done it anyway."

All right

You made me wait for it

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