Beautiful Trigonometry - Numberphile

Summary

This script explores the fascinating relationship between straight lines and circular motion through animations. The presenter demonstrates how dots moving in straight lines can create the illusion of circular orbits, sparking varied interpretations from viewers. The core concept is further elucidated by revealing the construction of the animation using a blue dot's coordinates to control the yellow dots' movements. The script delves into the mathematical principles behind sine and cosine waves, illustrating them as projections of circular motion. It creatively explains trigonometric functions as aspects of a circle's motion, rather than just ratios in triangles. The video concludes with a three-dimensional visualization of these principles and a mesmerizing optical illusion of multiple dots creating a rolling circle, emphasizing the beauty and utility of mathematical concepts.

Takeaways

• 🔁 The animation demonstrates how dots moving in straight lines can create the illusion of circular motion.
• 📊 Different viewers perceive the animation in various ways, some seeing orbiting motion while others see straight lines.
• 🟡 The two yellow blobs in the animation are actually moving in straight lines, controlled by a blue dot's vertical and horizontal coordinates.
• 📐 The animation is built on the concept of a trammel of Archimedes, which uses two points moving in straight lines to draw a perfect circle.
• 📈 The vertical and horizontal movements of the yellow blobs trace out sine and cosine waves, respectively.
• 📉 Sine and cosine functions are essentially the y and x coordinates of a point moving in a circle, which is a fundamental concept in trigonometry.
• 🤔 The animation raises the question of what happens when tracking the position of the blue dot over time, revealing a sine curve for the vertical movement.
• 📊 The sine and cosine waves are shown to be aspects of the same motion, with the cosine wave being a shifted version of the sine wave.
• 📐 The tangent function is introduced as the ratio of sine to cosine, which is also the gradient of the radius line to the tangent line at a point on the circle.
• 📈 The reciprocal trigonometric functions (secant, cosecant, and cotangent) are also related to the circle, as they are derived from the primary sine, cosine, and tangent functions.
• 🎨 The script concludes with an optical illusion animation of multiple dots creating the appearance of a rolling circle, despite all dots moving in straight lines.

Q & A

What is the main concept discussed in the animation presented in the script?

The main concept discussed is the visual illusion of circular motion created by dots moving in straight lines, which leads to a deeper exploration of trigonometric functions and their geometric interpretations.

What is the initial observation made about the animation involving the yellow blobs?

The initial observation is that the yellow blobs appear to be rotating around each other, creating an orbit-like or circular motion.

How does the script describe the motion of the dots in the animation?

The script describes the motion of the dots as straight lines, with the illusion of circular motion being a result of the way the dots' paths are constructed.

What mathematical concept is used to explain the motion of the dots in the animation?

The mathematical concept used to explain the motion is the sine and cosine functions, which are related to the y-coordinate and x-coordinate of a point moving in a circle, respectively.

How does the script connect the animation to trigonometry?

The script connects the animation to trigonometry by demonstrating that the sine and cosine waves can be visualized as the vertical and horizontal coordinates of a point moving in a circle.

What is the significance of the blue dot in the animation?

The blue dot in the animation controls the position of the yellow dots. Its vertical and horizontal coordinates directly influence the paths of the yellow dots, creating the illusion of circular motion.

How does the script explain the relationship between sine and cosine functions?

The script explains that sine and cosine functions are essentially the same, with the cosine wave being a sine wave shifted along. They represent the y and x coordinates of a point moving in a circle.

What is the three-dimensional extension of the animation discussed in the script?

The three-dimensional extension involves projecting the point's motion along an axis, resulting in a spiral. This spiral can be viewed from different angles to observe sine and cosine waves.

What is the tangent function in the context of the animation?

In the context of the animation, the tangent function is represented as the gradient of the radius line where it intersects a tangent line drawn to the circle, which is defined as sine divided by cosine.

How does the script use the animation to illustrate the concept of 'circular functions'?

The script uses the animation to show that all trigonometric functions, including sine, cosine, and tangent, are fundamentally related to the circle and its properties, hence the term 'circular functions'.

What optical illusion is created in the final part of the script involving multiple dots?

The optical illusion created involves multiple dots moving in straight lines that, when viewed together, appear to form a rolling circle due to their alignment and sinusoidal motion.

What is the purpose of the Brilliant.org reference in the script?

The Brilliant.org reference is a promotional mention, offering a discount to their premium subscription. It is used as a platform for further exploration of mathematical concepts like the ones discussed in the script.

Outlines

00:00

🔵 Exploring Motion Perceptions

The narrator discusses how different people perceive the motion of two yellow dots. Some see a circular motion, while others see straight lines. An animation built with Visual Basic and later in GeoGebra illustrates this dual perception. The narrator explains that one dot moves vertically and the other horizontally, both influenced by a blue dot moving in a circle. This creates the illusion of circular motion, despite each dot moving in a straight line. The explanation touches on trigonometry and how the sine and cosine functions describe this motion.

05:03

🔴 Three-Dimensional Motion

The narrator extends the two-dimensional motion into a three-dimensional view using GeoGebra. By projecting the circle into a third dimension, a spiral motion is revealed, demonstrating how sine and cosine waves are projections of this circular motion. The concept of tangent is introduced, explained as a line that just touches a curve, leading to a deeper understanding of how trigonometric functions are related to circular motion. The narrator emphasizes the geometric nature of trigonometric functions beyond their use in triangles.

10:03

🟢 Mathematics and Its Beauty

The narrator highlights the intrinsic beauty and usefulness of mathematics, emphasizing how mathematicians are often driven by the elegance of math before considering its practical applications. The segment transitions to a sponsorship message, promoting Brilliant.org, which offers daily challenges and deep-dive courses to stimulate the mind. The narrator encourages viewers to take on these challenges and explore more through Brilliant, offering a discount code for their subscription service.

Keywords

💡Animation

Animation refers to the process of creating a visual representation of movement by displaying a series of images in rapid succession. In the context of the video, the animation showcases two yellow blobs moving in such a way that they appear to orbit each other. This is a key element in illustrating the concept of circular motion being created by straight-line movements.

💡Orbiting

Orbiting is a term used to describe the motion of one object moving around another in a circular path. The video script mentions that some viewers interpret the animation as one object orbiting the other, highlighting how different people can perceive the same visual information differently.

💡Straight Lines

Straight lines in this context refer to the actual paths that the dots in the animation are following. The script emphasizes that despite the appearance of circular motion, each dot is moving in a straight line, which is a fundamental concept in the video's exploration of visual perception and geometrical principles.

💡Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right-angled triangles. The script discusses how sine and cosine, key functions in trigonometry, are derived from the y and x coordinates of a point moving in a circle, which is a central theme in the video.

💡Sine Curve

A sine curve, or sinusoidal wave, is a smooth, periodic oscillation that resembles a wave. In the video, the speaker describes how tracking the vertical position of a point over time results in a sine curve, illustrating the connection between circular motion and trigonometric functions.

💡Cosine Wave

The cosine wave is similar to the sine wave but is phase-shifted by a quarter cycle. The script explains that the horizontal movement of the point in the animation traces out a cosine wave, further demonstrating the relationship between circular motion and trigonometric functions.

💡Trammel of Archimedes

The Trammel of Archimedes is a historical device used to draw perfect circles using two points moving in straight lines. The script mentions this as an example of how straight-line motion can be used to create circular forms, tying into the video's theme of geometrical illusions.

💡Projection

In the context of the video, projection refers to the method of mapping points from one coordinate system to another, such as projecting a point's position onto a vertical or horizontal axis. The script uses the concept of projection to explain how the appearance of circular motion can be created by the projection of straight-line movements.

💡Tangent

Tangent, in the script, refers to a trigonometric function as well as a geometric concept. The tangent function is defined as the ratio of the sine to the cosine of an angle. Geometrically, it represents a line that touches a curve at a single point without crossing it. The video explains how the tangent function relates to the motion of a point on a circle.

💡Three-Dimensional

The script introduces a three-dimensional perspective to explain how the two-dimensional circular motion can be extended into a 3D space, creating a spiral. This helps to visually demonstrate how trigonometric functions can be represented in multiple dimensions, enhancing the viewer's understanding of the concepts.

💡Optical Illusion

Optical illusions are visual phenomena where the brain perceives an image differently from the objective reality. The video concludes with a demonstration of an optical illusion created by multiple dots moving in straight lines to form the appearance of a rolling circle. This ties back to the video's theme of exploring how our perception of motion and shape can be influenced by simple geometric principles.

Highlights

The animation shows two yellow blobs moving in a way that can be perceived as either straight lines or circular orbits.

Different viewers interpret the motion as either orbiting or straight lines, demonstrating the subjective nature of visual perception.

The animation was created using a blue dot that controls the movement of the yellow blobs through vertical and horizontal projections.

The circular motion illusion is driven by the circular path of the controlling blue dot.

The animation reveals that the perceived circular motion is actually a result of straight lines moving in a specific way.

The concept of a trammel of Archimedes is introduced, which uses two points moving in straight lines to draw a perfect circle.

The animation demonstrates the mathematical relationship between the perceived circular motion and the actual straight-line motion.

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

Trigonometry is shown to be deeply connected to circular motion, rather than just right-angled triangles.

The animation provides a visual representation of how sine and cosine waves are derived from circular motion.

A three-dimensional version of the animation is presented, showing a spiral that represents sine and cosine waves.

The tangent function is introduced as the ratio of sine to cosine, derived from the geometry of a circle and a tangent line.

The reciprocal trigonometric functions secant, cosecant, and cotangent are shown in relation to the circle and tangent line.

The animation concludes with an optical illusion of multiple dots creating the appearance of a rolling circle, despite each moving in a straight line.

The optical illusion demonstrates the power of perception and the beauty of mathematical concepts.

The transcript ends with a challenge from Brilliant, encouraging viewers to engage with mathematical puzzles and problems.

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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