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How Radians Work in 30 Seconds (googleusercontent.com)
474 points by _pius on Jan 19, 2014 | hide | past | favorite | 100 comments

This is by LucasVB, who is prolific at animating math for Wikipedia: https://en.wikipedia.org/wiki/User:LucasVB/Gallery

His other works are also worth checking: http://1ucasvb.tumblr.com/ (for example, he has a super-nice explanation of Fourier series).

The first time I was able to properly visualize sine and cosine was from an old 8mm film reel produced in the 1950s by IBM which showed essentially this same animation:


After seeing that, trig really just "clicked".

I was introduced to sine in elementary school. Then we had 6 months of trigonometry in high school. We even touched infinite series at the end of high school. Then at univ we had some trigonometry again but not much, some infinite series and stuff.

Nowhere along the way I was shown that sine is actually how high above the ground you are when traveling around the circle. It was always some length of side in right triangle divided by length of another side. That was technically the same thing but nowhere close as intuitive and natural. Even when at some point I got interested in math and start thinking about sine as y coordinate it was always in relation to angle of a triangle (rooted in the center of the circle, one point at radius, the other at x axis below it) and not in relation of actual distance traveled along the circle. At some point radians appeared and we were just taught that's other way to measure angles, so you know: 0 degrees is 0, 90 degrees is 1/2 of PI and 360 degrees is 2 PI. I thought it was kinda weird to this that way but w/e probably people have their reasons...

The concept is so beautiful and natural yet it was made as artificial as possible during math education I had. No wonder so many kids don't get or like math :(

My pet theory is that people usually are shown all these things, they just happen to lack a piece or a few of the underlying concepts, so it floats on by.

(a decent set of textbooks should have lots of these didactic gizmos in it, but that doesn't solve the problem of getting the student ready to look at it or making sure they look at it after they are ready)

> sine is actually how high above the ground you are when traveling around the circle.

That makes total sense, and I wish it would have been explained like that to me, too.

My math teacher in HS spent an hour and a half explaining how derivatives work by just working through stuff on the board after he insisted no one take notes, and it made everything so clear. And reading through the Tau Manifesto finally made me understand what was going on in trig at a basic level.

I totally agree with you about how artificial math education can feel, and I think if larger concepts were introduced more like that many more people would be able to get much better at it.

I read something a couple years ago about how some school was experimenting in a math program starting in 8th grade where they would teach algebra, geometry, and calculus concepts all at the same time. I guess the students were really picking things up fast, and understanding how everything fit together better.

>> sine is actually how high above the ground you are when traveling around the circle.

I learned my trigonometry when trying to figure out how to rotate and move objects around in 2D space for a computer game. I learned then, that sin is responsible for y axis, and cos for x axis.

But I never ever phrased this as "how high above the ground you are when traveling around the circle". I like the elegance of that; things would probably have clicked for me immediately back then, had I heard a sentence like this (and ditto for cosine being "how far to the right are you").

Bottomline, I'm in agreement with both OP and GP here.

> That makes total sense, and I wish it would have been explained like that to me, too.

Did you (and everyone else) really never learn the (paradigmatic) parametric equation describing circular motion, (x, y) = (cos t, sin t)?

the guy is great with his animation, may be we can pitch and keep his work going.. http://1ucasvb.tumblr.com/post/57176345908/im-now-open-for-d...

He answered a few questions in a reddit thread: http://www.reddit.com/r/oddlysatisfying/comments/1vcm7j/this...

Thanks, saw the image linked on Twitter without this context.

I first heard about LucasVB from Empirical Zeal, amazing and intuitive animations.

Here is the article if you want to take a look: http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-h...

The illustration of line integrals on a vector field took me four goes to figure out but is brilliantly clear.


It's not an animation, but when I saw the visual illustration of eigenvalues/eigenvectors on this page I was blown away:


I studied linear algebra in college and computed plenty of eigenvalues/eigenvectors, but until I saw that graphic I had no idea what they actually were. I can't believe in retrospect that a textbook would explain them without an illustration like this.

I was also blown away the first time I saw that animation, it really helped to clarify the notion of what eigenvalues and eigenvectors are. However, that is only a geometrical interpretation which is a very particular case, as eigenvalues/vectors appear in many many other applications which don't really have a geometrical representation or meaning. That's probably why books tend to stay away sometimes from particular examples, even though I have to say that this one specifically is very useful.

Isn't the geometric formulation equivalent to the algebraic one? What makes the geometric representation any less general? Of course in higher dimensions you can't formulate it in terms of a drawing, but for 2 it seems to capture the entirety of the concept.

Degrees of freedom don't necessarily have to have a geometric meaning in some (many) systems! For example, in basic analysis of mechanical vibrations you can use modal analysis to decompose a coupled system (two harmonic oscillators with viscous dampers in series for example) into two independent systems. The eigenvalues here correspond to the natural frequencies and the eigenvectors are the modal shapes, which give you an idea of how the oscillators actually end up moving after some disturbance. There is some geometrical significance here but is not quite as easy to interpret.

But how do you teach anything without examples? Pedagogically, this stuff is invaluable.

I made a little interactive web app for visualizing 2-d linear transformations some time ago. My original intention was to see what a 2-d DFT did. Turns out its pretty nifty for seeing the action of eigenvectors, singular matrices, rotations, reflections... anyway, feel free to check it out:


Not to be dismissive or condescending, but do a lot of people have trouble understanding radians? I was under the impression that they were standard material in high school trig/physics.

I got excellent grades in high-school math. I took a year of math in college. I knew how to calculate with radians, but never understood their basis until viewing this animation, despite asking people on a number of occasions what the benefits were vs. degrees. (They never gave me a real answer to that question, and certainly never derived the basis for radians.)

I now feel enlightened; too bad it has been more than 20 years since I last needed to calculate with them! However, it'll probably come in handy when I help my children with their math homework in the coming years...

So here's a question -- is it just the animation that helped, or is it the combination of the animation and being 20 years older? Or, more specifically, having the luxury of being able to throw brain power at the subject without having to worry about learning material for several other subjects, and many sub-topics within all those subjects, all at the same time.

I'm sure that it all helped. But I definitely remember asking, on several occasions, why we would ever want to use radians rather than degrees, and why they seemed to be in such weird increments. And no one ever really told me.

The animation was helpful in that it explained everything very quickly and obviously, obviating the need for a numeric or verbal explanation.

I believe when we were first introduced to the radian, it was explicitly defined in terms of the radius, i.e. 1 radian = an angle whose arc length is equal to the radius of the circle.

Was a radian just defined as an angle of 57.3 degrees in your class? I imagine that would seem really arbitrary to students.

> I knew how to calculate with radians, but never understood their basis until viewing this animation, despite asking people on a number of occasions what the benefits were vs. degrees. (They never gave me a real answer to that question, and certainly never derived the basis for radians.)

Well, the way I think of the basis for radians goes like so:

1. All circles are similar to all other circles (assumed without proof), so there's a relationship between the radius of a circle and its circumference.

2. Angles, being scale-free, ignore the size of a particular circle; if an angle captures 1/3 of a small circle, it will capture the same 1/3 of a larger one, or a smaller one.

3. We can use the combination of an angle ("How much of the circle are we talking about?") and a radius ("How big is this circle?") to specify an arclength along the edge. We do it by multiplying the radius by the angle, and then by a conversion factor.

4. The radian measure is chosen so that the conversion factor mentioned in step (3) is equal to 1, which is easy to multiply by.

Here's the problem: out of a normal-sized group of randomly selected math students, expect between 0 and 1 of them (generally 0) to be interested in hearing you say this.

There's another argument for using radians (related, but not obviously so, to the first), which goes like this:

1. When measuring in radians, the trig functions behave well when taking the derivative: d/dx (sin x) = cos x and so forth.

2. If we used degrees (e.g., call dsin x the sine function for degrees instead of radians), we'd get all these awful constants mucking up our computations. d/dx (dsin x) = (180/π) dcos x, and d/dx (dtan x) = (180/π) dsec^2 x. The fourth derivative of sin x is sin x, but the fourth derivative of dsin x is (1049760000/π^4) dsin x. So we use radians to measure angles for the same reason we use e to measure logarithms -- they are more correct.

That one has the double disadvantage of being something no one wants to hear and drawing on applications they haven't yet heard of.

There's a shorter, less accurate version of the first approach I presented, that goes like so:

1. A circle's radius and circumference are both linear measures; they both have units of length to the first power.

2. Therefore, the conversion between them is a dimensionless (= unitless) number, a constant.

3. Radians are the unique correct constant for that "conversion" (there can't be more than one, any more than there can be more than one constant that converts meters to kilometers).

This is short and easy to say, but is sloppier and unsatisfying to precisely the students who care.


I'm another story of someone who's good at maths, got excellent grades in University, etc. etc.

During High School I didn't study as hard, and while I knew how to do calculations with Radians, never "understood" them.

One day, while in University, someone explained Radians to me much as this gif does. It took one sentence, and something I'd been working with for years without understanding immediately became clear.

Why don't schools/textbooks offer this? I have no idea. But it's a shame.

Have you checked whether they do? In my experience, textbooks often contain more information than you remember they do. Also, for the radians thing, how else would you introduce them? As a magic scaling factor, in the same vein as explaining the difference between degrees and grads?

I got through calc in high school and college and I really wish somebody had told me that radians were based on the radius like this. I had no idea.

Maybe somebody told me and I didn't take it in.

Shoddy math teaching was par for the course where I grew up.

It wasn't a guarded secret... "Radian" is almost the same word as "radius", and 2pi * radius = circumference = 2pi radians.

But unless you take a moment to think about it, that connection isn't so obvious. And in HS, unless math is your interest (and even if it is) there are hundreds of other things running through your brain that are probably more interesting than pondering the relation between the words "radian" and "radius". There are also many other words that people use frequently that they don't realize are connected until much later. And as others have said, HS math (in the US) often presents them as: radians = degrees * pi/180. The calculation has zero relation to the radius of the circle, as presented.

Yeah, you could easily get though by knowing the method of calculation without ever having understood a model of what they actually are.

High school math is often taught as simple facts, without proper explanation of the significance or how it's derived. It's entirely possible to learn how radians correspond to degrees and what you can do with them without understanding where they come from or what they mean.

I learned all the rules of logarithms in 10th grade, but didn't actually know what a logarithm was until chemistry in college. They just.. didn't really tell us. And how often are integrals explained as "the area under a curve", which is a definition that makes absolutely no sense?

Can you elaborate on your understanding of logarithms and integrals? Curious to know if it tallies with mine!

Do you want to know a disturbing secret? Most people don't understand the fundamentals of most of the subjects they have studied. People get by because the perhaps even more disturbing fact is that cargo cultism often actually works in practice. A lot of the time you can get away with just going through the motions.

And I believe it's a problem of the education system. Back in high school they had me solve dozens, if not hundreds, of derivates and integrals, it was not until college that a fantasic teacher had the guts to come up with his own syllabus, threw away all the "traditional" calculus and taught us what was a diffeeential, what were we doing when getting a derivative and an integral, completely changed my mind.

"And I believe it's a problem of the education system."

That's just a cop out. We all had teachers and subjects that didn't connect, for whatever reason. Ultimately, the responsibility to learn is personal.

I don't mean to be harsh, but basically every comment on this subthread is someone saying "I was a great math student, but I never understood what radians MEANT until now!" That's crap. If you do 10,000 practice problems and get high marks and still don't understand the concept, you haven't done anything. Good teacher or bad, your job as a student is always, forever, to figure it out on your own.

The responsibility of a teacher is to instill and assess knowledge. Repetition of busywork does not require a teacher, it only requires a textbook. The purpose of having a teacher in the loop is to ensure that students are actually understanding the material, which is why they are graded. If a student can get through a class without acquiring any degree of competency or basic understanding of fundamentals that's a failure on the teacher's part.

If "teaching" was just a matter of reading and completing exercises then we could replace teachers with scantron machines. Yet college students pay teachers thousands of dollars per course to ensure that they receive more value than that. If they don't get it then they are being mis-served and cheated.

So let me summarize your argument: "Teachers cost money. Therefore, if I do not learn the subject matter, it must be because the teacher was ripping me off."

That's an excellent way to externalize personal failure.

A teacher cannot teach a student who doesn't make an effort to learn. Full stop. But it's a symptom of our ever-increasing sense of entitlement that we assume that our role as students is that of the audience for a play. We intend to lie back, and let the education wash over us. If it takes effort to learn, they must not be teaching correctly!

Let me flip around your supposition: if the people in this thread -- all of whom were apparently high-achieving math students -- cannot manage to understand a core principal of elementary trigonometry that is stated clearly in every textbook, the problem is more likely to be with the unwillingness of the students to think independently than the failures of a diverse group of teachers from all over the world to shove the knowledge into their head.

It's far easier to blame the teacher than to do the work of learning.

> A teacher cannot teach a student who doesn't make an effort to learn.

Quite so. But a poor teacher can do quite a lot to impair a student who does make an effort to learn, and, at least in the public schools of the United States, there is little meaningful incentive for a teacher to be other than poor. Not to say that all US public school teachers are poor, of course; there are at least a few really excellent ones, or were when I was a student. Excellence, though, is anything but a requirement.

I'd also argue that a really good teacher can elicit from her students the sort of effort you describe; I've seen it happen -- indeed, most of my basic facility with the English language, I gained under the tutelage of precisely such a teacher. That's a more complicated proposition, of course, but I start to wonder whether or not it's at the core of the current argument.

It's not hard, or so the US public school system would have us believe, to teach someone enough of a given subject that she can qualify to teach it herself. But is that the essential feature of pedagogy? Or is it instead to lead a class full of students, in such fashion as to overcome the basic distaste, for complex and difficult learning without immediate reward, which is part of the human heritage?

If a teacher fails to impart understanding and they give the student a passing grade then that's a failure of the teacher. If a teacher is a hard ass but has open office hours and students don't take advantage of them and end up failing then that's a failure of the students as well.

The problem here is that it's so common for students to be able to glide through course after course using rote memorization and a willingness to do busywork without attaining understanding and in so doing not only passing courses but also getting high grades. And that's a failure of student and teacher alike, but it's a failure that the students have been trained for and guided to by the system.

Again, the problem is that schools do not reward or generally even bother assessing learning. And certainly part of the fault for that lies on the student, but given that they are being passed through the system and actively rewarded for not learning I think the bulk of the blame lies on teachers and institutions. If the value of an institutional education is to provide an opportunity for auto-didactism and whether one takes that opportunity or not is not reflected in grades or degrees granted then I have to question the value of such education.

"Yet college students pay teachers thousands of dollars per course..."

No. College students pay colleges and universities thousands of dollars per course. There's a difference.

Exactly. In many subjects it's far easier to concentrate on busywork than on understanding. And rushing through lots of "material" that tests only short term memorization rather than knowledge is a good way for teachers to feel like they are doing a lot more than they actually are. Yet the value of an education 1, 5, 10, or 20 years down the road is always in the fundamental understanding, not the breadth of material covered. Understanding sticks, the details of busywork fades away.

Should an education concentrate on short term knowledge of trivia that fades away rapidly or on long-lasting understanding of fundamental concepts? Is it even remotely reasonable to charge tens of thousands of dollars to students for something that is only temporary?

It's not so much that cargo cultism "works" (however defined); it's that if everyone is doing it you're obviously not going to do worse than they are by doing the same thing. So from a relative standpoint, you're home free.

I studied Maths to the second highest level available in high school (so higher than the majority of students) and although I used e.g. 2PiR all the time this is the first I've actually understood how it works. It isn't something we were taught. We were taught how to get the answer, not why/how that is the correct way to get the answer.

New ways of explaining things should always be welcome

This is a 30 second animation that could be shown to a junior high school class room and most would get some of the idea (if not all).

The real beauty of this is that it could just as well be broadcast to the stars as a quick primer in linking earth based, fundamental mathematical symbols with universal constants.

It actually is quite condescending to think that people upvoting this actually have trouble understanding the concept. It's an elegant animation and young students would benefit from seeing it.

When I was in high school in the late 90's/00's in Australia we were taught in degrees. It wasn't until university that I relearned in radians.

Yeah, I think for many students radians are that thing you have to turn off if you want your calculator to show the correct result.

Things like this were often the downfall of math classes: Yeah, everything was explained, somehow, but if you didn’t really pay attention that one time it was easy to miss the subtleties. If something is then additionally not frequently used it’s easy to get by not really understanding it (and maybe only knowing that radians are an alternative way to represent degrees, where 2π are 360°).

It’s not that hard to understand, but if you don’t need to understand it to succeed, why bother if you aren’t inherently interested in the topic anyway? (Ok, the percentage of people who are inherently interested in math is probably higher on this website than elsewhere. Or at least of people who needed to work with radians to the degree that they had to really understand them.)

I learnt radians in Australia at high school in the same time period.

These animations make more sense than any math teacher or textbook ever did in my entire life. There, a data point of one.

This being 'Hacker' News, I think this kind of material is an awesome way to 'hack' the 'understanding' of concepts and way better than spouting convoluted terminology used by textbooks and teachers who care shit about making sure students 'get' the concept rather than just show how smart they are and how dumb all those are who don't just memorize and regurgitate whatever they say by taking it on face value as some gospel truth. Not everyone sees (literally!) the world the way you do and just because they cannot see what you see does not make them any less smart or incapable than you (and that road goes both ways). If there is a way to enable others see the world your way, it empowers them to bring their abilities to solve such problems and enriches the world by tapping into a previously untapped resource.

I say, bring more such hacks to HN.

I just remember radians being taught as a "thing", an alternative to degrees. We learned how to convert between the two and that was that. Seeing this animation, and seeing that radius bend to wrap itself around part of the circle was pretty breathtaking for me and now I must study it more.

I didn't do trig until after I had finished BS in psychology. Rather I learned it from Khan academy, Wikipedia, and the like. During my second BS (this time in math) a colleague was amazed at my inside and intuition to trig, even though he had taken as much math prior to Uni as he could, and I hadn't.

Just to give an anecdote of the inverse.

"Not to be dismissive or condescending"

but you are... there's nothing wrong on trying to make concepts more understandable.

>but you are

How? I simply asked whether people had trouble with this material because I don't remember it being an issue with any of my classmates in high school or in my 24 hours of undergraduate math.

Saying "anyone who doesn't understand this is an idiot" would be condescending and dismissive, but I fail to see how asking an innocuous question is equivalent in your mind.

Very nice.

Practical: take a 15cm ruler and drill small holes at each end. Thread a 1.2m ish loop of string through the holes so there is a radius of 57cm. Tie the string together about 20cm from the ruler.

The result lets you measure angles on the horizon, e.g. width of Moon (not Sun with children please!) and the altitude of navigation stars near rise and set.



Needs a version with tau: http://tauday.com/

I was skeptical, but tentatively convinced by the end of the section on the hypersphere volume/surface area.

I don't think it's worth actually changing things, but if we can accept that 2pi is a more natural constant than pi, then it should help us write more sensible formulas, even if the symbol tau isn't used.

Thanks for this. Tau seems nice, I'll try using it from now on and see if it makes things simpler (judging from what I've seen so far, I'm optimistic about it).

Nice visualization. Many confusions could have been avoided had "radians" been named "radiuses". Measuring angles in spanned "radiuses" minimizes abstractions and explains the 2*pi factor. (Radii works too.)

I muddled through it learning electronics not realizing I was not quite understanding it, this helps.

Who was it that said something along the lines of " If you can describe a complex thing to someone briefly in simple terms that shows knowledge" Feynman? Einstein?

Whoever said that I find it profound because most people can learn something eventually but to know something, knowledge, I find close to impossible, very hard.

This is probably not what you are thinking, but C.S. Lewis said, "Any fool can write learned language: the vernacular is the real test. If you can't turn your faith into it, then either you don't understand it or you don't believe it."

Similar and it's the gist of it. Understanding something well enough to teach it to someone who doesn't, some people seem to be able to learn enough to get by but not teach others what they are supposed to know.

> If you can't explain it to a six year old, you don't understand it yourself.

Perhaps this one? I pulled it from memory, even though I couldn't remember the author. Google attributes it to Einstien. But the interwebs misattributes quotes all the time, so take it with a grain of salt.

I think the term complex is massively overused. With a clear explanation many things are really quite simple. The difficulty in understanding something is far more likely to come from confusion in the instructor's mind than the student's.

I think this statement is complex.

Nice! They should use this in schools.

Some do, that's how I learned them.

edit: I still think the most confusing thing about radians when learning them is the 2 pi. This seems so arbitrary.

We now use Tau at work and it makes everything easier to understand.

You use τ where you work? Where do you work (in general terms)?

Why arbitrary? C=2pi*r. It's degrees that are arbitrary.

C = tau * r. Why divide the circle in half?

You aren't diving the circle in half, you are using the radius. Radius vs arc length makes perfect sense; diameter vs arc length doesn't.

Which is why tau is a better constant to use than pi.

A = 2taur^2. Why divide the circle in half?

The early use of pi was calculating area and it was derived based on that. At least as far as history can tell us. Records are a little sketchy that far back.

The argument in favour of tau in the formula for the area of a circle is this: To find the area of a circle with radius r, you integrate the circumference of every circle with radius between 0 and r. The integral of tau*r with respect to r is 1/2 tau r^2. Since 1/2 ax^2 is such a common form when dealing with integrals, it can be easier to remember this way.

That's a weak argument. Integrating 2a * x is also insanely common in dealing with integrals. And really if you're deriving the area of a circle and integrating 2pi * r gives you pause you're not on a good road to begin with and removing a 2 from some equations isn't going to help much.

edit: forgot that * would make everything italicized.

Good. Now try sphere surface.


Sphere surface area: 4pi * r^2

Sphere volume: (4/3) * pi * r^3

Both would still have a coefficient if tau replaced pi...

C = pi * D

where D = 2r

I prefer "pau". http://xkcd.com/1292/

lol. I always wonder how people find these things.

Incidentally, I recently read an article about how the creators of The Simpsons actually held math degrees. They hid mathematical easter-eggs all over their episodes. One gag was a "solution" to Fermat's last theorem where 1782^12 + 1841^12 = 1922^12. It's valid up to like, the twelfth significant digit.

It's even more funny when you know that "pau", in Catalan, means "peace".

Think of it as radiUS and radiAN, the AN being angle. Since there is PI angle in a radius, there has to be 2PI in a diameter since d=2r.

> Since there is PI angle in a radius

The only part that doesn't make sense.

His comment took me a while to parse too. The way I ultimately interpreted his comment was "Since 1(d) = 2(r), it's only natural that 1(rotation) = 2(PI radians)". Which is ironic. I.e. the inequality between the coefficients is actually the tau-ists' primary argument against pi. Also, I don't know how it's relevant to its parent comment, which was about "1 rotation being defined as 360 degrees" being arbitrary.

The 360 isn't arbitrary, rather boils down to 60 arc-minutes and 60 arc-seconds, which of course is 3600 arc-seconds in circle. Where the 60 comes from:


Wouldn't surprise me if the minutes and seconds relates to navigation time in the past.

We do. And silly songs on YouTube. And any other good content people want to make.

This being HN, how about someone whipping up a Web application where teachers can post 'want ads' for small things like this and creative visual people can respond by posting links to bits of work? For the lutz and to help?

For those who prefer a text definition:

   radians = arc length in a circle with R=1

Length of which arc?

The measure of an angle in units of radians is equal to the length of the arc subtending that angle in units of radii.

An arc with the same length as the radius. This is what the animation illustrates.

This article is so rad!

I love when something is explained in such an excellent simple way and was a good refresher for me. This is how things should be taught.

Interesting to note that the author, LucasVB, uses a custom PHP library using GD to make these animations. He explains why he finds alternatives like Processing, Matplotlib, etc. insufficient here:


According to Carl Munck in "The Code" (http://www.youtube.com/watch?v=Xw9lTB0hTNU), radians were the unit of measure used to build and position the megaliths.

30 seconds? More like 206264.

What is that little angle after the 3rd r that completes the pi-r?

pi is 3.141...

so it's the .141... part

Man...why wasn't I shown this gif in calculus?

Rocket science.

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