11 ms·
How Radians Work in 30 Seconds
- chris_wot 13y agoNice! They should use this in schools.
- Coincoin 13y agoSome 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.
- srl 13y agoYou use τ where you work? Where do you work (in general terms)?
- RogerL 13y agoWhy arbitrary? C=2pi*r. It's degrees that are arbitrary.
- sp332 13y agoC = tau * r. Why divide the circle in half?
- alayne 13y agoC = pi * D
- backlava 13y agowhere D = 2r
- ccleve 13y agoI prefer "pau". http://xkcd.com/1292/ http://xkcd.com/1292/
- Double_Cast 13y agolol. 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.
- iagooar 13y agoIt's even more funny when you know that "pau", in Catalan, means "peace".
- rtkwe 13y agoA = 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.
- shawnz 13y agoThe 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.
- rtkwe 13y agoThat'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.
- est 13y agoGood. Now try sphere surface.
- rtkwe 13y agoSure... Sphere surface area: 4pi * r^2 Sphere volume: (4/3) * pi * r^3 Both would still have a coefficient if tau replaced pi...
- RogerL 13y agoYou 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.
- yen223 13y agoWhich is why tau is a better constant to use than pi.
- madengr 13y agoThink 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.
- shurcooL 13y ago> Since there is PI angle in a radius The only part that doesn't make sense.
- Double_Cast 13y agoHis 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.
- madengr 13y agoThe 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: http://mathforum.org/library/drmath/view/65468.html http://mathforum.org/library/drmath/view/65468.html Wouldn't surprise me if the minutes and seconds relates to navigation time in the past.
- keithpeter 13y agoWe 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?
- cjdrake 13y agoNeeds a version with tau: http://tauday.com/ http://tauday.com/
- ronaldx 13y agoTau version: https://en.wikipedia.org/wiki/File:Circle_radians_tau.gif https://en.wikipedia.org/wiki/File:Circle_radians_tau.gif
- yetanotherphd 13y agoI 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.
- shurcooL 13y agoThanks 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).
- shurcooL 13y agoLooks good so far. https://github.com/shurcooL/Conception-go/commit/78c4bad7219ccf92c5a217a274f1ba1f3ef4f3c3 https://github.com/shurcooL/Conception-go/commit/78c4bad7219...
- ronaldx 13y agoThis is by LucasVB, who is prolific at animating math for Wikipedia: https://en.wikipedia.org/wiki/User:LucasVB/Gallery https://en.wikipedia.org/wiki/User:LucasVB/Gallery His other works are also worth checking: http://1ucasvb.tumblr.com/ http://1ucasvb.tumblr.com/ (for example, he has a super-nice explanation of Fourier series).
- _pius 13y agoThanks, saw the image linked on Twitter without this context.
- machbio 13y agothe 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-donations http://1ucasvb.tumblr.com/post/57176345908/im-now-open-for-d...
- virtualSatai 13y agoHe answered a few questions in a reddit thread: http://www.reddit.com/r/oddlysatisfying/comments/1vcm7j/this_gif_of_how_radians_work/ http://www.reddit.com/r/oddlysatisfying/comments/1vcm7j/this...
- imranq 13y agoI 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-homer-simpsons-face http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-h...
- Patrick_Devine 13y agoThe 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: http://1ucasvb.tumblr.com/post/43524004530/drawing-process-for-the-sine-function-since http://1ucasvb.tumblr.com/post/43524004530/drawing-process-f... After seeing that, trig really just "clicked".
- dhughes 13y agoI 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.
- ollysb 13y agoI 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.
- source99 13y agoI think this statement is complex.
- hcrisp 13y agoThis 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."
- dhughes 13y agoSimilar 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.
- Double_Cast 13y ago> 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.
- deleted 13y ago[deleted]
- MatthewWilkes 13y ago30 seconds? More like 206264.
- dbbolton 13y agoNot 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.
- javert 13y agoI 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.
- robryan 13y agoYeah, you could easily get though by knowing the method of calculation without ever having understood a model of what they actually are.
- judk 13y agoIt wasn't a guarded secret... "Radian" is almost the same word as "radius", and 2pi * radius = circumference = 2pi radians.
- Jtsummers 13y agoBut 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.
- deevus 13y agoWhen 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.
- haberman 13y agoIt's not an animation, but when I saw the visual illustration of eigenvalues/eigenvectors on this page I was blown away: http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors 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.
- elteto 13y agoI 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.
- haberman 13y agoIsn'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.
- elteto 13y agoDegrees 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.
- 13y ago
- ivan_ah 13y agoFor those who prefer a text definition: radians = arc length in a circle with R=1
- espeed 13y agoAccording to Carl Munck in "The Code" (http://www.youtube.com/watch?v=Xw9lTB0hTNU http://www.youtube.com/watch?v=Xw9lTB0hTNU), radians were the unit of measure used to build and position the megaliths.
- deleted 13y ago[deleted]
- bluejes 13y agoNow just $140 http://www.vosgeschocolate.com/ http://www.vosgeschocolate.com/
- eulern 13y agoNice 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.)
- keithpeter 13y agoVery 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. http://en.wikipedia.org/wiki/Kamal_%28navigation%29 http://en.wikipedia.org/wiki/Kamal_%28navigation%29 http://kaloujm.com/page_kamal_uk.php http://kaloujm.com/page_kamal_uk.php
- leke 13y agoWhat is that little angle after the 3rd r that completes the pi-r?
- edtechdev 13y agopi is 3.141... so it's the .141... part
- colund 13y agoThis 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.
- deleted 13y ago[deleted]
- forlorn 13y agoRocket science.
- alok-g 13y agoInteresting 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: http://1ucasvb.tumblr.com/faq http://1ucasvb.tumblr.com/faq
- tommydiaz 13y agoMan...why wasn't I shown this gif in calculus?
- dimadima 13y agoRadians were confusing for a bit, and then somone explained that a radian is a measure of angle just like the degree, except convently set up such that there are 2pi—or approximately 6—radians in a circle. And why is this convenient? Because, among other reasons, that's the circumference of a unit circle. This makes expressing the results of certain calculations cleaner. 360 degrees, or about 6 radians. Sometimes degrees are easier to use, sometimes radians are easier use. Holy fuck, what's confusing here? If you divide 2pi by 2pi you get 1. But if you divide 2pi by 360, well that is some irrational number. Can't believe our educators have freaked so many people out about thinking and are so freaked out themselves that radians are some bogey man to so many people, myself included for a few months in high school.