# Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are mathematically beautiful at the same time.

Do you know of any other concepts like these?

• It looks like mathpop or demand for math entertainment)
– rook
Apr 2, 2014 at 12:59
• Something that has always annoyed me is irrational numbers. They are easy to understand (like the proof for $\sqrt{2}$), but not being able to have a rational representation just irks me. And don't even get me started on transcendentals like $\pi$ and $e$. Apr 4, 2014 at 16:12
• @ColeJohnson the 'transcendentality" is the beauty of it!
– Guy
Apr 4, 2014 at 17:07
• There is a considerable overlap with mathoverflow.net/questions/8846/proofs-without-words Jun 20, 2014 at 22:13
• @TheGuywithTheHat That's the reason for the second spike of visits, on August 27. The comment by LTS is from April; back then the traffic was driven by Ycombinator. As a result, this same question made both April 7 and August 27 the two days with most visits to the site.
– user147263
Aug 29, 2014 at 18:37 I think if you look at this animation and think about it long enough, you'll understand:

• Why circles and right-angle triangles and angles are all related.
• Why sine is "opposite over hypotenuse" and so on.
• Why cosine is simply sine but offset by $$\frac{\pi}{2}$$ radians.
• @joeA: it's not as smooth as regenerating at a higher framerate, but gfycat.com allows one to view gifs at different speeds: gfycat.com/TintedWatchfulAxisdeer#?speed=0.25
– Max
Apr 2, 2014 at 20:51
• If high school math just used a fraction of the resources here, we'd have way more mathematicians. Apr 3, 2014 at 17:00
• Is the source of this animation available? (It looks like it's in $\mathrm\TeX$.) Apr 3, 2014 at 23:31
• This is the normal way of introducing sine and cosine at high schools. At least in our country. Apr 7, 2014 at 15:36
• @joeA: I've made a programmatic version of this on Khan Academy; feel free to change the speed to your liking: khanacademy.org/cs/circle/4597064320155648 Apr 16, 2014 at 10:48

My favorite: tell someone that $$\sum_{n=1}^{\infty}\frac{1}{2^n}=1$$ and they probably won't believe you. However, show them the below: and suddenly what had been obscure is now obvious.

• My first intuitive visualization of this sum was a circle. I wasn't $100\%$ sure that the answer was $1$(long time back, I had never seen an infinite series before, and 1 was just my first immediate thought) :)
– Guy
Mar 31, 2014 at 17:38
• I still don't believe you. Apr 2, 2014 at 8:51
• Another way to think of this is that 1/2 + 1/4 + 1/8 = 0.111...binary = 0.999...decimal = 1. Apr 7, 2014 at 3:02
• You don't need to divide the square into such complicated fragments, just use vertical lines (assuming the x coordinate ranges from 0 to 1) at x=1/2, x=1/4, x=1/8 etc. Each time you add 1/2^n to the area, and in the limit you obviously get 1. Apr 7, 2014 at 4:05
• @MaximUmansky: That way you'd just get lines that get closer together and it'll be not as obvious. Here, you see the fractions $\frac{1}{2}$ and $\frac{1}{4}$ in their "standard shape", so what remains must obviously be $\frac{1}{4}$. Then, put the same shapes inside the remaining square which is of the same proportions as the initial one (and it's easily checked that $\frac{1}{2}\cdot\frac{1}{4}=\frac{1}{8}$); you'll get the next smaller square, hidden deeper in the corner. Repeat, and the square will shrink to a tiny dot (not a whole line, which may intuitively seem larger). Apr 9, 2014 at 20:09

This visualisation of the Fourier Transform was very enlightening for me: The author, LucasVB, has a whole gallery of similar visualisations at his Wikipedia gallery and his tumblr blog.

• @gekkostate You're a bit off. f is the sum of multiple simple waves, all with different frequencies and phase angles. The fourier transform takes a complex wave from a given time period, and gives you the phase angles and frequencies of all of the component waves. f^ is the amplitude of each component wave.
– Joel
Apr 7, 2014 at 2:15
• I'm down-voting this because this animation doesn't include phase information. The phase is the difference between a pulse and cw. If you don't want it, you should use another basis (wavelets). Apr 7, 2014 at 3:19
• this is pretty cool too: bl.ocks.org/jinroh/7524988 Apr 7, 2014 at 20:54
• I saw this animation a long time ago, and it helped me finally understand what the Fourier transformation does, but I still am lost as to how it works... Apr 11, 2014 at 1:21
• I don't find this graphical example helpful in explaining the Fourier Transform. If anything, it is going to add to existing confusion people have with the Fourier Series. The sin and cos expression shown is too general to be meaningful, and the graphic shown for the "transform" is really just a visualization of the magnitudes of the Fourier Series coefficients you would have if the partial sum function were presumed to be infinitely periodic. When I calculate the actual Fourier Transform of the segment of function shown, I get something quite different: imgur.com/eQuerJa Aug 31, 2014 at 18:53

Here is a classic: the sum of the first $n$ positive odd numbers $= n^2$. We also see that the sum of the first $n$ positive even numbers $= n(n+1)$ (excluding $0$), by adding a column to the left. • If you remove the leftmost column, you get a proof that the sum of the first $n$ non-negative even numbers (counting 0) is $n(n-1)$ Apr 3, 2014 at 16:29
• I read something about how Galileo noted that the distance a falling body covered over a unit time went as the series of odd numbers and was confused until I realized this. Apr 3, 2014 at 17:37
• I have noticed an interesting method to calculate a square root based on this fact: disqus.com/home/discussion/geeksforgeeks/… Sep 7, 2015 at 23:46

The sum of the exterior angles of any convex polygon will always add up to $360^\circ$. This can be viewed as a zooming out process, as illustrate by the animation below: • I always found the easiest way to think about this was that if you move around a convex shape such that you eventually end up at the start, you must have rotated through exactly 360º. If you walk around any shape, and count anticlockwise rotations as negative, but clockwise rotations as positive, I imagine the angles will still add to 360º (or -360º depending on which direction you take). Apr 5, 2014 at 12:27
• I like to view this as a limiting process. Imagine that the picture on the right is the picture on the left zoomed out a great distance. Apr 5, 2014 at 14:10
• This would be better as an animation (which achieves the zooming effect). Apr 7, 2014 at 10:31
• Edited. Do have a look. (It may look horrible, since this is the first time I made an animation.) Apr 7, 2014 at 13:36
• this is really,really clever Aug 3, 2017 at 9:07

A well-known visual to explain $(a+b)^2 = a^2+2ab+b^2$: • Always liked this one
– Almo
Apr 2, 2014 at 14:11
• why area of a^2 is greater tham area of b^2? Apr 3, 2014 at 7:34
• @jaczjill Because a and b are not the same number. Apr 3, 2014 at 11:24
• @jaczjill the areas of a and b are arbitrary. They could be any size and it would make no difference. It's just an example. Apr 4, 2014 at 8:29
• Have you seen the Montessori binomial cube?
– MJD
Apr 4, 2014 at 14:30

While attending an Abstract Algebra course I was given the assignment to write out the multiplication table modulo n. I forgot to do the homework until just before class, but it was so easy to write the program I was able to print the result between classes.

The circular patterns in the tables fascinated me, and compelled me to replace the numbers with colors. The result is a beautiful illustration showing the emergence of primes and symmetry of multiplication.

The colors were chosen to start blue at 1 (cold) and fade to red at n (hot). White is used for zero (frozen), because it communicates the most information about prime factorization.

The interactive version can be found here: https://web.archive.org/web/20140830110358/http://arapaho.nsuok.edu/~deckar01/Zvis.html

Multiplication of the integers modulo 15: Multiplication of the integers modulo 512: • This is awesome. Looks very much like a fractal. Is there a name for this fractal? Apr 7, 2014 at 15:52
• @OmnipotentEntity it doesn't qualify as having fractional dimension because the underlying space is discrete. If you use a real space instead - the set of x, y such that x * y = 0 mod n, you get a set of hyperboles (linear in dimension). If you take a limit towards infinite squares, then the count of white squares is equal to the length of the edge whenever the length is prime. Apr 7, 2014 at 18:48
• I made the exact same picture when I took group my first group theory class. Wonderful.
– Seth
Apr 7, 2014 at 20:24
• Although not colored WolframAlpha also does a nice job of illustrating this: wolframalpha.com/input/?i=Z_200 Apr 9, 2014 at 23:05
• The 'standing up too fast' fractal Apr 22, 2015 at 14:32 Logarithmic spiral and scale: • What program produces these images? Aug 11, 2014 at 19:49
• @rschwieb They're made by 1ucasvb, he's got an faq here: 1ucasvb.tumblr.com/faq Aug 13, 2014 at 14:40
• I can't believe how long I used radians in college without realizing what a radian really is. Jan 9, 2015 at 0:28
• The exponential curve has curvature-arc length relation like cycloidal? What does the animation show? Apr 9, 2016 at 2:31
• Thank you. I sort of knew that radians are just different representation of degrees but never knew where exactly their value comes from. Sep 11, 2021 at 9:48

When I understood Fourier series visually- • OH. Wow. I just took my signals course and even after I did well on the exam I still didn't actually UNDERSTAND fourier series. Thanks, wow. Apr 2, 2014 at 22:25
• This diagram is a particularly good illustration of the Gibbs Phenomenon, too. Nice! Apr 4, 2014 at 6:09
• I don't understand what this is showing: is it showing hte approximation of a square wave by a Fourier series? Apr 5, 2014 at 5:18
• @ShreevatsaR Yes. It's an illustration of the Gibbs Phenomenon. Apr 5, 2014 at 6:01
• I like this one more to show the difference between pointwise and uniform convergence: this Fourier series only converges pointwise and not uniformly. The total result will always be off (known as Gibbs phenomenom): this wouldn't happen if there was uniform convergence. Apr 9, 2014 at 3:09 Here is a very insightful waterproof demonstration of the Pythagorean theorem. Also there is a video about this.

It can be explained as follows. We seek a definition of distance from any point in $$\mathbb{R}^2$$ to $$\mathbb{R}^2$$, a function from $$(\mathbb{R}^2)^2$$ to $$\mathbb{R}$$ that satisfies the following properties.

• For any points $$(x, y)$$ and $$(z, w)$$, $$d((x, y), (x + z, y + w)) = d((0, 0), (z, w))$$
• For any point $$(x, y)$$, $$d((0, 0), (x, y))$$ is nonnegative
• For any nonnegative real number $$x$$, $$d((0, 0), (x, 0)) = x$$
• For any point $$(x, y)$$, $$d((0, 0), (x, -y)) = d((0, 0), (x, y))$$
• For any points $$(x, y)$$ and $$(z, w)$$, $$d((0, 0), (xz - yw, xw + yz)) = d((0, 0), (x, y))d((0, 0), (z, w))$$

Suppose a function $$d$$ from $$(\mathbb{R}^2)^2$$ to $$\mathbb{R}$$ satisfies those conditions, then for any point $$(x, y)$$, $$d((0, 0), (x, y))^2 = d((0, 0), (x, y))d((0, 0), (x, y)) = d((0, 0), (x, y))d((0, 0), (x, -y)) = d((0, 0), (x^2 + y^2, 0)) = x^2 + y^2$$ so $$d((0, 0), (x, y)) = \sqrt{x^2 + y^2}$$ so for any points $$(x, y)$$ and $$(z, w)$$, $$d((x, y), (z, w)) = \sqrt{(z - x)^2 + (w - y)^2}$$ Now I will show that $$d((x, y), (z, w)) = \sqrt{(z - x)^2 + (w - y)^2}$$ actually satisfies those properties. It's trivial to show that it satisfies the first 4 conditions. It also satisfies the fifth condition because for any points $$(x, y)$$ and $$(z, w)$$, $$d((0, 0), (xz - yw, xw + yz)) = \sqrt{(xz - yw)^2 + (xw + yz)^2} = \sqrt{x^2z^2 - 2xyzw + y^2w^2 + x^2w^2 + 2xyzw + y^2z^2} = \sqrt{x^2z^2 + x^2w^2 + y^2z^2 + y^2w^2} = \sqrt{(x^2 + y^2)(z^2 + w^2)} = \sqrt{x^2 + y^2}\sqrt{z^2 + w^2} = d((0, 0), (x, y))d((0, 0), (z, w))$$

As a result of this, from now on, I will define the distance from any point $$(x, y)$$ to any point $$(z, w)$$ as $$\sqrt{(z - x)^2 + (w - y)^2}$$ and denote it as $$d((x, y), (z, w))$$. I will also use $$d(x, y)$$ as shorthand for $$d((0, 0), (x, y))$$ Since distance satisfies condition 5, for any right angle triangle at all, not just ones whose legs are parallel to the axes, the square of the length of its hypotenuse is equal to the sum of the squares of the lengths of its legs. This image shows that using that definition of distance, for any right angle triangle whose legs are parallel to the axes and have lengths $$x \in \mathbb{R}^+$$ and $$y \in \mathbb{R}^+$$, the area of a square with the hypotenuse as one of its edges is $$(x - y)^2 + 2xy = x^2 + y^2 = (d(x, y))^2$$. Combining that result with the fact that distance satisfies condition 5, we can show that for any right angle triangle, even with legs non parallel to the axes, the area of a square with its hypotenuse as its edge has an area equal to the sum of the squares of the lengths of its legs. Sources:

• This does not actually demonstrate the pythagorean theorem. It is showing that for one single right triangle, the theorem holds. This could be a coincidence. Someone seeing this might think that 3,4,5 is the only pythagorean triple. Apr 7, 2014 at 16:06
• @Sparr is it not? </sarcasm> May 15, 2014 at 23:17
• It looks cool, but provides very little (no?) insight about why PT is true. Aug 29, 2014 at 1:25
• @GrumpyParsnip Before you learn why something is true, you might want to do a sanity check to verify that it probably is. This does that, while looking cool. The only thing that I really care about is that it looks cool, though. Mathematics is a very aesthetic thing, so I'm allowed to like something just because it looks nice. Sep 5, 2014 at 0:06
• I saw this example in a science museum when I was in high school back in the early 90's. I have always remembered it. Nov 29, 2016 at 6:08 This is a neat little proof that the area of a circle is $\pi r^2$, which I was first taught aged about 12 and it has stuck with me ever since. The circle is subdivided into equal pieces, then rearranged. As the number of pieces gets larger, the resulting shape gets closer and closer to a rectangle. It is obvious that the short side of this rectangle has length $r$, and a little thought will show that the two long sides each have a length half the circumference, or $\pi r$, giving an area for the rectangle of $\pi r^2$.

This can also be done physically by taking a paper circle and actually cutting it up and rearranging the pieces. This exercise also offers some introduction to (infinite) sequences.

• This doesn't work for me :( Sadly, I find it non-obvious that the length of the bottom really will converge helpfully to $\pi r$. Apr 8, 2014 at 16:50
• @NicholasWilson By definition, the circumference of a circle is π times the diameter of the circle. Here, we've cut the circle into slices, half flipped one way, half flipped the other. Therefore half the circumference (πd) is on the bottom. Half of the diameter is the radius, r. Does that help? Apr 8, 2014 at 19:29
• @ghoppe Nice try! But it's all wiggly. We're lucky that $\sin{\theta} \sim \theta$ as $\theta \rightarrow 0$, so the actual perimeter of the bottom (which is known to be $\pi$) does converge to the width of the rectangle. But - that nice property of $\sin$ is exactly what we're trying to prove! Apr 9, 2014 at 8:20
• @ghoppe What would you visually infer about the perimeter of a Koch snowflake? I've had people assure me "It must be bounded!" Visual inferences are susceptible to error :( You need to invoke some limit arguments to convince me those arcs on the segments do converge. Similarly, inscribing polygons to determine the circumference sounds very dangerous (think what would happen if you tried that with a Koch snowflake) -- but bounding the area above and below by inscribed/exscribed polygons is definitely a sound proof. Apr 9, 2014 at 17:56
• @NicholasWilson coming back to this discussion rather late, but perhaps I can convince you. The length of the bottom is always πr, it doesn't change with iteration so all we have to show is that the shape ends up as a rectangle. This is equivalent to stating that the line between two points on a circle tends to the tangent as the two points get closer together, which I believe is at least one definition of the tangent. Dec 16, 2014 at 22:16

A visual explanation of a Taylor series:

$f(0)+\frac {f'(0)}{1!} x+ \frac{f''(0)}{2!} x^2+\frac{f^{(3)}(0)}{3!}x^3+ \cdots$

or

$f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots$ When you think about it, it's quite beautiful that as you add each term it wraps around the curve.

• It is interesting to note that Taylor series are relatively very precise near their center, which makes them especially useful in computer science when you need a precise but still fast approximation. For example, processors do not have a cosine function built-in, but will fly through an equation such as x => 1 - (x^2)/(2!) + (x^4)/(4!) - ... and from this equation, you could build a function that accepts a precision parameter so that the call recurses until higher accuracy is achieved. (note: there are many more efficient ways to do this) Apr 3, 2014 at 21:55
• @sebleblanc It is just amazing that petty much any function can represented with a Taylor series locally Apr 4, 2014 at 8:15
• @sebleblanc Should have said "[some] processors" instead. x86 processors have had a cosine function for some time now with the FCOS instruction. It doesn't do it with a taylor series as far as I can tell. What I assume it does is use a lookup table followed by some sum/difference formulas. Apr 4, 2014 at 16:05
• You are right that X87 co-processors are omni-present, but there are far more embedded CPUs in the world than there are X86/87 CPUs. Just think of how many devices you interact with that are not strictly computers. ///// From my desk, I can see three computer monitors, one TV, two video game consoles, one oscilloscope, one modern refrigerator, two cordless phones, a wireless mouse, a microcontroller dev board, a Raspberry Pi, my cell phone, my credit card. All of these have a processor that does not have the FSIN instruction... If anything, I should have said [most] instead. ;-) Apr 12, 2014 at 14:31
• @sebleblanc The x86/87 FPU uses CORDIC which lets you compute trigonometric functions in hardware using only bit shifts and addition. Aug 27, 2014 at 16:57

When I look up "area of a rhombus" on Google images, I find plenty of disappointing images like this one: which show the formula, but fail to show why the formula works. That's why I really appreciate this image instead: which, with a little bit of careful thought, illustrates why the product of the diagonals equals twice the area of the rhombus.

EDIT: Some have mentioned in comments that that second diagram is more complicated than it needs to be. Something like this would work as well: My main objective is to offer students something that encourages them to think about why a formula works, not just what numbers to plug into an equation to get an answer.

As a side note, the following story is not exactly "visually stunning," but it put an indelible imprint on my mind, and affected the way I teach today. A very gifted Jr. High math teacher was teaching us about volume. I suppose just every about school system has a place in the curriculum where students are required learn how to calculate the volume of a pyramid. Sadly, most teachers probably accomplish this by simply writing the formula on the board, and assigning a few plug-and-chug homework problems. No wonder that, when I ask my college students if they can tell me the formula for the volume of a pyramid, fewer than 5% can.

Instead, building upon lessons from earlier that week, our math teacher began the lesson by saying:

We've learned how to calculate the volume of a prism: we simply multiply the area of the base times the height. That's easy. But what if we don't have a prism? What if we have a pyramid?

At this point, she rummaged through her box of math props, and pulled out a clear plastic cube, and a clear plastic pyramid. She continued by putting the pyramid atop the cube, and then dropping the pyramid, point-side down inside the cube: She continued:

These have the same base, and they are the same height. How many of these pyramids do you suppose would fit in this cube? Two? Two-and-a-half? Three?

Then she picked one student from the front row, and instructed him to walk them down the hallway:

Go down to the water fountain, and fill this pyramid up with water, and tell us how many it takes to fill up the cube.

The class sat in silence for about a full minute or so until he walked back in the room. She asked him to give his report.

"Three," he said.

She pressed him, giving him a hard look. "Exactly three?"

"Exactly three," he affirmed.

Then, she looked around the room:

"Who here can tell me the formula I use to get the volume of a pyramid?" she asked.

One girl raised her hand: "One-third the base times the height?"

I've never forgotten that formula, because, instead of having it told to us, we were asked to derive it. Not only have I remembered the formula, but I can also even tell you the name of the boy who went to the water fountain, and the girl who told us all the formula (David and Jill).

Given the upvoted comment, If high school math just used a fraction of the resources here, we'd have way more mathematicians, I hope you don't mind me sharing this story here. Powerful visuals can happen even in the imagination. I never got to see that cube filling up with water, but everything else in the story I vividly remember.

Incidentally, this same teacher introduced us to the concept of pi by asking us to find something circular in our house (“like a plate or a coffee can”), measuring the circumference and the diameter, and dividing the one number by the other. I can still see her studying the data on the chalkboard the next day – all 20 or so numbers just a smidgeon over 3 – marveling how, even though we all probably measured differently-sized circles, the answers were coming out remarkably similar, “as if maybe that ratio is some kind of constant or something...”

• ^^1 million upvotes for the stories about maths education. I am writing both of these down for future reference. In fact, I think we should start a website with instructions to help primary teachers teach mathematical concepts in interesting ways. Apr 5, 2014 at 12:34
• @daviewales It's been started - the mathematics educators SE.
– user122283
Apr 5, 2014 at 14:45
• Great stories. I do, however, think your first rhombus is fine. I actually find it easier to understand the result using that image than the second. Either diagonal splits the rhombus into two triangles, each of area $$\frac{1}{2}\frac{d_1}{2}d_2=\frac{1}{2}d_1\frac{d_2}{2}.$$ (Is there some aspect-ratio issue with your image? The white rectangle should be a square, but doesn't look like it on my screen.) Apr 6, 2014 at 2:36
• The first rhombus picture is fine. Just draw a rectangle around it, and it's plain to see that the four triangles that form the rhombus cover half of the rectangle. In contrast, the second picture is so confusing that you yourself got it wrong: you write that it "illustrates why the product of the diagonals equals half of the area of the rhombus", while the ratio is the opposite.
– LaC
Apr 7, 2014 at 3:47
• @Peter - I'll admit that I'm using the word derive a bit loosely for a mathematics forum (it's a soft answer to a soft question), but it's too bad you see this as "dumbing down" and "a waste of time." Most 7th-grade teachers charged with teaching volume would simply write the formula on the board, and it would be well-forgotten by the end of summer. Her technique might have been weak insofar as mathematical rigor goes, but the pedagogy was very strong. I assure you, this woman was not one to "dumb down" anything; I remember re-learning concepts in 11th-grade that she taught us in Jr High.
– J.R.
Feb 22, 2015 at 9:44

As I was in school, a supply teacher brought a scale to lesson: He gave us several weights that were labeled and about 4 weights without labels (let's call them $A, B, C, D$). Then he told us we should find out the weight of the unlabeled weights. $A$ was very easy as there was a weight $E$ with weight($A$) = weight($E$). I think at least two of them had the same weight and we could only get them into balance with a combination of the labeled weights. The last one was harder. We had to put a labeled weight on the side of the last one to get the weight.

Then he told us how this can be solved on paper without having the weights. So he introduced us to the concept of equations. That was a truly amazing day. Such an important concept explained with a neat way.

• I do the same thing... except with drawings instead of an actual scale and weights. I need to get this set. Apr 7, 2014 at 0:34
• @MeemoGarza although drawings are nice, too, I think that an actual device has some advantages: Pupils rather get excited by a real device than by a drawing; it's easier to accept something that you can touch and play with where you instantly get feedback if what you did was correct; as soon as pupils got familiar with the concept of equations, you can explain the physics behind scales ($a_1 \cdot m_1 = a_2 \cdot m_2$ where $a$ is the length of the part of the scale and $m$ is the mass on that side). That way students can see that math is also about describing the real world in a formal way. Apr 7, 2014 at 5:58
• I agree completely, but I don't have the apparatus. I'll look into purchasing one before the next semester. Thanks for helping me make up my mind! Apr 8, 2014 at 0:50
• @MeemoGarza Meanwhile you might be interested in this: walter-fendt.de/ph14d/hebel.htm - it's a site created by a physics teacher I had in school. This website contains lots of Java applets that show things in physics. Apr 8, 2014 at 1:59

How about a line integral of a scalar field by http://1ucasvb.tumblr.com: • Awesome — this will be a great introduction to fields & the usage of integrals which I am about to explain to a friend of mine! ;) Aug 27, 2014 at 17:44
• Nice visual intuition, but why is it true that $|\mathbf{r}'|$ is the right integrator to use? (i.e. the 'flattening/projection to the real line' step) Aug 11, 2016 at 5:18
• Isn't this how students usually learn line integrals? Mar 9, 2017 at 1:50
• @enthdegree the integral is with respect to arc length ($ds$= arc length). $$\int_C f(x,y)ds=\int_{a}^{b}f(x(t),y(t))\sqrt{x'(t)^2+y'(t)^2}dt=\int_{a}^{b}f(r(t))|r'(t)|dt$$ Oct 6, 2018 at 1:54
• It's just "u-substitution." Jan 28, 2019 at 4:12

This is similar to Aky's answer, but includes a second drawing (and no math.)

To me the second drawing is key to understanding why the $\mathrm c^2$ area is equal to the sum of $\mathrm a^2+\mathrm b^2$. Edit: comments requested an animation, so a simple gif is attached... • I've always felt this one worked better in .gif format. Apr 3, 2014 at 20:47
• This would be amazing if you animated the change from the first to the second by having the triangles on the right flop over onto the ones on the left to create those rectangles. Apr 7, 2014 at 0:31
• I think this one is lacking because it is not explained in the left picture that the blue and red areas are the same size. Apr 7, 2014 at 15:58
• @Sparr they are not. The salmon area is just 4 times the size of the triangle. The only equivalence is bewteen the blue area on the left (c²) and the blue area on the right (a² + b²) which is just Pythagoras' formula. I will upload a better picture to show it. Apr 7, 2014 at 16:33
• Anyway...THIS was my favourite, so, good that you already did it! May 12, 2014 at 21:49

Similarly to eykanal's answer, although demonstrating some interesting facts about medians and geometry as well. It demonstrates that $\displaystyle\sum_{n = 1}^{\infty}\frac{1}{2^n} = 1$: • I think this one better. In the square one you have to make a choice between dividing horizontally or vertically. With the triangle there is only one dividing cut. Apr 5, 2014 at 13:55
• Just nitpicking, but you got the areas of the triangles wrong. For example, the one of are $\frac{1}{2}$ according to you has actually an area of $\frac{1}{2}\times\frac{\sqrt{2}}{2}\times\frac{\sqrt{2}}{2}=\frac{1}{4}$, so what this figure is proving is that $\sum_{n=2}^{\infty}\frac{1}{2^n}=\frac{1}{2}$.
– S4M
Apr 7, 2014 at 7:12
• @S4M: The area of the first triangle is defined as $1/2$; how is the calculation wrong when there is no calculation... only definition? Apr 7, 2014 at 16:51
• @AndrewCoonce I was wrong in my previous comment. I asscumend that the side of the main triangle was $1$. It's all fine if its side is $\frac{\sqrt{2}}{2}$.
– S4M
Apr 9, 2014 at 12:27
• @S4M do you mean just rad two? Dec 9, 2016 at 10:32

Steven Wittens presents quite a few math concepts in his talk Making things with math. His slides can be found from his own website.

For example, Bézier curves visually:  He has also created MathBox.js which powers his amazing visualisations in the slides.

• These are beautiful! May 15, 2014 at 23:16
• Also, if you read the text out loud you get to say "lerp lerp lerp lerp", which has its own aesthetic appeal. Sep 23, 2014 at 21:43
• beautiful, but I don't understand XD Apr 20, 2015 at 11:20
• Jun 21, 2016 at 9:24

Fractal art. Here's an example: "Mandelbrot Island". The real island of Sark in the (English) Channel Islands looks astonishingly like Mandelbrot island: Now that I think about it, fractals in general are quite beautiful. Here's a close-up of the Mandelbrot set: • Are complex numbers easy to explain? Mar 31, 2014 at 17:15
• Probably not. But the idea that an object "looks the same if you zoom in on it" is easy to explain, I think. Mar 31, 2014 at 17:16
• Whereas complex numbers might not be easy to explain, it is easy to explain the idea idea that each pixel is calculated by repeating a simple calculation on two numbers (which are initially the coordinates of the pixel), and the color is a measure of how the resulting pair of numbers "escapes" (grows large).
– Kaz
Apr 1, 2014 at 4:57
• @adam.r I don't recall precisely when I learned about complex numbers in school, but it was around the 9th grade (maybe 8th, maybe 10th). To understand the Mandelbrot set, nothing more is needed than basic arithmetic with complex numbers. That should put it within reach for anyone who graduated high school. It's certainly a good candidate for this list. Apr 1, 2014 at 16:12
• It is a close up. The Mandelbrot set is not completely self-similar, but there are some self-similarities as you zoom. See this zoom sequence for an example. Apr 3, 2014 at 3:12

This animation shows that a circle's perimeter equals to $2r*\pi$. As ShreevatsaR pointed out, this is obvious because $\pi$ is by definition the ratio of a circle's circumference to its diameter

In this image we can see how the ratio is calculated. The wheel's diameter is 1. After the perimeter is rolled down we can see that its length equals to $\pi$ amount of wheels. Source

• But can you show that a sphere with radius, $r$, has a volume of $V_3(r) = \frac{4}{3} \pi r^3$ and a surface area of $SA_3(r) = 4 \pi r^2$ Apr 4, 2014 at 16:09
• That the circle's perimeter is $2\pi r$ is the definition of $\pi$, so I wouldn't say this is an explanation of the fact; rather it's an illustration of what the definition means, and that the value of $\pi$ is about $3.1$. Apr 5, 2014 at 5:24
• @ColeJohnson There's a neat way to find the volume of a sphere with Cavalieri's principle and Pythagoras. Just look up Cavalieri's principle on Wikipedia, I think it's on there. Feb 23, 2015 at 1:45
• @ColeTobin Yes, for surface area: math.stackexchange.com/a/2398385 Nov 16, 2021 at 12:11 An iron pendulum is suspended above a flat surface, with three magnets on it. The magnets are colored red, yellow and blue.

We hold the pendulum above a random point of the surface and let it go, holding our finger on the starting point. After some swinging this way and that, under the attractions of the magnets and gravity, it will come to rest over one of the magnets. We color the starting point (under our finger) with the color of the magnet.

Repeating this for every point on the surface, we get the image shown above.

• How is the shading of the image given? Apr 21, 2014 at 14:27
• @DavidZhang, the longer the pendulum swings, the darker the point. I don't know the exact function used for this image, but it you click on the link, you'll find the algorithm. Apr 21, 2014 at 15:27
• Deterministic chaos? Mar 8, 2015 at 11:51
• @PeterMortensen Absolutely. In fact, you can see how (in certain areas) a minute change in initial conditions can cause a drastic change in outcome. Mar 9, 2015 at 12:36
• with this i think that all things have a hidden pattern, some of they can be very hard to see May 26, 2019 at 2:36

Simple,visual proof of the Pythagorean theorem. Originally from Pythagorean Theorem Proof Without Words 6). • Perhaps I'm trying to oversimplify, but this visual proof would be way easier (perhaps even trivial) if the legs of the big straight angle (S.A.) triangle (hypotenuse = the circle's diameter, third vertex on the top of that leg of length $\;b\;$) were drawn, and then from basic geometry: " In a S.A. triangle, the height to the hypotenuse divides the triangle in two triangles similar to each other and also to the big triangle". Nicely brought. +1 Mar 31, 2014 at 12:31
• Another great pictural proof is 4 triangles in a square: mathalino.com/sites/default/files/images/01-pythagora.jpg Mar 31, 2014 at 15:10
• I don't get the original equality. Are you relying on the similarity of two triangles? How is that obvious from the diagram? Mar 31, 2014 at 17:10
• It's not immediately obvious to me why $\frac{c+a}{b} = \frac{b}{c-a}$ I'm sure it's very simple and I'll kick myself for asking, sorry. Mar 31, 2014 at 17:12
• Well, I looked at that picture and thought: "Err ... how does this work?" I figured it out, but it was far from obvious for me. Apr 3, 2014 at 18:33

Here's a GIF that I made that demonstrates Phi (golden number) • That's Beautiful! Sep 4, 2016 at 12:36
• Please tell logic behind taking $\sqrt{1.25} +.5$. Sep 14, 2020 at 2:15

Topology needs to be represented here, specifically knot theory. The following picture is from the Wikipedia page about Seifert Surfaces and was contributed by Accelerometer. Every link (or knot) is the boundary of a smooth orientable surface in 3D-space. This fact is attributed to Herbert Seifert, since he was the first to give an algorithm for constructing them. The surface we are looking at is bounded by Borromean rings. • I do not know much about topology so I will take your word for it that this is a beautiful idea/concept. However this picture and your description explain nothing to me. It seems like you missed the "easy to explain" bit in the question.
– dfc
Apr 7, 2014 at 5:25
• @dfc I don't know, it seems like you can convey most of the meat here using soap bubbles. Apr 7, 2014 at 20:10

Take a look at this great example of Fourier series visualizations written in JavaScript. • The link is brilliant! Jun 14, 2015 at 9:50
• If you had captured an animated gif, this would be the top answer! Jul 10, 2015 at 0:14

Fourier transform of the light intensity due to a diffraction pattern caused by light going through 8 pinholes and interfering on a wall, for different choices of parameter:       The best thing about them is, they satisfy periodic boundary conditions, and so you can pick one of them and set it as a desktop background by tiling it, resulting in a far more spectacular image than just the single unit cells posted above!

The images seem to be a vast interconnected network of lines once you tile them, but in fact the entire picture is actually just a single circle, which has been aliased into a tiling cell thousands of times.

Here is a video of the first couple thosand patterns: http://www.youtube.com/watch?v=1UVbUWuyNmk

Here is the Mathematica code used to generate and save the images. There are two parameters that are adjustable: mag is the magnification and must be an integer, with 1 generating 600 by 600 images, 2 generating 1200 by 1200 images, etc. i is a parameter which can be any real number between 0 and ~1000, with values between 0 and 500 being typical (most of the preceding images used i values between 200 and 300). By varying i, thousands of unique diagrams can be created. Small values of i create simple patterns (low degree of aliasing), and large values generate complex patterns (high degree of aliasing).

$HistoryLength = 0; p = {x, y, L}; nnn = 8; q = 2.0 Table[{Cos[2 \[Pi] j/nnn], Sin[2 \[Pi] j/nnn], 0}, {j, nnn}]; k = ConstantArray[I, nnn]; n[x_] := Sqrt[x.x]; conjugate[expr_] := expr /. Complex[x_, y_] -> x - I y; a = Table[k[[i]]/n[p - q[[i]]], {i, nnn}]; \[Gamma] = Table[Exp[-I \[Omega] n[p - q[[i]]]/c], {i, nnn}]; expr = \[Gamma].a /. {L -> 0.1, c -> 1, \[Omega] -> 100}; ff = Compile[{{x, _Real}, {y, _Real}}, Evaluate[expr], CompilationTarget -> "C", RuntimeAttributes -> {Listable}]; i = 250; mag = 1; d = 6 i mag; \[Delta] = 0.02 i; nn = Floor[Length[Range[-d, d, \[Delta]]]/2]; A = Compile[{{x, _Integer}, {y, _Integer}}, Exp[I (x + y)], CompilationTarget -> "C", RuntimeAttributes -> {Listable}] @@ Transpose[ Outer[List, Range[Length[Range[-d, d, \[Delta]]]], Range[Length[Range[-d, d, \[Delta]]]]], {2, 3, 1}]; SaveImage = Export[CharacterRange["a", "z"][[RandomInteger[{1, 26}, 20]]] <> ".PNG", #] &; {#, SaveImage@#} &@ Image[RotateRight[ Abs[Fourier[ 1 A mag i/ nnn ff @@ Transpose[ Outer[List, Range[-d, d, \[Delta]], Range[-d, d, \[Delta]]], {2, 3, 1}]]], {nn, nn}], Magnification -> 1]  • Those images are incredibly beautiful, but can you explain what exactly they represent and how they were generated? "Fourier transform of the light intensity due to a diffraction pattern caused by light going through 8 pinholes and interfering on a wall" isn't very clear for me. Apr 7, 2014 at 9:11 • "Visually stunning math concepts which are easy to explain" – user856 Apr 8, 2014 at 1:06 • @Rahul: It's aliasing of a circle. Aliasing is easy to explain. Draw a big circle on a clear plastic sheet. Cut the image into little squares. Stack the squares on top of each other, and look at it. That's the image. The different images above were done using little squares with various side-lengths. I can post the code if you'd like, there are literally tens of thousands of visually distinct diagrams which can be formed. Apr 8, 2014 at 1:17 • Could you post a link to the code here? Apr 9, 2014 at 18:46 • > "Here is a video of the first couple thosand patterns: youtube.com/watch?v=1UVbUWuyNmk"  Unless you're referring to the finite number of frames in the video, the continuous nature of the real numbers that the radius of the circle comes from, means that video shows the first "several" uncountably infinite patterns. ;-) Apr 13, 2014 at 18:54 Francis Galton's Bean machine is interesting as it demonstrates Central Limit Theorem : • It's one of the best answers. Mar 16, 2016 at 3:42 • it probably demonstrates it... Aug 11, 2016 at 5:41 • That's why the grand prize is always at the corner Oct 20, 2021 at 7:06 A very satisfying visualization of the area of a circle.  • I like this, but one tiny criticism is that it's not visually obvious why the unrolled rings should form a triangle. It's obvious if you can see that the unrolled length of the rings is linearly proportional to their radii, but not visually obvious. Sep 1, 2014 at 10:52 • Is it obvious that you can straighten a ring out and get a rectangle when the ring started out bent? I think a lot of these "bang! pi r squared!" ones leave you with real analysis-induced heebie jeebies :) Sep 26, 2014 at 0:27 • Kalid Azad and BE are quite amazing. Apr 15, 2020 at 3:17 Math is always fun to learn. Here are some of the images that explain some things beautifully visually     • In E-M wave propagation, I thought the E and M components were supposed to be$90$degrees out of phase. The figure appears to have them in phase or$180$degrees out of phase, depending on how you interpret the directions of the axes. Aug 28, 2014 at 21:25 • @DavidK That's only for circularly polarized photons. This is illustrating a plane-polarized wave, whose components are in phase but offset$90^\circ$. Jan 9, 2015 at 1:42 • When I read the number 142857 it was like a dejavu for me and I was like "OH MY GOED!! I'VE KNOWN THIS NUMBER!" Jan 11, 2015 at 4:20 • Magic 9 hiding on it. 142857 -> (1 + 4 + 2 + 8 + 5 + 7 = 27); ( 27 -> 2 + 7 = 9 ) Mar 16, 2016 at 10:12 • @NateGlenn It's really just a visualisation of long multiplication. (Try writing out the long multiplication of$13 \cdot 12$and see how it corresponds to the diagram.) Mar 16, 2016 at 15:57 I do not know if this meets your criteria of "visually stunning", but nonetheless - I like this proof of Pythagoras' Theorem (image taken from www.wisfaq.nl): The key to understanding this is to realize that the inner quadrilateral must be a square - the sides are equal in length (obviously) and each of its angles is$90^{\circ}$because the two angles on either side sum to$90^{\circ}$, and the sum of the three angles is$180^{\circ}$. The area of this square is$c^2$. The outer square's area is$(a + b)^2$, which is$c^2$plus$2 a b$which is the total area of the four triangles, each of area$\frac{1}{2} a b$.$(a + b)^2 = c^2 + 2 a ba^2 + b^2 + 2 a b = c^2 + 2 a ba^2 + b^2 = c^2\$, which is Pythagoras' theorem.

• +1 finally i can prove Pythagoras theorem without looking into internet or books :) Jul 31, 2014 at 20:58
• yes it is the most common proof, the easiest to remember Apr 22, 2016 at 21:13