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?
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123 Next $\begingroup$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.
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.
$\endgroup$ 17 $\begingroup$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.
$\endgroup$ 13 $\begingroup$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.
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:
A well-known visual to explain $(a+b)^2 = a^2+2ab+b^2$:
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:
Multiplication of the integers modulo 15:
Multiplication of the integers modulo 512:
Simple answer for "what is a radian":
Logarithmic spiral and scale:
When I understood Fourier series visually-
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:
$\endgroup$ 12 $\begingroup$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.
$\endgroup$ 15 $\begingroup$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.
$\endgroup$ 5 $\begingroup$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...”
$\endgroup$ 27 $\begingroup$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.
$\endgroup$ 4 $\begingroup$How about a line integral of a scalar field by :
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...
Similarly to eykanal's answer, although demonstrating some interesting facts about medians and geometry as well. It demonstrates that $\displaystyle\sum_{n = 1}^{\infty}\frac{1}{2^n} = 1$:
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.
$\endgroup$ 4 $\begingroup$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:
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.
The magnetic pendulum:
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.
$\endgroup$ 5 $\begingroup$Simple,visual proof of the Pythagorean theorem. Originally from Pythagorean Theorem Proof Without Words 6).
Here's a GIF that I made that demonstrates Phi (golden number)
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.
Take a look at this great example of Fourier series visualizations written in JavaScript.
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:
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]Francis Galton's Bean machine is interesting as it demonstrates Central Limit Theorem :
A very satisfying visualization of the area of a circle.
Math is always fun to learn. Here are some of the images that explain some things beautifully visually
One of my favorites - I've seen it somewhere on the web but can't find it again now, so had to reconstruct myself. It is not as pretty but suffices to convey the idea.
It gives good grasp both for $e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n$ and for $e^{2k\pi i}=1$
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