# 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 '14 at 12:59
• @ColeJohnson the 'transcendentality" is the beauty of it! – Guy Apr 4 '14 at 17:07
• There is a considerable overlap with mathoverflow.net/questions/8846/proofs-without-words – Martin Brandenburg Jun 20 '14 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 '14 at 18:37

One of my favourites is from Littlewood's Miscellany, where he amicably mentions that "for the professional the only proof needed" for the one-dimensional fixed point theorem is the following figure. The theorem is:

If $f:[0,1]\rightarrow [0,1]$ is continuous and increasing then, under iteration of $f$, every point is either a fixed point or else converges to a fixed point. Check out the "Proofs Without Words" gallery (animated) here:

http://usamts.org/Gallery/G_Gallery.php

And the related proofs here:

http://www.artofproblemsolving.com/Wiki/index.php/Proofs_without_words

Many of these are similar to the ones already listed here, but you get a bunch in one place.

The fact that the graph of inverse of a function is nothing more than its image in line $y=x$ but still finding inverse is so difficult is a math concept I really find amazing. Also inverse of some functions have special name and are really special and useful.

In group theory, "visual" explanation of the group $D_4$ (or sometimes called $D_8$, which is dihedral group of order $8$ or degree $4$) was really exciting to me. Even with some elementary knowledge about groups, if someone defines $D_4$ as $$\langle x, a\ |\ a^4 = x^2 = e, axa = x \rangle$$ it might seem meaningless or too abstract to understand. When I first saw this, I simply thought it as something to memorize. But when I saw the visual explanation, it was a "stunning" moment for me because it was really easy to understand and there was nothing to memorize at all:

First let us take a square $S$ with vertices named as $A, B,C, D$. Then, if we let the element $a$ rotate this square $90^\circ$ clockwise direction and let the element $x$ flip the square through the first diagonal, i.e. $x=y$ line, we can have all $8$ elements of $D_4$ as the following: $e$: Identity element. It does nothing to $S$.

$a$: Rotates $S$, $90^\circ$ clockwise direction.

$a^2$: Rotates $S$, $90^\circ$ clockwise direction twice, i.e., rotates $S$, $180^\circ$ clockwise direction.

$a^3$: Rotates $S$, $90^\circ$ clockwise direction three times, i.e. rotates $S$, $270^\circ$ clockwise direction.

$x$: Flips $S$ through its first diagonal, i.e., interchanges $A$ and $C$.

$ax$: Flips $S$ through its first diagonal first, then rotates the flipped square $90^\circ$ clockwise direction.

$a^2x$: Flips $S$ through its first diagonal first, then rotates the flipped square $180^\circ$ clockwise direction.

$a^3x$: Flips $S$ through its first diagonal first, then rotates the flipped square $270^\circ$ clockwise direction.

Here, only thing we need to be careful is that rightmost function is applied first (for example, $ax$ means: first flip, then rotate). Now, we can verify the properties that are mentioned in the definition:

First of all, we have $a^4 = e$ because $a^4$ rotates $S$, $90^\circ$ clockwise direction four times, which means $360^\circ$ clockwise direction, which doesn't change the place of any vertex. So it is as same as the identity element $e$.

Secondly, we have $x^2 = e$ because if we flip $S$ through its first diagonal (interchanging $A$ and $C$) and then flip it again, we get $S$ again, which also corresponds to identity element $e$.

Finally, we have $axa = x$, which can be verified by rotating $S$ first, then flipping it and rotating it again. In the end what we get is as same as flipping $S$, which is done by $x$.

Riemann integration has always amazed me. Its simple yet extraordinary. Transpose of a matrix column, this gift shows the easiest proof ever made In plane geometry Morley’s theorem is a stunning fact in my opinion: In any triangle, the points of intersections of adjacent trisectors of the angles form an equilateral triangle : In analytical geometry: The generalization of triangles or tetrahedra in n-dimensions is simpleces. And the formula for the simplex volume is a beauty, for example, the volume of four-dimensional simplex which is called pentachoron, pentatope or 5-cell (using the coordinates of its vertices): $$\text{Four dimensional volume} = \pm\frac{1}{4!}\;\begin{vmatrix} \;x_1-x_5 && y_1-y_5 && z_1-z_5 && w_1-w_5\;\\ \;x_2-x_5 && y_2-y_5 && z_2-z_5 && w_2-w_5\;\\ \;x_3-x_5 && y_3-y_5 && z_3-z_5 && w_3-w_5\;\\ \;x_4-x_5 && y_4-y_5 && z_4-z_5 && w_4-w_5\;\\ \end{vmatrix}$$ In the case of triangles we get $$A=±\frac{1}{2}\;\begin{vmatrix} \;x_1-x_3 && y_1-y_3\;\\ \;x_2-x_3 && y_2-y_3\;\\ \end{vmatrix}$$

Of course, we can write it even shorter if we use vectors.

And 4-dimensional spheres are truly amazing, as are tesseracts

And some things are truly arcane (Ramanujan summation formula): $$1+2+3+4+5\,+\,...= -\frac{1}{12}$$ You can get a bit of information about it in this and this Wikipedia articles, but divergent series like this (please refer to Wikipedia again) are not an elementary topic to easily understand. (It's usually met with misunderstanding and downvotes. It's currently 'attacked' in Wikipedia because it is not understood by lay mathematicians. The simplest way to get at least some sort of idea is probably to treat it (it's not completely arbitrary but actually one of the most beautiful concepts in math) as an abstraction, something abstract like, say, square roots of negative numbers. That's a bad comparison, but Ramanujan's idea is slightly tougher than pentachorons or tesseracts. It's not arbitrary assigning $$-1/12\,$$ to the zeta function. This arcane formula found its way into physics (Casimir force). Euler is considered the first to derive this formula more than two hundred years ago.

As to calculus, my vote would go to the beauty of Euler’s formula already posted in this thread ($$\,\boldsymbol{e^{ix}=\cos{x}+i\sin{x}}\,$$)

• **WARNING!** $$-\frac1{12}=\zeta(-1)\ne1+2+3+4+\ldots$$ – Simply Beautiful Art Jul 2 '17 at 22:52
• I would remove my downvote if concerns in the previous comment was addressed. – Frenzy Li Nov 13 '17 at 18:16

$$\sum_{i=1}^{\infty}\frac{1}{x^n}=\frac{1}{x-1}$$

In base x this sum equels to 0.1111111.... and if you multiply it by x-1 you get 0.(x-1)(x-1)(x-1).... which equels to 1.

An important concept im math is $$\infty$$: the Bernoulli's lemniscate is very similar to its sign. Here the GIF of its construction from an hyperbole:

The equation is very simple:$$\left ( x^2+y^2 \right )^2=2a^2(x^2-y^2)$$

Take a times table with only 8 rows and 8 columns where for for each positive integers m and n from 1 to 8, the entry in the $$m^{th}$$ row and the $$n^{th}$$ column is the fourth last binary digit of $$(2m - 1) \times (2n - 1)$$ represented as a black square when it's 1 and as a white square when it's a 0. Then you get the following image. All the numbers from 1 to 10 are a factor of $$71^2 - 1$$. This can be explained as follows. $$8 \times 9 = 72$$ so $$7 \times 10$$ must be 2 less than it which is 70. 5 is half of 10 but 5 is also 2 less than 7. 70 can also be expressed as $$2 \times 5 \times 7$$. Now $$2 \times 6 \times 6$$ must be 2 more than that which is 72. Now 4 is half of 8 so it must be a factor of 72 and 3 is half of 6 so it must be a factor of 72. I learned from the YouTube video https://www.youtube.com/watch?v=2JM2oImb9Qg that $$7! = 71^2 - 1$$ and I just verified the accuracy of that statement.

I don't know the explanation for what I have to say next but I suspect there is one. $$71^2 = 3 \times 41^2 - 2$$ and $$41^2 = 2 \times 29^2 - 1$$. In addition to that, all the numbers from 1 to 7 are a factor of $$29^2 - 1$$ and all the numbers from 1 to 8 are a factor of $$41^2 - 1$$.