# 记录一次Stack上关于“数学之美”的brainstorm

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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 ?/2π/2 radians

My favorite: tell someone that

∑?=1∞12?=1∑n=1∞12n=1

and they probably won't believe you. However, show them the below:

and suddenly what had been obscure is now obvious.

or:

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.

Here is a classic: the sum of the first ?n positive odd numbers =?2=n2.

We also see that the sum of the first ?n positive even numbers =?(?+1)=n(n+1) (excluding 00), by adding a column to the left.

A well-known visual to explain (?+?)2=?2+2??+?2(a+b)2=a2+2ab+b2:

The sum of the exterior angles of any convex polygon will always add up to 360∘360∘.

This can be viewed as a zooming out process, as illustrate by the animation below:

This is a neat little proof that the area of a circle is ??2πr2, 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 ??πr, giving an area for the rectangle of ??2πr2.

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.

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 ℝ2R2 to ℝ2R2, a function from (ℝ2)2(R2)2 to ℝR that satisfies the following properties.

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

Suppose a function ?d from (ℝ2)2(R2)2 to ℝR satisfies those conditions, then for any point (?,?)(x,y), ?((0,0),(?,?))2=?((0,0),(?,?))?((0,0),(?,?))=?((0,0),(?,?))?((0,0),(?,−?))=?((0,0),(?2+?2,0))=?2+?2d((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),(x2+y2,0))=x2+y2 so ?((0,0),(?,?))=?2+?2‾‾‾‾‾‾‾√d((0,0),(x,y))=x2+y2 so for any points (?,?)(x,y) and (?,?)(z,w), ?((?,?),(?,?))=(?−?)2+(?−?)2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√d((x,y),(z,w))=(z−x)2+(w−y)2 Now I will show that ?((?,?),(?,?))=(?−?)2+(?−?)2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√d((x,y),(z,w))=(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), ?((0,0),(??−??,??+??))=(??−??)2+(??+??)2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√=?2?2−2????+?2?2+?2?2+2????+?2?2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√=?2?2+?2?2+?2?2+?2?2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√=(?2+?2)(?2+?2)‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√=?2+?2‾‾‾‾‾‾‾√?2+?2‾‾‾‾‾‾‾‾√=?((0,0),(?,?))?((0,0),(?,?))d((0,0),(xz−yw,xw+yz))=(xz−yw)2+(xw+yz)2=x2z2−2xyzw+y2w2+x2w2+2xyzw+y2z2=x2z2+x2w2+y2z2+y2w2=(x2+y2)(z2+w2)=x2+y2z2+w2=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 (?−?)2+(?−?)2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√(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 ?((0,0),(?,?))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 ?∈ℝ+x∈R+ and ?∈ℝ+y∈R+, the area of a square with the hypotenuse as one of its edges is (?−?)2+2??=?2+?2=(?(?,?))2(x−y)2+2xy=x2+y2=(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:

• The validity of the proofs of the Pythagorean Theorem and the concept of area
• https://www.maa.org/press/periodicals/convergence/proportionality-in-similar-triangles-a-cross-cultural-comparison-the-student-module
• https://thept.weebly.com/the-theorem.html

A visual explanation of a Taylor series:

?(0)+?′(0)1!?+?″(0)2!?2+?(3)(0)3!?3+⋯f(0)+f′(0)1!x+f″(0)2!x2+f(3)(0)3!x3+⋯

or

?(?)+?′(?)1!(?−?)+?″(?)2!(?−?)2+?(3)(?)3!(?−?)3+⋯

When you think about it, it's quite beautiful that as you add each term it wraps around the curve.

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, I can even tell you the name of 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...”

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.

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.

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:

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.

"3Blue1Brown", 这个频道用最直观的物理的方式描述了数学中许多的经典问题,比如为什么"直线的本质是圆?", 随便看几期视频就能感受到数学的美妙, 困扰多年的问题也许就迎刃而解了.

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