# The World of Mathematical Reality

This always blows my mind that this is true. If you take any four-sided shape at all, make it as awkward and as ridiculous as you want. Four-sided shape, any. If you take the middles of the sides… and connect them… it always makes a parallelogram. Always. In other words, these two lines are always in the same direction, and these two are always in the same direction. No matter what crazy, kooky thing you started with. That’s scary, to me! That’s a conspiracy. That’s amazing. That’s completely unexpected. I would’ve expected you make some crazy blob and connect the middles, it’s gonna be another crazy blob. But it isn’t. It’s always a slanted box, beautifully parallel. WHY is it that? That’s not a scientific question. The scientific question would be about ink and distances roughly approximate, da da da, test tubes and all that. The mathematical question is “why?”, its always “why?”. And the only way we know how to answer such questions is to come up, from scratch, with these… narrative arguments that explain. And you can have more than one explanation. You can compare them, you can critique them, you can contrast them. Just like you can poems. And so for mathematicians our aesthetic is: your argument, your explanation, your discovery, your deduction: it needs to make sense. It can’t be wishful thinking, it can’t be pie-in-the-sky, it can’t just be what you want. It has to follow logically. Which is a tough demand. So your reasoning needs to be tight and correct. But, more importantly, really, is that your argument has to be beautiful. It has to be astounding. It has to be revelatory. It has to be charming. That’s really hard to do. So, what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod… known as problems. And, how we can possibly… craft these elegant reason poems to get at what we want to get at. And it may not be everyone’s cup of tea. Certainly I’m not suggesting, or even necessarily wanting it to be that everyone becomes a mathematician. Just like I don’t think everyone wants to be a photographer or a ballet dancer. But I think it IS important that everyone understand that there IS an art of photography, there IS an art of ballet, there IS an art of mathematics, and those things have their own challenges, frustrations, joys, etc. And that one can then choose to integrate such arts into one’s life to the extent that one wishes to. There’s nothing deeper than this mathematical act. There’s nothing that comes even close.

Wow! thanks. I love this. I teach math and am definitely going to check out your books. Students usually just want the 'how'; not the 'why'. But the 'why' always stays with you.

any two intersecting lines will form angles that add up to 360deg and the opposing angles will be of same degs. Process threw this one more time to make the sides of your inscribed quadrilateralsand threw the resulting symmetry the resulting figure has to be a parallelogram

eurrghhhh, so american.

Worthless comment.

Clearly you have no understanding of what math is.

All art can be studied through a mathematical lens, and its beauty often arises from unseen mathematical relationships that we experience but don't necessarily understand.

Some art is seen as beautiful because it breaks our expectation for those mathematical relationships.

Claiming mathematics is about crunching numbers and combining rules in novel ways, is like claiming art is about producing ink and combining photoshop filters.

The amount of ambiguous and abstract poetry in his speech made me lose interest. He's expressing wonder, which is great and all, but in the end he isn't really talking about anything. There is nothing to actually learn from his speech. This feels too much like a glamor piece.

No, it just went over your head.

Can yall stop arguing whether math is art or not with this guy? Whether something is 'art' is completely subjective to the individual, and clearly this guy doesn't appreciate the aesthetics of mathematics whatsoever…so let him be

Please stop expressing your opinion as fact. You say math isn't art, but to some people it is. Math gives us a way to explain the happenings in our universe and some people find that beautiful, hence math is art. You undermined your own argument at "Bleck."

Pardon us for the persecution, but you've touched on something that we all feel passionately about. That said, it seems like you're arguing from an incomplete view of mathematics. Theorems can be built in more than just the mathematical deduction you're used to. If you're not completely turned off by the subject I would recommend looking up what some of the open problems of mathematics are and how solutions to some have been found. Or just google "beautiful proofs".

One further point; math isn't always a purposeful study. Like Dr. Lockhart just demonstrated math can sometimes just be pattern searching. Yes, sometimes these patterns can turn out to be useful, but that's not the point. The point is to find something interesting or "beautiful" and if the expression of beautiful things, regardless of its media, isn't "art" then I'm not sure if they've been careful with their definitions.

I'm pretty sure you've never "tasted" mathematics.

Sorry for the misunderstanding but I meant that at the guy you were talking about, not the person in this video. He was saying how math wasn't art and I was replying that, that guy didn't understand art as math is art.

You totally misunderstood what I was replying to. ACaffeineAddict was replying to someone who said math wasn't art and I was replying to ACaffeineAddict's comment to that guy, not the guy in the video. At the time it made sense but the top comment changed too quickly so now it looks like I'm dissing math.

Hello there, I'm a mathematics student and have been thinking about this same issue for a while. I'm seriously considering making a long term project of completely re-writing the curriculum for uk mathematics. The more you study maths, the more you realise the creative mindset necessary for real mathematics could be taught and integrated into study from as young as year 1.

What is "art" to you, then? Drawing pictures? Making sculptures? Your interpretation of what is beautiful is not the same as everybody else's. Do you know what makes art what it is? People use the word "art" to describe it. That's it. The logic and elegance behind the proofs I've seen IS art to me.

>Saying otherwise diminishes the value and expression of real art in our culture.

No, that just allows for a broader definition of what is "art."

Go for it!

Again, you have a limited definition of what is "art."

You're right in that if you don't "follow the rules" you produce garbage, but the method at which someone makes a proof or thinks about a certain problem is pure creativity. The beauty is in the fact that you didn't botch something up! Pure mathematics is a subject that takes the most amount of creativity that I've ever seen.

Consider a concave 4 sided figure

Great!… That was the very same property I showed in my first video… š

9.42 is not a real number , it's a wannabe number that should be trying harder.

I like math

to which I would reply, what's your point? your statement doesn't negate the fact that there is an art to math.

Read Jo Boaler's "What's Math Got To Do With It?" She's also teaching a free course online right now on the Stanford website. š

well said. lockhart's lament, anyone?

worrydream dotcom/refs/Lockhart-MathematiciansLament dot pdf

Now imagine slowly untightening the side from a point, by moving one of the corners, to make a four-sided shape. Then it is clear that both of the centre points inscribed in the four sided shape, that are moving, are moving at half the velocity at which the corner is moving. Since they are moving at the same velocity, the lines of the inscribed shape remain parallel, and you have a parallelogram.

postimg.org/image/5xhizg2vj/

Spoiler alert, my solution to the parallelogram question.

Imagine shrinking one of the sides of the four-sided shape, to a point. That gives you a triangle, with two of the sides of the inscribed shape flush with the edges of the triangle. Then it is easy to show that the inscribed shape is a parallelogram by an argument involving similar triangles.

How do you think about mathematical questions and puzzles?

The 4 sides of the inner shape are parallel to the diagonals of the shape.

Suggestion:

You should cap your marker and stop inhaling its fumes before mind blown.

Paul-ie, although the respect I have for you about some of the things you've said in Lochart's lament, it is obvious to me (as I previously doubted) that you're weak in math conceptual understanding. The fact that you mention in the video can amaze a layperson, but not a mathematician. There is a simple explanation for the question you asked. The joint midpoints of any quadrilateral form a parallelogram because any such quadrilateral has only 2 diagonals. And then additionally because in any triangle, the line that joins the midpoints of two sides is parallel with the third side of the triangle. That's all. Mystery solved!

It's because the world is flat

Iām reading his book measurement, one of the best math books I read

Paul is a great teacher. He taught me how to play Go.

I love how excited he is

Iām following your work for some time now and itās amazing how the Lament put everything I thought but couldnāt say, in perfect words.

My question is this. You have this amazing statement how we should teach kids math and though Iām using your examples (wonderfully I must add), I donāt have much to work with. Please write a book ālament – how to, with examplesā.

I wish that someone would do for physics and chemistry what Lockhart has done for mathematics. High School texts on physical science, physics, and chemistry are just awful.

I love Paul Lockhart. He is a great example for any budding mathematician.

Unfortunately, I wasn't aware of his work during my school days. This is a gem that I found.

I feel blessed…

May God bless you, Sir. You are an inspiration to many…

Every mathematics curriculum should include Paul's books as compulsory reading. I can say for sure that there will be more math lovers than any in history.

'Mathematics is an art like music and painting'. Bravo, Sir. Thank you for those wonderful words.

Dear respected sir Paul Lock heart,

I have seen 3blue & one brown's videos on surface area of sphere and I was very much astounded by the interpretation. It's simply genius.

While on the other video where he proves that slicing the cone with slope less than that of the slope of sides of cone results in eliptical cross section and it is anologous to the elips drawn by keeping thumb tacts at each foci, in this demonstration he had mentioned your name so Keeping that in mind please help me reveal the secrets of conics. I have read many books but it did not satisfy me. I want to know why distance from foci to any point on parabolla is equal to the distance from the diretrix to that point. If focus is a light source and has a parabolic reflector then why reflected beams are parallel to each other , is this phenomena have anything to do with equidistance part. All the books I have read takes this this things for granted stating that it's just a property of parabolla. Even in ellipse how eccentricity has two definition and how they are co related. Kindly shed lights on my taught.

Regards,

Victor Maxwell peters

I seems the reason it's a parallelogram is because there's always two pairs of lines, each with some degree of non-parallel-ness (divergence/convergence). And each par is connected to the other pair, in the same way, with straight lines, in a 2D space (paper) therefore each pair must have a measure of convergence/divergence that is 180Ā°.

Why? Because if you cross two straight lines on a piece of paper the total of the angles of those two lines must be 360Ā°.

In other words, cross two lines. If one angle is theta then the adjacent angle is 180Ā°-theta. So when you draw two lines that aren't crossed and then two other lines that aren't crossed the corners, the connection, is where they cross.

Each pair of lines is a 180Ā° complement to the other pair … their 'convergence' is going to be proportion to other pair of lines making them average, become parallel … a parallelogram.