IRA FLATOW: This is Science Friday. I’m Ira Flatow. How well do you recall your high school geometry? I loved high school geometry. Mr. Caballero was a great, great math teacher.

Given isosceles triangle ABC where side AB is equal to side BD, then angle A is also equal to angle C. Of course, you remember that. It’s not something your math teacher invented.

It’s a proof written down in 300 BCE by the Greek mathematician Euclid of Alexandria. You know that. He’s also the founder of geometry. Euclid’s pie in the sky vision of an ordered and methodical universe ruled by geometric equations– it struggled to catch on for centuries until Renaissance painters and French monarchs found a way to connect the ancient science of geometry to the real world.

And the story of how geometry went from a philosophical concept to a system for designing cities to a staple of high school mathematics is carefully laid out like a geometrical proof in the new book Proof! How the World Became Geometrical written by my next guest, scientist-historian Amir Alexander. Welcome to Science Friday.

AMIR ALEXANDER: Thank you so much for having me on. Yeah, I’ve been a listener for many years, so it’s great to be on.

IRA FLATOW: Well, great. Yeah, you know, I never really realized the great history of geometry. I knew the math. I loved the math. And you say in your book that geometry was only discovered once in the course of human history. What does that mean?

AMIR ALEXANDER: Well, that means that they were actually there are different great mathematical traditions in the ancient world. There were traditions. In Babylon, they developed forms of algebra.

In Egypt, they used math for measurement, but there were only one. They’re only in one place that anyone have the idea that mathematics was not just a tool for counting, measuring, or looking at astronomical observations but actually, simply a way of finding absolute truth. Mathematics can lead us to something that is absolutely irrevocably, irrefutably true. And that happened in the Greek world and one of the Greek cities that dotted the shores of the Mediterranean in the 5th century BC. And we don’t know.

We don’t know when exactly it happened and what exactly that first proof was. But we do know that it was the very first time. It was the only time that was discovered. And that, as I argue in my book, it really changed the world in a profound way.

IRA FLATOW: And you say that Euclid was the one who actually– I’m going to quote you. “Euclid is likely the most influential mathematician who ever lived.” I mean, there have been a lot of mathematicians.

AMIR ALEXANDER: That is true, but I think that I think there is a good argument to be made that there was no one more influential than Euclid because what Euclid did was really remarkable. He took those already existing proof that were sort of a disjointed array of different theorems about lines and angles and circles and so on, things that we know from geometry today, and he turned them into not just a single set of different truths. He turned them into a whole world– a pure, abstract geometrical world in which you start out with a set of postulates, and then you deduce step by logical step. You deduce absolute necessary truths that are absolutely incontestable.

And not only that, but they’re all dependent on each other, and first level proofs are then the basis for higher level proofs have been the base a higher level proofs. And all of these truths that are incontestable are interrelated. Plato lived before Euclid, but he saw what the project was.

And he saw it as an absolute beautiful world of absolute interconnected truths that there was nothing like it. That’s where truth was. The world around us, as the Greeks thought, was chaotic, was changing was transitory, was unreliable, but geometry, that gives you a world of absolute incontestable truths.

IRA FLATOW: But you also say that the Greek geometers we’re not looking for any practical applications of geometry but for truth for the sake of truth, and then along comes Euclid and changes that whole thing.

AMIR ALEXANDER: Well, Euclid did not change that. He was also he created a world, an entire world of truths of intricate, of interconnected truth, something that is the closest thing to Plato always thought that there is a world of the forums at which the perfect truth all reside all reside that is better and more pure and more beautiful than our very corrupt and transitory world. Euclid claimed the closest to actually creating that world, making it real, an interconnected world of absolute truth. It is all interconnected.

Everything has its place. Everything has its perfect relationship to other truth. Everything is perfectly known, and everything is also hierarchical because it starts from those general statements postulate. And then it goes step by step by step by step to higher and higher levels.

IRA FLATOW: But you also say that a while they were all interested philosophically in these truths. It was the artists that brought math back down to earth.

AMIR ALEXANDER: Exactly right, exactly right. Yes, what happened was that first that the Greeks built– they were very impressed with Euclid’s accomplishment, but they didn’t believe it described the real world. Our world is messy, too messy, changing, transitory. It’s nothing like that perfect world, the world of geometry.

And that was also the case in the Middle Ages when the church thought that the world is corrupt. It has fallen. It can’t be described by those perfect truths.

And what happened in a particular place at a particular time in Florence around the first half of the 15th century, the 1400s, a group of men that we know well, they took those concepts of geometry and showed that geometry is not just up in the sky. It’s not just up there in the abstraction. Geometry pervades our world– that the world space or the space that we live in itself is geometrical. And they invented the science of perspective. That was really that was really a turning point because that’s saying–

IRA FLATOW: Yeah, I want to get into that. Tell us how that was discovered. You had you describe a really fascinating scene using a mirror and a hole punched. You describe it, how is this whole fascinating idea came about.

AMIR ALEXANDER: Yes, the experiment the famous experiment the demonstrated the principles of linear perspective. That was accomplished by the Florentine, Filippo Brunelleschi, who is also famous for building that beautiful giant dome over the Duomo in Florence. But years before he built the dome, he conducted experiments in perspective. And he stood before in the front gate of the Duomo, the Cathedral in Florence, looking at that octagonal structure– that famous octagonal Baptistery of St. John in Florence, just across the square, about maybe 100 feet apart.

And in his hand, he had a painting. He had a painting. And the painting was not one of what you’d expect, like a Madonna and Child or the traditional thing.

The painting was, in fact, a picture of what he was seeing in his actual life. That is, it was a picture of the Baptistery of St. John. And he looked through a hole from the back of the painting.

And in front of the painting, in the direction of the Baptistery, he held a mirror. And because he drew the painting according to the geometrical principles of perspective, what he could see in the mirror was also a three-dimensional picture of the Baptistery. And then he removed the mirror and looked at the Baptistery.

And his hope– what he was trying to accomplish– was, it would look exactly the same. Why exactly the same? Because on a flat surface, he had managed to create a three-dimensional image of the Baptistery.

So it was really a turning point, not because this was just a nice trick for painters, which it also was a practical thing, but because he showed that you can actually recreate the geometry of space itself, that space itself is structured by geometrical principles, and that every point in space can be defined geometrically. And that was really the first time that was done– geometry brought down from the sky.

And now, suddenly, people started looking. It’s all around us. It’s in the geometry of space. Where else is geometry?

IRA FLATOW: He sort of did the first virtual reality experiment.

AMIR ALEXANDER: I guess so. Yeah, yeah.

IRA FLATOW: But I want to move on to that next point that you make, which you say is very important, because geometry then became a symbol of power. Right?

AMIR ALEXANDER: It did indeed, because think about it. What did Euclid do? Euclid created a world that was perfect. Every single thing in that world had its precise place. It was exactly true.

It has an exact proper relationship to other things. And it was perfectly hierarchical. And if our world is, in fact, like that, well, that tells us some things about space. That tells us some things about science. All true.

But it also tells us some things about ourselves and what kind of world and what kind of order human beings should be living in. And those who first realized the enormous political power of geometry were the Kings of France. And they tried to create what you could really call Euclid’s kingdom– a monarchy that is perfectly ordered, in which everything has its exact proper place.

It is true. It is irrevocable. And it is hierarchical. And it is undeniable.

Who could argue against– who could try to overthrow a king that rules not just because he has a big army? He rules because he is an expression of the eternal laws of geometry– the deepest order of the universe.

IRA FLATOW: So you create Versailles, which is the symbol of that.

AMIR ALEXANDER: Exactly. Versailles, it was a long process. It started quite humbly by a French king called Charles VIII, who went with a– at the head of the great army, he went to Italy and brought back a few gardeners– geometrical gardeners from Italy. And over 200 years, the French kings identified the monarchy closer and closer with geometrical order, and especially with their geometrical gardens. And no garden was as spectacular, as powerful, as the geometrical gardens of Versailles.

IRA FLATOW: Very interesting. I’m Ira Flatow. This is Science Friday from WNYC Studios, talking about geometry with Amir Alexander, author of Proof– How the World Became Geometrical. And of course, keeping up with this theme of power, if you weren’t going to build a garden, you could build a whole city that was geometrically sound.

AMIR ALEXANDER: Indeed, indeed. So when you go to Versailles, which is a great garden, but also a capital of a great nation of France, you walk those geometrical path, those straight lines, all leading uphill to the royal palace. And you learn that the rule of the king of France is based on those geometrical principles, because you see them all around you. It tells you that the deep order of the world points at the king’s palace.

But like you said, it’s not just gardens. It is cities, as well. And there are quite a few examples of cities that were designed according to geometrical principles.

But none is more, I think, more famous, perhaps even more beautiful and more astounding, than our very own capital of Washington DC, which was designed by a Frenchman– a Frenchman who grew up in the gardens of Louis XV and Louis XVI and knew Versailles well.

IRA FLATOW: L’Enfant.

AMIR ALEXANDER: Yeah, L’Enfant– yeah, yeah. Pierre Charles L’Enfant. And he brought the principles– the geometrical principles– of Versailles to America. But L’Enfant, of course, America is not a monarchy. It was everything but.

Of course, it was created as a rebuke to all monarchy. But he– in Washington, DC, he managed to use the geometrical principles that he had learned at Versailles and present them in the geometrical order, as a Republican geometrical order. And that’s really what Washington, DC is. It is, in a way, the Constitution. The streets of the city are the Constitution set geometrically.

You see, if you look– for example, if you stand on the Mall, looking up at the houses of Congress on Capitol Hill, that is Versailles. That is Versailles, the palace on the hill. All roads lead to it. Those straight geometrical roads pointing at the palace, telling you that this is the fixed, eternal, unchanging order, looking up at the palace– in this case, the palace of the people.

But in Washington, DC, unlike in Versailles, that is not the end of the story because there’s also other centers of power. So you have what L’Enfant called the President’s Palace, and we know as the White House. He wanted it probably be grander than it actually is. He wanted it something on the same scale as the house of Congress. But again, a palace, a great house on the hill with a garden at a right angle to the mall, leading up to it– and again, if you look at it alone, it also reflects Versailles.

But in Washington DC, those are two poles of power. They intersect at right angles. It was in the Washington Monument. And they are connected by Pennsylvania Avenue in this intricate balance of rivalry and cooperation, two centers of power.

IRA FLATOW: Instead of one, which would be at Versailles and the king, we have a balance of power. And this is just fascinating, Amir. Reading the book about geometry and then seeing how it is applied to gardens and cities, who would have thunk this is how geometry branched out?

I want to thank you for taking time to be with us. It’s a fascinating book. It’s called Proof– How the World Became Geometrical. Amir Alexander, author of the new book, he teaches the history of science at UCLA. Good luck with the book. It’s a great read. Thank you for taking time to be with us today.

AMIR ALEXANDER: Thank you so much for inviting me.

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