C (2:53)

That's correct. As a matter of fact, what we did here, which I believe is not very popular, is we've included quite a few women mathematicians who heretofore have been pretty much ignored, as they were during their lifetime. And their stories are also quite unusual, as we'll find out as we go through some of them.

B (3:18)

Yeah, and that's one of the things that I appreciated about your book. You have, you know, you not only have women, but you have people from other cultures. And we'll get to them, and we'll get to them in time. But one of the problems that you must have faced was it must have been incredibly hard to decide who's in and who's out. Of course, there's a certain amount of subjectivity in any top 50 list. But did you have criteria to help you decide?

C (3:43)

Well, actually, we made a decision first. We had quite a few more than the 50, and then we had to narrow it down to the required 50 at the publisher. Accepted. And as it is, the book is quite long, which is good. But what we did was to take those mathematicians who had an effect on the mathematics that we learn today, rather than just to go after those folks who came up with these very difficult to follow areas of mathematics that the general readership wouldn't understand. There are certainly some of those there. But what we tried to do is to talk about their life, their lifestyle, the unusualness of their life. And I can tell you, just about every single one of them was unusual. Because if I could be a little blunt about it, anybody who is as brilliant as those folks that we have selected for this cannot have lived a normal social life as we, as most people know it, because you can't think the way a normal person thinks and still come up with these outrageously incredible discoveries. So we began with Thales, who is probably responsible in most part for the geometry that we talk about in the high school course. Things at vertical angles and how deal with parallel lines and so on. But more importantly, there's what's known as Thales Theorem, which we don't refer to it as such here, but in Europe, almost all the countries refer to Thales Theorem. The triangle that is inscribed in a circle where one side is the diameter and the angle opposite is a right angle. That's known as Thales Theorem. And it's a very simple thing, but it's basic to the study of geometry.

B (5:47)

I'm sorry, go ahead.

C (5:48)

We then go to Pythagoras and everybody knows about the Pythagorean theorem, but not so much about the person or about the cult that he led. And you know the famous theorem that everybody knows, the Pythagorean theorem. A squared plus B squared plus C squared. You talk to people on the street and they say, what do you remember from your high school Geometry? They'll tell you it was that. And yet in the last hundred years or more, we found that he was not necessarily the first one to come up with that relationship of a right triangle. That the Chinese knew it and other old cultures outside of the European circle did already know it. But because we are so, I guess, caught up in the United States and European background, it seems not have been the case. A perfect example of that is we talk about calculus. And calculus supposedly was discovered by Newton and Leibniz simultaneously. As a matter of fact, I once in the British Museum read a correspondence between the two gentlemen because he was in their library there and they argued who came up with what, and so on. It turns out that Leibniz's method of writing the stuff, the nomenclature and so on is the one we use today, although Newton's ideas should not go by the wayside. However, in the last 150 years or so, it has been discovered that they were not necessarily the first ones to discover calculus. That it was Eudoxus the Greek, who lived approximately 400 to 350 before the common Era. And he had a thing called exhaustion, which was very, very similar to the notion of calculus as we know it today. So he doesn't get credit for it. We still refer to Newton and Leibniz and we're not sure who did what. It's been controversy for a long time. And so rather than to spend much time with Archimedes, everybody knows about him, we went on to Eratosthenes. Now, Eratosthenes did something very interesting. He measured the size of the Earth. And he did it in such a simple way that a high school student in the first semester of geometry would be able to understand what he did by just using a shadow of a pole set up at Alexandria and then moved a bit further away to see what the shadow angle was and distance and so on. And he was able to get the size of the circumference of the sphere, the Earth sphere, very, very close to correct. We then went out to someone and again we're still in the ancient days, so to speak, 2000 years ago or so. We talked about people like, let's say, Diophantus of Alexandria, who came up with a way of, or dealt with solving problems where the only acceptable results were integers. And that's. We have diophantine equations today, which we study and deal with, where only integers and integer solutions are accepted. And so we go right down the line and then we get to. Finally, we get into Europe. The place to start in Europe, I hate to say it, because it's pretty far down the road, was with Leonardo Pisano, who was better known today as Fibonacci. Now, what made him so special? Well, we know about the famous Fibonacci numbers. Matter of fact, I think, you know, I've written a book on it. And the Fibonacci numbers are everywhere in our existence, from the spirals on a pineapple or on a pine cone to the way the branches of a tree are ordered, to the stock market. I hate to say it that way, because right now we're not in a very good stock market change. And so it's everywhere. However, what really is interesting is that Fibonacci wrote a book called Liber Abaci in 1202. And the first sentence of the book, he indicates that he came across these Indian numbers. 9, 8, 7, 6, 5, 4, 3, 2, 1. And also this thing called Zephyr, which we call zero today. And that was the first time in history of the European continent that these numerals were ever shown. It took 50 to 100 years before they actually caught on, but it's fascinating to see where it actually began. Now, how did he get it? As a kid, he worked on the African coast because his father was a tax collector from Pisa and he hung around with some Arabs down there who were doing mathematics using these Indian numbers. So since he got them from Arabia and they were from India, we refer to them as the Hindu Arabic numerals. And that's where that came from. It's fascinating. He also is the one who came up with the fraction bar. When we draw a fraction, one half that line. And he's in a number of other things for which he doesn't really get much credit, but it is an important aspect, so we went right there. Now, once we have the numbers, we look at John Napier, the Scottish mathematician, who we credit as having one of the first calculators, if I can use that. We call Napier's rods. These were rods with numbers on them, and you slid them in a certain Way you could say it was a forerunner, to some extent, of a slide rule.

B (11:52)

Yeah, that's what hit me.

C (11:55)

Right. And so we went down the line. We did talk about Kepler. Now, Johannes Kepler, German mathematician, lived around the late 1500s, early 1600s. And he came up with three laws, which are absolutely astonishing. He came up with laws of how the planets travel in an elliptical path and what that. The three properties of those of the pads. And it's just incredible that he came up with this. And the truth of the matter is he did this with observations, with a very, I'll use the word primitive telescope. And it was maybe about 25 years ago or so, it was written up in the New York Times where someone found his calculations and that he made a few corrections. In other words, he saw.

B (12:53)

He fudged the data.

C (12:55)

Exactly, he fudged the data. But he was right to do it because his machinery, so to speak, wasn't that accurate. Then we get to Descartes and Fermat, and the two of them, brilliant mathematicians, French, corresponded with one another, and the correspondence was what the chances of winning a game are and how you can figure that out. And they argued back and forth in writing. Now, you have to remember they didn't have the Internet, and writing took a while until the letter was transmitted and so on. And so it was one of those things that led to what we call probability theory, and that's how it began. Pascal mixed in as well. And so you have that going on later on. We now get to Newton and Leibniz. Newton English, Leibniz German. Both brilliant and both independently, so to speak. Rediscover, and I'm going to use that word, the calculus. And of course, we use Leibniz's notation today. Moving right along.

B (14:06)

Go right ahead.

C (14:07)

Okay, we get to Giovanni Siva. Now, he's not very well known.

B (14:11)

He's not known at all to me.

C (14:14)

Yeah, well, he did a very interesting thing. He found a relationship that if you have any triangle and you draw lines from each vertex to the opposite side, and they intersect in one point, in other words, they are concurrent. When that happens, the alternate segments along the sides of a triangle, regardless of what it is, are equal. The product of the alternate segments of one is equal to the product of the alternate segments of the other, which is an incredible thing. And of course, you can apply it to altitudes, angle bisectors, medians, and so on. But again, it's one of those things that was never really popularized in the high school course. But it's so simple and yet it provides a lot of insight into triangle properties. Now we come to where did the high school geometry course that we all worship come from? Well, the best we can do is start with Robert Simpson, also a Scottish mathematician who lived in the late 1600s into the mid-1700s. Now, he was actually an MD, strangely enough, but also he wrote a book which was a rough, so to speak. It's called a translation of Euclid's Elements. Now, I didn't mention Euclid because everybody knows that Euclid wrote the Elements, which was a summary of all of the geometry that was supposedly known to that day. Of course, he would have some problems with certain things that he did not define, like the concept of betweenness. But we can talk about that another time. In any case, here's Robert Simpson writes his huge tome, which was the first English version of Euclid's Elements, and it was published in the early half of the 1700s, and it was continuously available into the mid to late 1800s. And I've got copies of a book from the 17 and from the 1800s of his. Of his book on geometry. And it's interesting because the same words, except the quality of the paper 100 years later was significantly better than the quality of the paper in the earlier version.

B (16:33)

Can I interrupt briefly to ask you a question about paper?

C (16:36)

Go ahead.

B (16:37)

Okay. One of the things that always impressed me about the Greek mathematicians is they didn't have paper, as far as I know. So they're coming up with these brilliant ideas without the ability to just scribble and do things and, you know, just throw away the paper. And I'm wondering, what did they do? Do you happen to know, did they write stuff down in the sand and then later transcribe it onto parchment or what?

C (17:04)

Yeah. Well, we all know that Archimedes did that in the sand. That's something everybody laughs about. But they etched on stone, they wrote on parchment, they wrote whatever they could. I mean, it varied from time to time and era to era. But moving right along, there are times when a mathematician was not famous for anything except one thing, and that's the famous Goldbach conjecture, which is an interesting relationship with prime numbers. I won't go into it just now, but it's a very, very simple thing. And he is known for one thing and one thing only, and that's this conjecture, which to this day has not been proved. The interesting thing there is that in many fields, and I'm being a little bit brutal here, there are people who have become famous for one thing. They did largely like Bizet wrote Carmen. He did other things too, which are good, but if it weren't for Carmen, we wouldn't know about it. Salinger wrote Catcher in the Rye. That made him famous. It's just an interesting feature. Engelbert Humpedinck wrote Haslin Gretel. If he hadn't done that, we'd never know who he was today. So, I mean, this is an example here in mathematics where Christian Goldbach came with a very, very simple thing and wrote mathematicians asking them, hey, can you tell me if this is correct and can you prove it? And no one has, to this day, ever proved it. And this is. We're talking now from the early 1700s. Now we get to an amazing mathematician, Leonard Euler. Leonard Euler, Swiss mathematician, who spent a bit of time in Berlin, a bit of time in Moscow, but it is well known that he has written more than any other mathematician in history. Now, here comes the kicker. He, as a child was blind in one eye, but that didn't stop him. He wrote well, he saw what he did and so on. In the middle of his lifespan, he lost sight of his other eye. He was now virtually blind. And it is said that he wrote more while he was blind than when he was able to see. And the question is now, how the hell could he write when he couldn't see? What he wrote, he dictated. And now if you think about it, and you deal with the mathematics that he did and you wonder about it, you say to yourself, how could he dictate something that would have been 10 steps, 20 steps? His memory was so phenomenal that he was able to remember everything he wrote and go right along.

B (19:51)

That's just fantastic.

C (19:52)

Yeah. And now let's talk about a woman. Now we have Maria Gatania Agnassy. And here's a woman who lived in the early 1700s, 1718 to 1799, so most of the 1700s. And she's remembered largely for cubic curve, which is known in Italian as the mercia. And it is very close to the word. And which means in the curve means which of agnasi is how we refer to it, because the word is so close to the word of witch. And so the curve today is known as the witch of agnasi. Now, what makes this unusual is she so impressed Pope Benedict XIV that he provided her a professorship at the University of Bologna, which we know is the oldest university in Europe.

B (20:54)

Well, you may have known that, but very few of our listeners probably do.

C (20:58)

Okay. Well, it's quite amazing that this Happened because women were not allowed to even walk into a university. Moving along, we get to someone like Lorenzo Mascaroni. Now, what makes this guy interesting is he came up with a scheme for doing constructions using only a straightedge. Now, if we can recall back to our days in high school, we did geometric constructions with a pair of compasses and a straightedge or a ruler. And there were various things we could do, and there are certain things we couldn't do. For example, it was not possible to trisect an angle, a general angle, with just those two tools, but you could do a lot of things with them. However, this guy Mascaroni proved that you can do all the constructions with a straight edge alone, and you don't need the compasses. Then you say, well, how did he draw a circle? Well, he was able to. No, I'm sorry, wrong way. He did it with the compasses and not with the straightedge. My error. With the compasses, he draw a circle, but he had no way of drawing a line. Well, how did he get a line when you couldn't. You didn't have a straight edge because using the compasses he was able to get as many points as he wanted to be collinear. In other words, lying on the same line. So virtually he could have created a line with enough points. And so he's famous for that. However, here comes something that's very interesting, and that is that about 100 some odd years ago, someone found a book by a fellow by the name of George Moore, M O H R, a Danish mathematician, who came up with the same thing 100 years before mascaroni. And you say, well, why do we refer to it today as the Mascarni constructions? Well, we do, because that's a habit. But it should be called the Moore constructions, although it is well known, or so far someone has researched it, that Mascaroni did not know about Moore's work. So he did independently, so we can give him credit for that. A similar thing happened also with Robert Simpson we spoke about before, the supposed father of geometry. He came up his credit with a famous theorem called Simpson's Theorem. And he did not discover it because it was discovered after he died by somebody else, by a fellow named Wallace. So again these misnomers occur and we leave it the way it was popular and not necessarily correct. We get to Sophie Germain, another woman. Here's a woman who is French, lived from 1776 to 1831. French. And she taught herself Latin and Greek, and she was able to read the works of Newton and Euler yet her mother discouraged her from studying mathematics, but eventually she relented. Here again is a situation where women could not be allowed into the university, but she was able to get the notes and then created her own ideas from these notes of lectures and sent them to Legendre, who was so impressed with them that he admitted her to the university. She again used a pseudonym to contact and once again impressed people, so that she was able to put a great deal of effort into what we today know as Fermat's Last Theorem. And most of her honors were bestowed to her posthumously because of her gender. Interestingly enough, though, when I speak of Postern Elite, a street in Paris is named after her, as is an annual award offered by the Paris Academy of Sciences. So she got most of her credit after she died. Moving right along, we get to probably the most famous or most productive German mathematician, Carl Friedrich Gauss. Now, here's A guy again, 1777-1855, who did so much, and yet it is believed that much of what he did was a result of his ability to think in a most unusual way, of numbers and relationships in his head mentally. And it was these mental relationships that he was able to create a lot of theorems. There's a cute story that a good math teacher should have told all of their students somewhere along the line about young Gauss when he was in the third grade, and the teacher wanted to keep the class busy and told the class, take out your slates. In those days they didn't use paper, they used a slate and wrote with chalk on the slate. Take out your slates, and I want you all to add the numbers 1 to 100 and don't talk, because I've got to do something at my desk. And before the teacher finished the sentence, the little Gauss raised his hand and the teacher said, now keep quiet, Carl, and just sit there. And he sat there for about an hour till most of the others were finished, and at the end, he was the only one who got the correct sum immediately. Well, how did he do it? Everybody did. One plus two plus three plus four and so on. Not Gauss. He said, wait a minute, that's silly. If I add 1 and 100, I get 101. If I had 2 and 99, I get 101, 3 and 98, I get 101, I got 50, 100 ones. So I just have to multiply 50 times 101, I get 5050. And that's the answer. He was the only one who was right. But again, this was now third grade, so it's quite unusual. And this is just one story, of course. Gauss is famous for so many things, one of which he was most proud of. As a youngster, he found it possible to use straight edge encompasses and construct a 17 sided polygon. Now you say, well, 17 sided polygon. And he wanted it on his gravestone. Well, people said you can't do a 17 sided polygon because it'll look like a circle with 17 dots on it when you have so many sides. But he wanted it there. And there is a monument today with a 17 sided polygon honoring Mr. Gauss.

A (27:31)

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C (29:16)

Experian.

B (29:18)

Well, I'm happy to know. I'm happy that I would get classified as a good math teacher by you because I tell that story in every class to which I'm asking them to learn the sum of the integers from.

C (29:31)

1 through n. Well, you must be. I can tell the next one is Charles Babbage. Now, Charles Babbage is not too well known, but we should honor him because he is the first one, by some measure anyway, to have constructed a calculating machine. However, it was a difference calculator, or difference machine as he called it. And it was kind of limited what could do, but it was a mechanical machine that would subtract numbers. And here we come to another woman, Ada Lovelace. Woman with a sort of semi or quasi royal background. And she worked with Babbage, who is credited with developing the first calculator called the difference machines. We said he was hugely impressed with her brilliance. When she translated lecture notes from French to English, she added her own ideas which are the basis for what we today call computer programming. Therefore, we can consider her the founder of computer programming. She grew up in a broken household who was rather ill in her childhood. And as she got older, she worked her way into the academy and into the academic world and even married into royalty. As I mentioned before, she was both attractive and smart and made a big hit at the time when women were not even allowed to do anything beyond household work. So here's one. Ada Lovelace. We move along. Here's a crazy story, and this is one that I love to tell people when they just say, well, what did you write about in this book? Evaristi Galois, French mathematician. He was born 1811 and died 1832. You can see he did not live very long. He was brilliant and he was tricked into making a liaison with a woman who, unbeknownst to him, had either. We're not sure if it was a boyfriend or husband who was a sharpshooter. And he catches Galois with the woman and he tells him, as they did in those days, my honor is such that tomorrow we must meet for a duel. And Galois saying, oh, I guess it's over. That night he went home, did not go to sleep, and wrote down everything he knew about mathematics. The next day, gets up, beats him for a duel and get shot and killed. And here's this 20 year old kid got killed. And what he wrote that night is today a complete branch of mathematics called Galois theory. So here is again a crazy situation where a person has done something quite unusual and worth talking about. Another woman, Russian Sophia kovalevsky, is again 1850, 1891, again, one of these Russian women who sort of broke into the university. And how she did it was very curious. She noticed that people were covering a wall with wallpaper that was old notes, mathematical notes. And this was what motivated her to read these things, and it actually motivated her to do mathematics. And she's the first woman to receive a doctorate in mathematics and become a math professor. And so she gets credit for that. Moving right along, there's another woman, Emmy Nourtha, who struggled through her early years, ended up in the United States, because, again, she did not only suffer the female prejudices that were out there, but she was also Jewish. And when Hitler came, she had to leave Germany, but she had a rather successful life. And Einstein felt that she was one of the most brilliant women he's ever come across.

B (33:37)

I think she ended up teaching at Bryn Mawr.

C (33:39)

Yes, she did. She did very good. If anybody saw the movie the man who Knew Infinity, they would have come across Ramanujan, who was an Indian, young Indian man. Again, this is now born 1887, made it not very long in life, into his 30s, who was extraordinarily poor in India and married at a very young age and so on. And he corresponded with GH Hardy in England with the hope that he would recognize some of his ideas. And eventually Hardy invited him to come to England. And at first he was not well accepted because he was different from the rest. He was an Indian, very poor, not very well versed in the culture of England, but his mathematics was brilliant. And they were just astonished with his findings. And slowly but surely they began to accept him and invite him over, so to speak, into their small cliques. However, the funny thing is, if you see the film, they do this in the film as well, which to my mind, really shows the brilliance of the man. Hardy finds out that now Rahman John is in a hospital in England and not in great shape, lying in bed, and he goes to visit him, and as he gets to him, he says, I gotta make some small talk. So he says, you know, I just got out of this taxi, and the number, I think it was 1729.

B (35:26)

It was 1729.

C (35:28)

I can't make any sense out of this number. And without batting an eyelash, this guy is half dead, lying in bed. He was going to die. Soon anyway, says, oh my goodness, that's the smallest number that can be expressed as a sum of two cubes in two different ways. Now, it's incredible that someone could come up with that response that quickly from out of nowhere. And that gives you a sense of the brilliance of that man.

B (35:59)

I have something I want to add to this story that happened to me. I had a friend who was working with computers in the 1960s, and we decided to see whether or not we could find the smallest number that was expressible in two different ways as the sum of 2 4th powers. So we ran up some time on a computer and we got it. It's a 12 digit number. And then we decided to see whether or not we could find the number that was expressible as the smallest number that could be expressed in two different ways is the sum of 2 5th powers. He ran up a lot of computer time and they told him to stop doing it. And only later did we find out that it was an unsolved problem. And nobody knows that there is such, whether or not there is such a number.

C (36:46)

Very cute. So now we're getting into the 20th century. Now, John von Neumann, who put in the faun himself, because in the Austro Hungarian Empire, the Fon means of. And it was a sign of nobility or royalty. And you were given this honor by the Kaiser or king, whichever country you were in. But he decided to put it in himself. Matter of fact, I should tell you then, in Austria today, no one is allowed to use that Fon because that was given up at the end of World War I. The Austrians cannot use the Fon as part of their name. The Germans do have it, but Austrians do not. In any case, Neumann, coming out of Hungary, moved eventually to the United States and came up with a whole bunch of British ideas again. Once again showed his brigades. He really showed off a lot with doing mental calculations, which also included translating texts. We consider him the founder of game theory and perhaps best known for having developed the modern computing as we know today. There's a cute story about his mental capacity which I will just tell quickly and see if we can get it done. There's a very common or famous or well known problem that has a trick solution. Now, if your listeners can recall, in elementary algebra, they did these uniform motion problems where they learned that rate times time equals distance. So if you know a person's rate and the time they're traveling, you can find out how far they went. And the problem is that you have two trains traveling towards each other about to crash. But as they're traveling towards each other, there's a little bee, a buzzing bee, that's much faster than the trains and flies from one train to the next, back to the first train, back and back and back and back until the two trains meet and the bee gets crushed. And the question is, how far did the bee travel before it got killed? And you know, where they started from and so on, and what a person does if they know the trick. The trick is not to figure out the actual pieces of distance, because that would be incredible. But what a person does is say, okay, it took so and so long for the trains to. To touch, to collapse. And the speed they were going at, we can also tell. So we know the speed of the bee. And so the speed times the time will give us how far I travel. Now, he was given. When von Neumann was given that problem, he came up with the answer just as quickly as anyone would come up with it. And I said, well, how did you know the trick? Someone said, what trick are you talking about? Well, how did you. Don't you, you know, how did you do the problem? He says, I figured out each of the little travels that the bee made instantly, which is an incredible task. I mean, it's almost, you know, even a computer would struggle with that. And he did it in his head. So it's just one example. He was again, one of these brilliant people who has the capability of doing these calculations in his head. We're getting into the modern era. We get Alan Turing, also a film made about him where he's the guy who broke the German, the Nazi codes and allowed us to abbreviate the World War II, so to speak. We get to another, I have to say, this unusual mathematician called Paul Erds, a Hungarian mathematician, 1913-1996. So he's a rather recent person, but this is one of the most unusual lives you can imagine. Again, he was Hungarian, moved to the United States when Jews were not allowed to flourish in Hungary during World War II. And he never had a home. He lived nowhere. He had no address. He had a suitcase. And he lived one week with this mathematician, two weeks with this university, another week or two with this mathematician someone once counted up. He lived with 500 mathematicians or places in his life of 80 some odd years, and he wrote 1500 papers. Now, the key thing in mathematical circles today is that he was so brilliant and so well known that people say having written an article with Mr. Erds is a real honor. So today, if you've written an article with Dr. Erds, you're given an Erds number of one. If you written an article with someone who wrote an article with Dr. Erds, you got an Erds number of two. And so it goes that you got Erdos number three, four and so on. In other words, he's become so famous.

B (42:20)

There'S actually a directory somewhere of people who have all the ERDS numbers. And I have an Erdos number of two.

C (42:28)

Oh, you're better than I am. I have an Erdish number three.

B (42:31)

So here's my wife.

C (42:34)

Okay, well.

B (42:37)

Better just know more. I just know the right people.

C (42:40)

That's right.

B (42:41)

One of whom is you.

C (42:42)

Oh, yeah, right. The next one. And I guess I more or less conclude I have a couple more. But the one that I have a personal interest in is Herbert Hauptman. Herbert Hauptman was the first mathematician to win the Nobel Prize. Now, of course we don't know why, but we do know there is no prize in mathematics in the Nobel realm. However, he solved a 40 year old problem in chemistry. And the story is very interesting and I'll just take two or three minutes to tell you the story. After he served in World War II in the meteorology area in the Navy, came out and worked in Washington and the research labs there, National Research Lab. And he bumped into, ended up working with Jerome Carl. Now both of these guys graduated City College of New York in 1937. One was a math major, obviously Haltman, who won all the awards in mathematics, and the other one was a chemistry major and they worked together. And Houptman and I knew him very well because we wrote a few books together. Nicest human being you'll ever meet. He died several years ago in 2011 at age 94. But it worked until the end. Nice guy, super nice guy and very generous now, but this is something. Well, I can say it. What the heck. It's not a. I got this from Hauptman's wife. There's never been a Nobel Laureate who was not first a member of the National Academy of Sciences. So Carl and Hauptman had done this work and it was constantly being nominated for the Nobel Prize in Chemistry. Carl, knowing this, tried to get his wife into the National Academy of Sciences as he was already in there, which he did and tried to, according to Hauptman's wife, blocked her from getting in there because he's a mathematician, not a chemist. And so he was hoping that when the Nobel Prize was awarded for the work they did, and I'll tell you what that was in a moment, it was in crystallography. It was actually a method that's still used today when antibiotics and so on have to be created by pharmaceutical corporations. They use his technique to do it. But anyway, he figured that he and his wife, Carl and his wife, were going to get the prize because they're there, the work is being constantly nominated. And it turns out that 1985, the Nobel Prize was awarded to the work they did to Hauptman and Carl and not to the wife. And it's amazing, according to Edie Hauptman, his wife, how fast the National Academy of Sciences called her and said, we want to have you as a member. We want to have you as a member. So he then became a member. But it was interesting because he, you know, he was a brilliant guy. And I'll give you one more example. In one of the books we wrote together, he came up with a technique for using fractional, I'm sorry, exponential factorial. And he worked up a sequence with that. And he said to me, you know, I don't know. I've never seen this before. I don't know if this is original or not. So I called a friend who is. I don't know if he's still alive at University of Wisconsin, to ask him, who is an expert in algebra, if he's ever seen this before. Because I hadn't. And he said, no, I don't think I've ever seen this before, but I think it was anticipated by Gauss. And so when I told Herb says, oh, I love to be anticipated by Gauss. And so he was perfectly happy with that.

B (47:15)

You know, we're coming to the end of the allotted time. And there's one person that I'd really appreciate that you talk about, and I think the audience would, too, is Maryam Mirzharkani.

C (47:26)

Yeah, she died early. Brilliant mathematician. She won all the prizes that we won. And the curious thing that if you ask me for a cute story, when we included her, she's the last one, because she's the. I could say the youngest of the lot. She died in 2017 at a very, very young age. Born in 1977, Persian. And the story was. I want to use a picture of her because we have. In our book, we have a photograph of everybody and many of the things they did, as well as mathematical things and historical things and so on. But I wanted to have a picture of her. The publisher said, no, you can't do that. It's against the copyright law. So I said, let me call her widowed husband. And I contacted him and he was elated. Please use a photograph. And he gave us a lovely picture of her. So that circumvented the publisher's concerns about copyright law.

B (48:33)

You know, one of the things that I think makes me appreciate Mirsha Khani more is that there's an award that's fairly well known in mathematics, the Fields Medal. And the Fields Medal is only awarded every four years. And if mathematicians don't feel that there's anything good done in the last four years, they don't award it. And Mirja Khani not only got it, she was the first woman to be awarded a Fields Medal. And so this is just a remarkable accomplishment. And it's so sad that she died.

C (49:03)

So young, age 40.

B (49:05)

Yeah, just, you know, one of the things that we've gotten, sort of a Brief History of Mathematics. But I really urge listeners to read the book because there are so many fascinating stories and there were so many instances that I wanted to interrupt alone just because there were fascinating stories that we could have talked about, but maybe it's better that I didn't, because that way, if you want to see the fascinating stories, read the book. Because some of the mathematicians had lives that you absolutely would not believe, and they're in the book. And as I say, one of the things that I really loved about the book is that I've never really liked biographies, because reading all the dull portions of someone's life in order to get to the good portions is sort of like sitting through an opera waiting for the one good aria. I don't like opera, by the way, but reading Al's book, and he does have a co author, but I think of it as Al's book, is that you get the cream of what there is. You get the best mathematicians, the work that they did, and also some of the fascinating stories of their lives. And as I said, There were about 20 instances where I wanted to interrupt, but, Al, you were going so strong, I couldn't bring myself to. But one of the things that I always do at the end of an interview is I'd like to ask listeners how they can get in touch with you.

C (50:28)

Well, the best way is email today is I check it regularly. My email is asp1818mail.com and I respond to everyone all the time. Matter of fact, someone just wrote me they found a typographical error. A half a zero was cut off in the printing process. But I will say one thing. My co author, Christian Spreitzer, brilliant young man, and I always love to give young people an opportunity to publish, because it's not easy to break into the publishing world today. So when I get a chance and I see someone who's really got something to offer, and I offer that opportunity so they can also benefit from that. I will say one thing, though, that when we published this book, there was one criterion that we had to follow throughout to talk about the person and all of the nuances about that person's life and so on, but we also had to talk about some of the things they did mathematically that the general readership not only could understand, but also appreciate. So when we did with Euler, for example, who wrote more than anybody else, we picked a few things that, like, for example, he's the one who came up with the famous Euler relationship of any convex solid where vertices plus faces equals edges plus two. And we talk about something similar where the average reader can go out and, hey, let me see if that's true with a cube. Let me see if that's true with a icosahedron or whatever. And it is, of course. And so we try to, with every one of the mathematicians we talk about, say something about what they did, not just their life. And of course, their lives, as Jim was saying, are so unusual because as I said earlier in our discussion, none of these people could be considered, I hate to use the word normal and fit into a society that we know today. They all were eccentric, they have to be, because you cannot think the way these brilliant people did and then act as though you're just one of the people in a social circle.

B (52:46)

You know, as I say, one of the things about this book is Al had to twist my arm in order to get me to read this book because, as I said, I don't like biographies, because. But three biographies in. You can read the biographies when you are, you know, you can read the biographies. One biography in 10 minutes. They're charming stories, they're fascinating people. And of course, it's the mathematics that they've done that made their lives. And you were talking about Euler a couple of years ago. There was a net, there were mathematicians had an Euler year. And a friend of mine said that, you know, one of the things that they did was they voted on Euler's 10 top accomplishments. And I think the Euler relationship with Polyhedra ended up as number three. But my favorite equation in mathematics is Euler's E to the I PI plus one equals zero. That ended up number one because it has the five most important constants in mathematics, all on our giant Stage together.

C (53:47)

However, the general readership would know what to do with it.

B (53:50)

It's true. But for mathematicians and a lot of the people who listen to these podcast, they have some deeper knowledge of mathematics. Do you have, what plans do you have for the future? Al? Because what we'd like to do is we'd like to find out if you have future projects that we can maybe talk about in the future.

C (54:08)

Well, I'm constantly working on books. One that's in the editing process now is called the Joy of Geometry, and that's a book that deals with geometric relationships with no proofs. Because in the United States, unlike in Europe, we have a year long geometry course, which as I said, started with Robert Simpson, went to Legendre, and went to Robert Davies, who then created the course that we now have in 10th gear math. And it's been modified over the years as well. But when you think back to, when most people think back to high school geometry, they think back, proof, proof, proof, proof, proof. And they don't really take time to see what was proved and the relationships. And what we've done here is come up with a book that deals with amazing geometric relationships spelled out very gently and very nicely so everybody can appreciate them. And yet we just say the proofs are left to the. That's kind of a nice thing because you can see all these phenomenal relationships exist, equal areas, concurrent lines, collinear points, just all kinds of things like that. The other thing is that another book I did is called Math Tricks. There's so many cute things you can do in mathematics that can be used to trick others. You know, it's the unusual aspects of mathematics. Another book that's in the process now, how you can entertain people with mathematics. You see, I'm always looking for ways that we can make mathematics not the subject that most people are proud to have been lousy in. In school. When I say to someone what I feel is mathematics, the first reaction 80% of the time is, oh my God, I hated mathematics. I was always terrible at mathematics. It's like a badge of honor to say I wasn't good in mathematics. But look at me, I'm a lawyer, I'm a doctor, I'm a businessman. I'm trying to dispel that.

B (56:16)

Al. It's a noble goal and I look forward to reading these books and talking with you about them. Thank you very much.

C (56:22)

My pleasure.

B (56:23)

Take care. Bye bye.