 Tonight I'm delighted to bring up our introducer for the evening Brian Hopkins He's a mathematician at st. Peter's University And he's also a recent editor of the college mathematics journal And he's been a friend to the museum for a long time And if you look carefully on our website, you might notice that he'll be back with a wonderful performance of his own in the winter So for now, I give you Brian Hopkins So I'm very pleased to be able to introduce to you my friend Dave Richardson He is a professor at Dickinson College over in Carlisle, Pennsylvania Just to give a shout out to some institutions. He went to undergraduate at Hamilton College upstate and Then Northwestern for his PhD and did some postdoc work at Michigan State University I think Dave and I first met about 2007 or so it was a good year to be a history of math person It was the 300th anniversary of the birth of Leonard Euler and there was lots of interest in looking at his mathematics and talking about it we ended up being in some books together, I think and Dave took that work and then made a book with Princeton University Press called Euler's Gem Which did very well. It had an interesting subtitle the polyhedron formula and the birth of topology And this is very well received it went on to get the Euler book prize And that's not because the content was about Euler instead the Euler book prize is a four I'll read the citation it recognizes authors of exceptionally well-written books With a positive impact on the way that people the public views mathematics so he did that and No good deed goes unpunished So after writing a good book he became the editor of one of the mathematics associations of America's journals called math horizons This one is especially intended for undergraduate students, but of course is open to anyone who likes to read Engaging lively articles about mathematics and come a magazine format I at the same time was as Cindy mentioned the editor of the college mathematics journal. So we shared a lot of time in tedious meetings and council Zoom meetings and all kinds of things talking about publications, but luckily we also had the chance to interact with some mathematics Dave wrote an article for the college math journal called on the topic of who first figured out That if you take the circumference of a circle and divided by the diameter of the circle You get this constant that we now called pie. Who's the first person who did that? Great article ended up going into a collection called the best writing in mathematics for that year and Then he went on after that And has another book in just in case he's too modest to push it. I brought it up here with me You'll be over at a table behind after back over there Actually, the Oilers gym is there too and also this tales of impossibility Which is the foundation for the talk that you are about to enjoy? so a lot of us here have in our profession the joy of trying to elucidate mathematics to bring this potentially complicated topic to a wider audience and explain why it's engaging and why it's fun and Some of us do that in teaching some of us do that in writing Some of us do that as the museum does very engaging exhibits Dave does this in writing and also with a attention to the history behind the mathematics who did things why they did things bringing in the human element so I hope that you'll enjoy Dave as much as it went well in the first talk and We're lucky that he's here. He left his wife two children and 43 real analysis students Back in Dickinson College to be here with us. So Dave Richardson Thank You Brian, and thank you Cindy for the invitation. It's a pleasure to be here. I've been a Fan of the Museum of Maths is before it was the Museum of Math I visited the Math Midway when it was traveling around the country and I've been back to the museum With my son who has really enjoyed it and I come back when I can to the moves conference, which happens every other year Unfortunately, I could not make it this particular year. So it's a real treat to be here to speak to all of you Before I get started on my talk everyone should have one of these handouts at the table we're gonna have a few activities and so While the talk is going on you can cut out all the tools here. I hopefully punched out the hole in all of the Compasses here. We don't have compasses. So I have what I'm calling a rusty compass here And these are gonna help you draw circles at some point during the talk so we have a Ambitious agenda for today's talk. We're gonna talk about 2,000 years of mathematics I'm gonna talk about these famous Geometry problems and the eventual proof that they are impossible. So we've got a lot to say so why don't we get started? So as we all know nothing is impossible, right? This is the American dream Robert Goddard had something to say about this. He wrote just as in the sciences We have learned that we are too ignorant to safely pronounce anything impossible So for the individual since we cannot know anything not we cannot just know what is limit are his limitations We can hardly say with certainty that anything is necessarily within or beyond his grasp It is often proved true that the dream of yesterday is the hope of today and the reality of tomorrow Even shell Silverstein got in on the conversation. He wrote listen to the mustn'ts child listen to the don'ts listen to the shouldn'ts The impossibles the won'ts listen to the never haves then listen close to me Anything can happen child anything can be right? This is what we tell our kids. This is what athletic coaches say This is what teachers tell their kids and I'm sorry to be the bearer of bad news that this is not true That there are things that are impossible and we can use mathematics to prove that they are impossible So just as one example You guys may know this puzzle which you can get at cracker barrel gift shops or wherever It's the 15 puzzle where you have to slide the tiles around and get them in numerical order Well, there's a famous example from the late 19th century the puzzler Sam Lloyd had one of these 15 puzzles but he deviously switched the 14 and the 15 in the puzzle and he gave a Monetary prize he offered a monetary prize for anyone who could solve His particular puzzle, but little did the his customers know this is impossible And you can use mathematics to prove that if you make that simple move of switching the 14 and the 15 Then this becomes impossible to solve and so he never had to pay out the prize money So that's one example of a mathematical impossibility. There are many others. So just a few examples famously Leonard Euler analyzed these seven bridges that crossed the River Praggle in Kernigsberg and the people from that city wondered whether you could find a walking route that would cross every bridge One and only one time and it turns out that that is impossible and Leonard Euler proved it I also want to give a plug to Brian who introduced me he He co-authored a lovely article about the bridges of Kernigsberg, which I would highly recommend Another famous impossibility problem is Fermat's last theorem which was proved in the mid 90s And this is sort of a generalization of the famous Pythagorean identity But in this case, we're looking at exponents n that are three or larger and we want to know whether they're integers that satisfy this relation and Famously you cannot so there are no solutions to this when n is greater than or equal to three Another famous impossibility problem We often ask our algebra students to solve Polynomials look for roots of polynomials and linear polynomials are easy if we have a quadratic Polynomial a degree to polynomial. There's the famous quadratic formula that helps you solve those It's a much messier procedure, but you can solve degree three. You can solve degree four But famously it is impossible to have a general formula for Degree five or higher. So that's a famous impossibility problem. We all just voted. Hopefully we all just voted yesterday There's a famous impossibility theorem in voting theory. So Kenneth Aro listed a bunch of very reasonable traits that you would like a voting system to have and He proved that it was impossible to find a voting system that satisfies all these reasonable criteria and Probably the most meta of all of these impossibility problems is something called girdle's incompleteness theorem and this lives in the realm between logic and mathematics and girdle proved that if you have any sufficiently sophisticated axiomatic system that there are true theorems True statements that cannot be proven. So this is a very interesting impossibility theorem So these are just some examples of impossibility theorems and mathematics But there are four of them that I want to look very closely at today and to talk about their history So we will see that it is impossible using only a compass and straight edge To double the cube to trisect every angle to construct every regular polygon and to square the circle And so the structure of my talk is we're going to spend quite a bit of time talking about each of these four just to make Sure we understand What the problems are? Then we're going to talk about a period where people tried to solve them using various means and then at the end We're going to talk about the eventual proofs that they are impossible and Just so that we're sure we're on the same page what these things constitute We can transport ourselves back in my case. It was 10th grade We're in the geometry class the only tools we have at our disposal are a compass and a straight edge And you want to solve some certain geometric problem So in this case let's say we have a line segment and you're asked to draw the perpendicular bisector for this line segment So I will show you the procedure which you may remember from High school so you can draw a circle with the radius equal to this line segment another one with the center at the other end That gives you these two bull's-eyes and you draw a line segment between them And this is the perpendicular bisector of the line segment that you started with and so this is a straightforward Procedure and something that a typical 10th grader would learn in a geometry class And so these four problems are in some ways just as elementary to state as this problem From 10th grade but these happen to be impossible to solve. So let's look at them one at a time I'm going to start with what looks at you know at a glance to be the weirdest one So this has to do with cubes, but we're really talking about planar geometry But this really is a question in planar geometry The question is if we start with just this line segment the side of a cube Is it possible to compute to construct the side of another cube that has twice the volume? Okay, so what I'm going to do for each of these problems is I'll tell you what the problem is Then I'll show you an easier version of it and then we can return to the the difficult one again the impossible one So let's drop down a dimension and let's talk about doubling a square So suppose you have the side length of a square Is it possible to construct the side length of a square that has twice the area? So let's just say our original square had side length one So the area of the square is one We would like to construct the side length of a square with twice the area area two Okay, so if this has side length one has area one this has area two so the side length is Square root of two right so really this is going to boil down to a question about line segments So if you start off with a line segment of length one Is it possible to construct a line segment of length square root of two using only a compass and straight edge? And so it turns out all of our problems are going to boil down to something like this a question about Constructing line segments of various lengths. So here's our square with side length one Is it easy or hard to construct a line segment of length squared of two from from this? Easy right That's it right that has length squared of two and so this square is gonna have twice the area of this square Right, so this seemingly similar problem is almost trivial Okay, so let's go back to our cube here. Let's say that our original cube has side length one so you start off with a Line segment this long we want a cube the side length of a cube of volume two And so now instead of square root of two our side length is going to be the cube root of two And so just like for doubling the square it boils down to line segments If you start off with a line segment of length one Is it possible using only a compass and straight edge to construct a line segment of length cube root of two? Okay, so if you can do that you can double the cube The second one is the problem of trisaking an angle So I want to clarify one thing about this problem The question is can you take an angle and divide it into three equal angles and here There are some angles where you can do this But the question is sort of a generic question if someone handed you a random angle an arbitrary angle Are you guaranteed to have a procedure for? Dividing it into three equal sub angles Okay, so again, let's look at an easier version of this if you're given an arbitrary angle Can you bisect the angle and so just like our Perpendicular bisector problem at the beginning. This is a typical high school geometry problem So you might even remember how this goes if we draw a circle with a center at the vertex And then you draw two circles with this radius then that will give you this bullseye and you draw the ray and that Splits our angle into two equal sub angles. So very easy to solve Our impossible problem is splitting it into three equal sub angles. So for this discussion, let's Work things in reverse. Let's assume that we can trisect the angle So we can draw this ray where this angle has size theta over three Then we can draw a circle any circle that has a center here. Let's just say it has radius one We can drop a perpendicular down from this point The distance from the center of the circle to the foot of this triangle if you do a little trig That's cosine of theta over three. So if you can trisect the angle then you can construct this line segment of length Cosine theta over three, but actually all of those steps are reversible. So If you can construct this line segment of length cosine theta over three You can draw this perpendicular and that gives you this point and we can trisect the angle And so just like our previous problem had a number associated with it So does this one so if you start off with a line segment of length one Is it possible to construct a line segment of length cosine theta over three for any theta that you're given Right so for a generic theta and if so you can trisect an angle of size theta The next problem is really a class of problems We want to know can you construct every regular polygon with only a compass and straight edge? Okay, so these problems date back to the the Greek period if you look at Euclid's elements there's a lot of Constructions in there of various Regular polygons in fact the very first proposition in Euclid's elements you construct a regular Inequilateral triangle, okay, so some of these we can do but the question is can we construct all of them and I Should point out here that I'm phrasing this in terms of regular polygons An equivalent way to think about this is that if you have a circle Can you divide the the circumference of the circle into n equal parts if you can divide it into n equal parts You can connect the dots and that gives you your regular polygon And so sometimes when you see people talking about this problem they call it the circle division problem So dividing into equal parts, so let's look at an easier version of this I think probably the easiest one is constructing a regular hexagon so We want to construct a regular hexagon inside of this circle and this is a particularly nice problem because that Hexagon is going to have a very interesting property the side length of the hexagon is going to be exactly the same length as the radius of the circle And that makes the problem Very easy to solve so if we put our the tip of our compass here on the circle and The other the pencil at the center of the circle so we draw a circle of exactly the same size it's going to cross here and here and Because of these properties of the hexagon these three points are automatically vertices of the hexagon and So once we have these we can find the other three very easily we can just draw these line segments and that is our Regular hexagon so that is very straightforward and actually almost for free here If you have the regular hexagon you could join every other vertex and that would give you an equilateral triangle Right, so that's pretty straightforward Okay, so back to our chart here. We just saw that you could construct an equilateral triangle on one of the earlier Slides I implied that we could construct a square and we can that's not too hard The Pentagon we can construct a regular Pentagon. It's a little bit difficult and probably for students who are working through Euclid's elements This is one of the stumbling blocks to figure out how to construct the regular Pentagon It turns out that the regular Pentagon has a very interesting property if you were to join These three vertices you get an isosceles triangle and this isosceles triangle has a property that this angle is I guess this angle is twice this angle this angle is twice this angle and that's sort of the key ingredient You need to construct the regular Pentagon so that is possible We also saw that you could construct the regular hexagon In fact once you have a regular n-sided polygon you can always double the number of sides So you can imagine having one of these Regular polygons inside of a circle you could draw rays from the center of the circle to two adjacent vertices You could bisect that angle and that will give you one of the new vertices of the regular n-sided polygon so From those four we could get the regular octagon the regular ten-sided polygon twelve gone Sixteen gone and all the doubles of those right so these are all possible There's one other trick that Euclid had here The number three and the number five are prime numbers And so we can take advantage of that to find to construct one additional regular polygon so here We've constructed an equilateral triangle and a regular Pentagon inside the same circle so that they share a vertex here And if you look at these two vertices right here This vertex is one-third of the way around the circle from a and this vertex is two-fifths of the way Around the circle from a and two-fifths minus one-third is one-fifteenth And so this right here is the side of a 15 Gone and so once you have the one side you can get the rest of them Okay, and so by the end of the Greek period this was all they were able to solve So the green ones they could solve the yellow ones had question marks in them So the first one the smallest one that was unknown is the regular heptagon the seven-sided polygon The non-agon the nine-sided polygon was unknown as are these other ones. So these were unknown and The oh, I guess we need a number associated to this The analysis is almost the same as it was for the Angle trisection problem so here if you start off with the line segment of length one Then you can construct the regular end-sided polygon If and only if you can construct a line segment of length cosine 360 over n So that's that's sort of the number associated with the regular end gone Okay, and the fourth problem is probably the most famous of all of these problems This is the problem of squaring the circle and so here we ask the question Can you take an arbitrary circle and use only a compass and straight edge to draw to construct a square that has the same area? Okay, so I should point out here that when the Greeks thought about squaring a figure this was sort of equivalent to Computing the area of the figure if you could draw us in a square that has the same area as a figure That means you've sort of captured all of the area And so the Greeks tried to square all manner of different figures And the circle one of the simplest geometric shapes was outside of their grasp. That was something they could not do So let's start with an easier version of this. This isn't all that much easier, but it is a little bit easier Let's try to square the triangle So let's say you're given an arbitrary triangle. We know you can compute the area it's one half base times height and so you could give the geometric version of that by dropping a perpendicular bisecting that perpendicular drawing this parallel line and drawing these perpendicular to get this rectangle and if you think about it This rectangle has the same base as the original triangle and it has half the height of the original triangle So they both have one half base times height of the triangle. Okay, so Turning your triangle into a rectangle is pretty straightforward You might think it's trivial to turn the rectangle into a square, but it turns out that's a little bit tricky So I will show you the steps here Sort of one at a time sort of quickly one at a time So we can extend this line segment we can draw this circular arc now. We want to bisect this line segment Draw this circle we extend this line segment up it hits the circle here and It turns out that this square if you follow all those steps This square has the same area as this rectangle Which has the same area as this triangle and so this gives us a means to square an arbitrary triangle And so it turns out you can square any polygon Euclid could square any polygon Shapes with curved boundaries were a little bit trickier There are some examples from the Greek period where they could square shapes that had curved boundaries So one famous example is that Archimedes Took a parabola and if you sliced the parabola In any way with a straight line He found a way to square that little sort of fingernail wedge shape from the parabola but these were difficult problems trying to square various shapes and Shockingly the most simple geometric shape the circle turned out to be the one that they really wanted to do and were unable to do So let's take our circle. Let's assume it has radius one so the area of the circle is Pi yeah pi r squared so in this case pi because we're talking about radius one so the circle has Area a and so we want the square to have area a and so the side length of our square is going to be the square root of pi Okay, so just like the other problems this boils down to a specific number Given a line segment of length one. Can we construct a line segment of length square root of pi? Okay Square to pi is a little bit messy It turns out if you can construct a line segment of length square to pi you can construct a line segment of length pi and vice versa So we're just going to have pi be a stand-in for square to pi or actually I took this picture a few hours ago pi from the Doors of Museum math Okay Okay, so just to recap all these slides In each case, we're going to start off with a line segment of length one and If we can construct a line segment of length cube root of two then we can construct Then we can double the cube if we can construct a line segment of length cosine theta over three then we can Trisect the angle theta if we can construct a line segment of length cosine 360 over n then we can construct a regular n-sided polygon and If we can construct a line segment of length pi then we can square the circle Okay, so these are the numbers that are going to be important to us I'm going to show you this slide again near the end of the talk when we're talking about the impossibility of these problems Okay, so some of you might be asking why compass and straight edge what you know Why is that why are those the rules of the game here? and so one answer is that if you think about the most elementary geometry that you can do with as few tools as Possible, let's just say a piece of string or something string with sticks at the end You could plant this stick in the sand and you could use the other end to trace out a circle Or you could stretch your strings straight and that gives you a line segment So in some sense these are the most elementary geometric tools that you could imagine another answer to this question though is that Solving these problems really have two parts One is here are the steps that you need to go through to solve the problem But also you need to prove that your steps actually do what you claim that they do right So you need sort of mathematical theory to back these things up and the the Bible for geometry from 300 BC until you know the present more or less is Euclid's elements Okay, and Euclid famously started his his book with five postulates and the first of the postulates is that you Must be able to draw a straight line from any point to any point so you can draw a line segment using a straight edge Second is that you can produce a finite straight line continuously into a straight line so you can extend your line segment into a line third is that You can describe a circle with any center in any radius and so this basically says you can use a compass so in order to prove these Problems have been solved you use Euclid's postulates and one two and three are the compass and straight edge postulates And so that's another reason why these problems were important I left some space here for four and five four looks weird to us today It says that all right angles are equal to one another and you can see I left a lot of room here for five I'm not even going to read this This is the famous parallel postulate which itself has a long and very interesting history And could be the subject of a fascinating hour-long talk in and of itself Okay Okay, so now let's talk about solutions to these problems and I put solutions with an asterisk because they're not really solutions These problems are impossible but for many years people would ask what if I had a compass and a straight edge and And something else so can I solve these problems if I add some new thing to my bag of tricks So let's talk about some of these so one of the things you might add is some new drawing device So these are all actual things that you might encounter. We have the carpenters square over here This right here is a tool that draws a curve called a concoid So that's a sort of a specialty compass This is a shape This is a tool where these triangles slide to the left and the right and this line here can kind of move up and down It's sort of hinged right here and you can use this tool to double the cube So in this picture that length is the cube root of two This So-called Tomahawk, you this is one of the things on your sheet there We'll come back to that a little bit later. You can use this to trisect any angle There's all these interesting linkages which can do various things like trisect an angle This tool down here. This was a drawing device that was Invented by Descartes and you can use this to solve some of these problems and the origami crane here is a stand-in for A fascinating field of math that I know Plays a big role here at the Museum of Math where people try to take origami and look at it Sort of at a foundational level. What are the postulates that come out of origami? And if we could have sort of equivalent postulates to Euclid's postulates You can ask what can you construct with origami if you have some Certain rules of what you can and can't do with origami, but it turns out you can for instance trisect any angle using origami Okay, so let's look at some examples. So this one I thought you could follow along if you want to so in your on your sheet here We have a marked straight edge So you can use this just like an ordinary straight edge. You can see there is one mark on the side And so it turns out that Archimedes the great Archimedes showed that if you had a marked straight edge and a compass You can trisect any angle so for this activity I want you to use the marked straight edge and The rusty compass number one so this is the bigger circle here I will add as a little parenthetical note here that I'm calling these rusty compasses There's a whole area of math where people look at what geometry you can do if your compass has rusted in place And can't open or close so it's set to a fixed radius And so this disc is acting like your rusty compass and just in case you're curious. I've done this in advance so that the Radius of the rusty compass is exactly the distance from the mark on those straight edge to the end of the straight edge I will also add that The marks straight edge Some people say it has one mark some people say it has two marks Really you have the mark on the straight edge and you know where the end of the straight edge is so it can measure exactly one distance Okay, and so what you can do is draw any angle you want on your blank piece of paper So here's my angle and we will I will show you Archimedes procedure for trisecting this angle using a marked straight edge Okay, so draw any angle So the first thing I want you to do is to extend extend one leg of your angle down here Just as far as you want doesn't matter how far it is The next thing I want you to do is To take your rusty compass and to draw a circle that has the center at the vertex of the angle and you can draw the The rusty compass number one. Yeah, rusty compass number one. Sorry rusty compass number one so draw along here I went through the Cardstock and punched out the the center with a with a nail So hopefully you can see that if you have to clear it out with your pencil You can so this is rusty compass number one the larger one And so I'm showing it here as a circle where the radius is the distance between the marks Okay So it looks like I'm already using the marked straight edge and in a sense I am but so far. I'm not doing anything that Euclid could not have done Right. I'm not doing anything that Euclid could not have done. It's the next step where we really use the rusty compass Okay, so this step is going to take a little dexterity on your part. Let me show you exactly what's going to happen You're going to take the mark is going to be at this end So this is the this is the short distance from the mark to the corner and what I want you to do is to Arrange it so that the straight edge passes through this point of intersection So that this corner is sitting on the horizontal line and so that your mark is exactly coinciding with the circle so we're really lining three things up this point of intersection the mark and the circle and the corner on the horizontal line Okay, and so if we do that So it's a little tricky. You can see why it's not Euclidean here. We're lining three things up So this is what you cannot do with an ordinary compass and straight edge Okay, and if you draw that line segment then this angle is exactly theta over three So this this angle is exactly one-third of theta and I realized after the fact I should have made a printable Protractor on your piece of paper so you could actually measure it If you have one at home, you can try measuring it But this this angle is theta over three if this angle is theta And if you want to prove it, I won't prove it for you But you can if we draw this line segment here from the vertex of the angle to this point where The line crosses the circle you see that this right here is an isosceles triangle And this is an isosceles triangle and if you just sort of chase around these angles You can see that that this angle is three times this angle Okay So this is Archimedes method of tricycling an angle with a marked straight edge. It turns out that the marked straight edge Lived on for a long time. There are people who were interested in this Geometric technique viet was very interested in this you can do all sorts of interesting things with this It turns out that Crockett Johnson Who you may know as the author of herald in the purple crayon and he's also He's author of the comic strict Barnaby He was interested in mathematics and he found a way to construct any Construct a regular heptagon seven-sided polygon using a marked straight edge. And so that was in the 1970s I believe so this is had a long life the marked straight edge Okay, so that is one of our specialty drawing tools. I Also want to share with you a construction that uses the tomahawk But I think I'm gonna postpone that until After that'll be our ending activity after talking about the impossibility of these problems So we will return to the tomahawk at the end of the at the end of the presentation Okay, so adding new tools is one way to solve these problems another way to solve these problems is to add new curves So suppose you have lines and circles like compass and straight edge and and then fill in the black fill in the blank with any Curve that you know about so can you solve any of these problems of antiquity if you have one of these specialty curves? Okay, so let me give you an example of how you might use a specialty curve to solve these problems So this curve is a very old curve. So this is late 5th century BCE so this is well before Euclid or Archimedes and this is a curve called the quadratrix and We can use this to trisect any angle and so let me show you how this works It's a little bit confusing how it's drawn. So I'll show you two times so we're gonna draw this square here and What I want you to imagine is that this top of the square is gonna fall like the floor of an elevator and This side of the square is gonna tip over like a tree falling over Okay, and they're both gonna happen at the same time and they're both gonna reach the bottom at the same time and Where these two line segments cross? That's what's gonna trace the quadratrix. So I'll show you twice here So these things are falling and that that curve is what we call the quadratrix so we can show that again So this is falling like an elevator. This is falling like a tree and That curve is the quadratrix Okay, so let's see how we would use this curve to trisect this angle Okay, so the first thing we can do is drop a perpendicular from here over to this side I wrote a here what I mean by a is that this distance right here is a and It's impossible to trisect any angle, but it's not too difficult to trisect a line segment So we can find the one-third point here on this on this line segment right here And so we can draw a perpendicular from there over to here And this gives us this point right here where this horizontal line crosses the quadratrix and that gives us exactly the point that we need to trisect the angle and so this might seem Weird and confusing, but I claim that the quadratrix was created Expressly for this purpose. So if you think about it Think about this line the floor of the elevator being here It falls two-thirds of the way down when it gets to here This is like the tree tipping over it falls two-thirds of the way down when it gets to here And so that's exactly what this point of the quadratrix means and so it exactly trisects this angle So that's why this curve was was created Okay, it turns out the quadratrix can also be used to square the circle That's a little bit more of an involved discussion and actually that's where the name came from and that was much later That was after Archimedes But here's an example of how you can use a new curve to solve one of these classical problems Okay Okay, so now we're gonna jump ahead we're gonna jump ahead like 2,000 years And one of the really fascinating things about these problems is they are problems in geometry But really we need we need something new to tackle them We need algebra and we need a good understanding of numbers And so the Greeks did not have the tools to prove them and they did not have the tools to prove that they were impossible We had to wait for geometry to be Invented or discovered okay, and this whole episode of geometry coming into being I found it extremely fascinating to learn about the history of algebra There was a lot of pushback at the time Mathematicians thought that geometry was the way that mathematics should be done and this new new fangled algebra Was sort of mathematically heretical they did not like it at all and to some extent people still feel that way today So I found this amazing quote by Michael Atia who just passed away a couple years ago And he had he said the following so this was in 2001 But it really could have been written in the 17th century So he wrote algebra is to the geometry what you might call the Faustian offer Algebra is the offer made by the devil to the mathematician the devil says I will give you this powerful machine It will answer any question like you like all you need to do is give me your soul Give up geometry and you will have this marvelous machine Of course, we would like to have things both ways We would probably cheat on the devil pretend We're selling our soul and not give it away Nevertheless the danger to our soul is there because when you pass over to algebraic calculation Essentially, you stop thinking you stop thinking geometrically you stop thinking about the meaning so there really was a lot So this this really was the sentiment when algebra was being introduced And this spring I gave a talk at an MA meeting in Tennessee And Tom Banchoff who is a retired Mathematician was also giving a talk at that same conference and he said the following thing he said I have nothing against algebra except when it takes the place of geometry So I wrote this down and added the slide to my talk which was later in the afternoon And I put it up there and afterwards like can you give me a copy of that slide? So this was this spring I also this has nothing to do with my talk, but during his talk he had this This is a donut shaped balloon so you could blow this up and it's a donut Which I think is just awesome and has nothing to do with my talk, but I want I want one of those You should sell them in the no math gift shop So so this was The introduction of algebra and one of our heroes of this whole story is Descartes and so Descartes really was a geometer but he started the process of using algebra to solve geometry problems and He had a lot of contributions to mathematics the way I want to phrase what he did was he showed how to add and subtract line segments to multiply line segments To divide line segments and to take square roots of line segments and I'll say more about what I mean by that And if you can do that then you can do algebra on these geometry problems So here's an example suppose we have these two line segments and we want to multiply them So if we multiply these two line segments, what do you think the classical object that you would get if you were to multiply these two line segments? Yeah, an area a rectangle right you get a rectangle, so that's sort of the obvious thing to do the problem is If we multiply two line segments, it would be great to get another line segment and not to get and not to get an area and so Descartes figured out how to do this and He had an invention that was ingenious and what he invented was The number one so he invented the number one so what I mean by that is Descartes took some line segment on his Geometrical drawing and he said I'm gonna call this one So for the rest of this problem This is one and it turned out that was the ingredient that that he needed to be able to do arithmetic with line segments Okay, so let's see how that would work. So let's say we want to multiply Line segments a and b so you can draw an angle any angle. It doesn't matter We're gonna draw here a line segment of length one whatever we have chosen to be length one We draw a line segment here of length a we draw a line segment here of length b Okay, we can connect these two dots. We can draw a line parallel to this line And let's call this x right here. So what geometric idea pops out at you when you see this picture Trapezoids. Yes. Yes similar triangles. Yeah, so similar Yes, trapezoids, but similar triangles is what we need here. So we have similar triangles and If you look at this the right way, you could say a is to one as x is to b That gives you this relation right here and therefore this length x is what we're gonna call the product of a and b and if you imagine making this with one of those online Geometry applets and let's say that this was this was movable that you could change the length of one as you move This length you can imagine this point is also going to change and so the length a and b is going to change It's not well-defined. It's only well-defined when you've chosen how long one is and then for the rest of your problem You have this idea of You know how long the unit is Okay, and in case you don't believe me and hopefully you do believe me Here is the the very first paragraph of Descartes geometry so the title of this chapter is Problems the construction of which require only straight lines and circles. So we're really talking about compass and straight edge problems That's what Descartes writing about and so this is literally the first page of his book. Let me read this to you It says any problem in geometry can easily be reduced to such terms that a knowledge of the length of certain straight lines is sufficient for its construction Just as arithmetic consists of only four or five operations namely addition subtraction multiplication Division and the extraction of roots which may be considered a kind of division So in geometry to find required lines It is merely necessary to add or subtract other lines or else taking one line Which I shall call unity so that was his big contribution here in order to relate it as closely as possible to numbers and which can in General be chosen arbitrarily and blah blah blah and then he talks about the procedures for multiplying and dividing and taking square roots And so forth So this is what Descartes wanted to do He wanted to take his geometry problem Turn it into an algebra problem solve the algebra problem and then go back and and show how the geometry problem is solved Okay, so here's one thing that one way to look at what Descartes did. He said that Let's say we start off with a line segment of length one and a is some positive number And let's say that we can write a using just the integers and addition subtraction to multiplication division and square roots Then by using his techniques, we can construct a line segment of length a Conversely if you think about it if we're using a compass and straight edge We're only ever intersecting lines and lines or circles and circles or lines and circles And if you try to solve for these points of intersection at worst, you're gonna be solving the quadratic equation So you're not gonna get anything worse than square roots in in your final answer Okay, and so here's a theorem. This is not the way Descartes stated it, but this is the way we are going to We're gonna Ask about constructing a line segment of length absolute value of a and so what he proved And we're gonna call those a's that can be constructed these line segments We're gonna call those constructable numbers So a real number a is constructable if and only if it can be formed from the integers using the four arithmetic operations and the square root Okay, so these last few slides are a little abstract. You might have dozed off here Let's do some concrete examples, and I apologize These are all gonna be at the bottom of the screen We do have TVs over there if you need to to see them. So here's an example of a number So is this a constructable number? Can we construct a line segment of length one plus a square to five over two? Yes, is this a famous number Yes, it's the Golden ratio, right? So this is a this is a famous number actually if we went back to our regular pentagon And if the side length of our pentagon was one then the diagonal of our pentagon Is the golden the golden ratio here? So this we know from geometry that this is constructable But we know from from Descartes work that it is constructable because it only uses integers the four arithmetic operations and the square root Okay, so this is a famous number. The next number is also famous Okay, I'm kidding. That's not a famous number, but is it constructable is this number constructable? It is right. It satisfies. It's written using the integers square roots and the arithmetic operations So this is a constructable number What about this is this a constructable number? It is right. It's a fourth root, but a fourth root is the square root of a square root. So that is constructable Now we're really getting somewhere so the cube root of two So we want to know is this a constructable number. Can we write it using the arithmetic operations and square roots? Probably not I mean it looks like you probably couldn't I don't know if that's a proof though But probably not But it but you can see we're getting closer to being able to answer these questions About the impossibility of these problems, but if you get too confident about this and you're like, yeah There's no way we could construct this then here's a cautionary example So this one looks just as bad as the previous one, right? but That number is actually just one plus a square root of two This is easy to verify if you cube the left-hand side you get seven plus five root two If you cube the right-hand side you can if you're bored you can do it on your paper if you cube that you're gonna get seven Plus five root two so even though it doesn't look like it that is a constructable number So that makes us a little bit uneasy, but at least at least it looks like we're heading in the right direction And finally can we write pi using these arithmetic operations and square roots only again? Probably not, but we definitely don't have a proof at this point Okay, so Descartes pushed us a long ways towards solving these problems So this is just a reminder of what our what our four numbers are here So remember we start off with a unit line segment and we want to know are these Constructible numbers or are they not constructable numbers if the cube root of two is constructable Then we can construct then we can solve the problem of doubling the cube if cosine theta over three is Constructable for every theta then we can trisect any angle if cosine 360 over n is a constructable number for all n Then we can construct every regular polygon and if pi is a constructable constructable number we can square the circle Okay, so we're rephrasing our problems in what seems to be a much better way Okay So let's talk about regular polygons. So this is one of the mathematical greats Gauss And out of curiosity how many of you are 19 years old or younger a? Few of you okay, so some of you some of you still have some time when Gauss was 19 years old he Proved that you could construct a reg you could construct a regular 17 sided polygon So he proved this when he was 19 years old Notice I didn't say that he constructed it what he did was he proved that you could construct it So remember what that means that means that cosine of 360 over 17 is a constructable number Okay, and it turns out what that number is is that number from the earlier slide? So I said it was famous, but then I said it wasn't famous, but actually it is a famous number So cosine of 360 over 17 is this long thing and when Gauss got to this point He knew that that the 17 gone is constructable Okay, and this picture here actually is a postage stamp that that I got on eBay Here's Gauss's 17 gone here is a compass and that's not really a straight edge But I guess it has a straight edge here So that's kind of cool Gauss was extremely proud of this result And he often said that this is what made him want to continue to pursue mathematics He actually wanted the 17 gone on his tombstone. I don't think that actually happened But years later they built a monument to Gauss and Brunswick and if you look at The side by his foot here if you zoom in it does have a 17 pointed star Which is in honor of his 17 gone. So that's kind of neat Okay, so here's what Gauss actually wrote he says it seems that one has persuaded oneself ever since Euclid's time That the domain of elementary geometry could not be extended at least I do not know of any successful attempts to enlarge its boundaries on this side It seems to me then to be all the more remarkable that besides the usual polygons There is a collection of others which are constructable geometrically for example the 17 gone So actually Gauss did more than just show that the 17 gone was constructable. He found other ones as well So if we go back to our chart here, we can color the 17 gone green It turns out if you read what Gauss did this is probably not a surprise to anyone It was very deep it had to do with geometry, but it also had to do with complex numbers And also surprisingly number theory fell into the to the conversation here So again, I apologize that this is at the bottom of the screen. You can look at the TV monitors if you need to see it We need to talk about something called a fair ma prime So Pierre de Fermat was looking for a prime generating function a function that would spit out prime numbers And he observed that two to the two to the m plus one Seemed to generate prime numbers. So for example when m is zero we get two to the two to the zero plus one Which is three two to the two to the first plus one is five two to the two squared plus one is 17 And then it continues 257 and 65,537. Those are all prime numbers And so he believed that this would just continue continue generating prime numbers And he was not able to prove it, but he left it out there as a conjecture It turns out unfortunately for Pierre de Fermat that the very next the very next Fermat number is composite and does anyone know who factored that next number Another one of the greats Euler so Euler was able to factor the next one and it turns out that as far as we know none of the rest None of the rest that we know are actually prime. They all seem to be composite, but we do have these five and what gauze discovered is that if n is a Product of distinct fair ma primes and then remember you can always double the number of sides of a polygon But if it has that form Then the regular n-gon is constructable Okay, so remember we had three and we had five and we had three times five which is a product of them 17 is another one like 15 times 17 would be another constructable one etc note that it says Distinct fair ma prime so nine is three times three, but it's not the product of distinct fair ma primes and Seven is not a fair ma prime at all. So that does not fall under gauze's theorem. So You might think all right. We're done with one of the four problems But actually we're not this doesn't go the direction that you want it to go This says that if the number has this form then it's constructable So it doesn't say that these are impossible. It just doesn't fit the hypotheses of the theorem and so as you might imagine gauze did think about the converse of this theorem and In typical gauze fashion. He gave a really annoying Comment about that. This is what he wrote. He said the limits of the present work exclude this demonstration that the others are not Constructible, but we issue this warning lest anyone attempt to achieve geometric constructions for sections other than the ones suggested by our theory For example sections into seven eleven thirteen nineteen etc parts and so spend his time uselessly So in other words gauze is saying the rest of them are not possible. Trust me Don't waste your time But he didn't prove it and as far as we know he never wrote it down. It has not been discovered in his notes anywhere So gauze was saying that this is an if and only if theorem and that these are impossible The red ones are impossible the green ones are possible, but he did not prove it So it turns out gauze was correct. That's also probably not a shocker But we had to wait a little while to get a firm proof that this really is an if and only if theorem Okay, and so it turns out that we can thank one person for Knocking off three of these impossibility problems and this is a mathematician who probably needs no introduction to you Because you probably all know him right. Let me here's Pierre one cell, right? He is he is one of the main heroes of our story, but he is not a very well-known mathematician I did a screenshot of his Wikipedia page, which is longer than my non-existent Wikipedia page But for someone who solved three of these four problems. That is an unbelievably small Wikipedia page. We don't even know what pure one cell looked like the math mathematical writer Brian Hayes wrote this quote and I love it. It says Brian Pure one cell is hardly a household name even in mathematical households It turned out pure one cell died very young He was not sort of part of the mathematical establishment. Here's what one of his contemporaries wrote after he passed away He wrote one cell was blame worthy for having been too rebellious to the countenance of councils of prudence and of friendship Ordinarily he worked evenings not lying down until late Then he read and took only a few hours of troubled sleep making alternately wrong use of coffee and opium and taking his meals at Irregular hours until he was married. He put unlimited trust trust in his constitution very strong by nature Which he taunted at pleasure by all sorts of abuse. He brought sadness to those who mourn his premature death So he is the hero of our story and we don't know all that much about him Okay, so let's just put things in context here Descartes geometry was 1637. So we are now 200 years after Descartes now algebra is really a Mature field and as you can see from this theorem the language the algebraic Lambert language is very modern so Pure one cell proved the following thing that if a is a constructible number then it is the root of an Irreducible polynomial f with integer coefficients and the degree of f is a power of two so in case some of these terms are unfamiliar to you Being the root of a polynomial means that you get zero when you plug it into the polynomial Irreducible means this has integer coefficients You can't factor it into a smaller into two smaller polynomials with integer coefficients. So you can't factor it any further The degree of a polynomial is the largest power of x and power of two just means two four eight sixteen Etc. Okay, so this was the theorem that we needed and actually we need the logical Contra positive of this so the contra positive says that if a is the root of an irreducible polynomial with integer coefficients Whose degree is not a power of two then a is not a constructible number And so with this information we can solve three of our four problems So let's look at the qubert of two The qubert of two is a root of this polynomial right x cube minus two if we plug it in here we get zero You can prove that this polynomial can't be factored any farther So it's it's irreducible and the important fact here is that this exponent is three and three is not a power of two and therefore It's impossible to double the cube right the qubert of two is not a constructible number Okay, the angle tri-section problem was the next one. So remember we can Trisect some angles the question is can you trisect every angle so to understand this we need This trig identity, which I don't know about you, but I never Memorized this when I was in school, but if you plug in 20 degrees for theta here then this is 60 degrees and if you play around with this a little bit You'll see that cosine of 20 is a root of this polynomial with integer coefficients Okay, it turns out that this polynomial is irreducible and again, it has a exponent of three a degree of three and therefore cosine of 20 is not a constructible number which means you cannot Construct a 20 degree angle, which means you cannot Trisect a 60 degree angle and so that solves that problem proves that it's impossible The proof for the regular polygons is a little bit more involved, but it's still the same kind of idea I'm not going to talk about that But I do want to share with you what I think might be the greatest page in all of mathematics So it turns out one cells article was pretty short And it turns out that all three problems were proven to be impossible on the same page So if you look here, this has the doubling of the cube up here near the top of the page a little farther down We see the angle trisection right you can see here's our polynomial for example, and here's our polynomial and Down at the bottom here is the division of the circle into the circumference into n parts So this is the regular n-gon problem So I think this is perhaps the greatest page in all of mathematics you guys probably know that Newton and Einstein had their famous Miraculous year and so I think we should call this the miraculous page this this one page here Okay Okay, so finally so that was three of the four problems Once I'll solved all three of these the fourth one the most famous one was the trickiest of them all and this one took Longer to solve and so you guys may know that Abraham Lincoln Was a fan of Euclid's elements he liked to work through these geometry problems and thought that this way of geometric thinking was good for a lawyer and future president and so I came across this amazing anecdote of his law partner Herndon Walking in on Lincoln doing some geometry and so this is when Lincoln was on the circuit court And so this is what Herndon wrote. I will read it to you He wrote Lincoln was sitting at the table and spread out before him lay a quantity of blank paper large heavy sheets a compass a rule numerous pencils several bottles of ink of various colors and a Perfusion of stationary and writing appliances generally he had evidently been struggling with a calculation of some magnitude For scattered about or sheet after sheet of paper covered with an unusual array of figures He was so deeply absorbed in his study that he scarcely looked up when I entered I Confess I wondered what he was doing when he rose from his chair He enlightened me by announcing that he was trying to solve the difficult problem of squaring the circle For the better part of the succeeding two days He continued to sit there and grossed in that difficult if not undemanding proposition and labored as I thought Almost to the point of exhaustion. So this at this point The square the problem the square in the circle was still an open problem. It had not been proven impossible I think mathematicians knew that it was probably impossible But it was not proven to be impossible when Lincoln was working on this and so our final question is is Pi a constructible number and so pi has a long and rich history, which I'd love to share with you I Believe that pi should be called Archimedes number he Contributed to our understanding of circles and of pi in so many different ways Brian mentioned that article I wrote about who first proved that the circumference over the diameter is a constant the punch line is Archimedes He also came up with these famous bounds for pi and and so many other things related to the problem of squaring the circle But Archimedes didn't solve that problem A little later not a little later 2,000 years later Johann Lambert proved that pi is an irrational number This is an amazing theorem and quite an accomplishment But this also isn't what we need to say that it's impossible to square the circle And so finally what we need let's go back to Pierre von cells theorem. Remember Pierre one cells theorem was all about When these numbers are roots of a polynomial and it turns out that there are certain numbers that aren't roots of any polynomial with integer coefficient These are called the transcendental numbers and so if a number isn't the root of any polynomial with integer coefficients Then it is certainly not the root of a polynomial of degree 2 to the end And so the final piece in the puzzle was proven by Ferdinand von Lindemann in 1882 He proved that pi is a transcendental number and therefore it's impossible to square the circle And so that was the ending point for these four very famous problems So that is the end of the story. I promised you we'd have one further activity And so we are probably running a little short on time, but I will I'll walk you through this this problem Which you can carry out at your table so you don't have to draw anything just yet this This device is called a tomahawk. We don't actually know who came up with this geometric shape the earliest we can find is that it was referenced in 1835 and So you can use this tomahawk to trisect any angle and so here is how it works. This is very specially designed device where if you line it up so that this corner of the tomahawk hits one line coming out of the angle this edge Passes through the vertex and then this circular arc right here is Tangent to this line segment. So if you line those three things up through this through this corner the edge passing through the vertex and This guy being tangent, then it turns out that this right here is One-third of the angle And so this might seem mysterious The math behind it is that if we take this the head of this tomahawk and divide it into three equal line segments And then this the radius of this circular arc is the same as this segment right here Then hidden in here you have this right triangle this right triangle and this right triangle And those are all congruent right triangles and therefore these three angles are all the same So you have to specially design actually someone point out to me What's interesting is you can you can make the tomahawk with a compass and straight edge So you can make a device with a compass and straight edge that can solve these problems But you can't use a compass and straight edge to solve these problems So as a final activity Let's walk through constructing the regular nonagon. So I'm going to walk you through this You can follow along at yourself if you'd like if you want to just try it on your own You can block your ears here if you want so For this one, I want you to start with the rusty compass number one so this is the big circle and You want to draw the big circle here and put a little mark where the center of the circle is and Our objective our objective here is to construct a regular nine-sided polygon Inside this circle. Okay, so the next step that I want you to do is pretend like we're constructing the hexagon Okay, so you can take that same circle and put the center anywhere on the circumference and So the circle is going to pass through the center of the original circle and you'll just mark off where it crosses the circle on the Your original circle so you can have this point which you can mark this point which you can mark and also mark the center So that's gonna be right there So right now we're gonna have four points this point this point This center and this point on this circumference Okay, so this was this was the way we started off constructing the regular hexagon And we're gonna do one more step that's like that So if we draw a line segment through these two points, that's gonna give us our point on the opposite side here So we draw that that gives us this point So I don't want to I want to get ahead of you guys Okay, so right now we have these four points on the circumference And if you notice this point right here this point right here and this point right here Those are vertices of an equilateral triangle which are going to be vertices of our eventual nine-sided polygon And so we could Trisect this angle using our Using our tomahawk you could turns out the geometry is gonna be a little bit weird because of the size of your tomahawk So it turns out what's gonna be easier for us is to draw these two line segments here. So let's draw This one and this one so draw these two join these two and join these two Okay, it turns out it's gonna be easier with your tomahawk to Trisect this angle right here. So so these were the vertices of our equilateral triangle We join these two to get a line segment. We join these two to get a line segment So so far we've done nothing that Euclid could not do okay, but it's the next step that is non Euclidean So now we want to use our tomahawk. So remember how this works you want to line it up so that The corner is on this ray the blade is tangent to this Line segment and the handle the left hand side of the handle passes through this center So when you get that Draw a line segment along here Okay, that it requires a little finagling to get this to work So what this is gonna do it's gonna trisect this angle right here Okay, so So that that gives us this point right here Okay, and I claim that this point and this point are adjacent Vertices on the regular non-agon. So let's let's look at this angle right here So this angle is 60 degrees and so this angle right here is two-thirds of 60 degrees. So that's a 40 degree angle So that's a 40 degree angle, but 360 divided by 9 is exactly 40 so this is 1 9th of the circle and if I did everything right on my printout This should be exactly the radius of the rusty compass number 2 so if you had an actual compass if you had an actual compass what you'd want to do is Sweep out this arc and that would find your next one and then sweep out the next arc and work your way around But now you should be able to put the center of rusty compass number 2 here And that's gonna give you a point over here and you can sort of march your way around And that should give you all of the vertices of your regular non-agon using this compass so if you have if you did everything correctly and if I did everything correctly making the worksheet your rusty compass number 2 should be able to give you all these edges here Okay, and so I think that is probably a good place to end Thank you guys so much for coming out and it was nice to speak to you All right, if anyone has any questions you can raise your hand. I'll bring a mic Given that we can use the the compass and straight edge to To draw a Mark straight edge and to draw a tomahawk in no different a manner to which we draw any ordinary geometric construction by just making a mark Just outside the paper How how is that cheating are going beyond the rules in some way Give given that we're using the same tools to make yeah the extra construction. That's a good question I mean, I think it's sort of it's sort of meta thing. So can you use these tools? To make new tools to solve some of these problems and you can do that Like you said you could do the you could create the the mark straight edge and you could construct the tomahawk and so forth so I mean, I think The way these problems are structured is you have a well-defined set of rules and making new tools is not one of the Is not one of the the options that Euclid gave gave you so I think it's a neat fact that you can Use the compass and straight edge to make tools that allow you to solve new things But that doesn't mean that you can solve them with just the the compass and straight edge So it's all about what the rules are and so that would be breaking the rules so to speak That's a great question So given that the algebra here Was able to solve a geometric problem. Is it now possible at all to trace backwards from the algebra and create a geometric proof? Not that I know of I mean, it's pretty Pretty elaborate. I mean the the amount of algebra that needs to be built up is is so sophisticated It would be hard to work your way backwards and I would say historically. This is what this is what geometers would have That this that's exactly the complaint that that generations of geometers would have is that the algebra hides all this geometry So it's true that you could use algebra to solve these geometry problems But it's sort of impossible to follow the geometry behind of the algebra and so that's that's what all these These issues were with people who are refusing to accept Algebra into mathematics. I also want to mention that this clash that I read about between algebra and geometry reminded me a lot of You know when I was younger, let's say in the 80s and in the 90s when computers and calculators were starting to enter the scene and being able to solve mathematical problems and You know the older generation did not like that. They thought this is not the way we do mathematics But the younger generation said, you know, this yeah, this is how we do mathematics This is fine. And so the older generation is retiring and the younger generation is coming along and So I think these kinds of complaints are very That is a very historical complaint. And that's you know that we can't follow the geometry while we do all this algebra Thanks. So let's give the speaker another hand