Category Archives: Geometry

Rise and Fall of Wasan

Since most would not be able to read my report on Sangaku (Artfully done tablet of Geometry) I thought I would do a blog post on the rise and fall of wasan (Japanese Mathematics) for you all to enjoy. Never mind it’s a topic I know a lot about now and have done a bunch of research on the topic.

The Japanese didn’t really have their unique math until about the year 1627 when Jink ̄o-ki was published. This was the first Japanese mathematics book published. The Jink ̄o-ki was a Japanese publication that explained how to use the soroban (Japanese name for the abacus) to do things like calculate pi, and would provide other math instruction and problems. Until then, much of the learning and study in math came from the classics of China, with heavy emphasis on The Nine Chapters and Cheng’s Treatise. I’ll explain a bit as to why there was such a long delay in developing mathematics.

Leading up to the late 1500s, most uses for math in Japan was to levy taxes on the land and for basic arithmetic for business transactions. The government of the time actually created The Department of Arithmetic Intelligence to go to each landowner and measure the property so the owner would know how much tax to pay.

These math specialists only knew just enough geometry to get the area of the land and calculate the tax required. The government saw math as a means to an end for acquiring money. This meant that math was a tool used by the government and only a special few were educated in mathematics as deemed necessary. But since there was no one to teach them, they had to rely on the Nine Chapters to be their teacher.

Around the early 1600s things began to change. A new set of rulers named the Tokugawa family took over all of Japan, uniting all the land under one government. Taxes were no longer tied to the amount of land owned and The Department of Arithmetic Intelligence was no more. This, in turn, led the farmers to no longer know how much land they had and, as consequence, the amount of food they could produce.

The Tokugawas also brought about another important change, the closing of the Japanese boarder. Iemitsu Tokugawa outlawed Christianity and closed the boarders. The problem was that a growing number of converts started a community together and began to band together. At the same time the Spaniards attempted to compete for converts and in a bold attempt to be the only missionaries in Japan told the Tokugawa family that the other nation’s missionaries were trying to create an army to conquer Japan. The Spaniards’ plan backfired and all missionaries were put to death along with those that would not give up on Christianity.

With the closing of the boarders and all the enemies of the government crushed, a period of peace was created called the Edo period in Japan that lasted until 1868 when boarders were opened again. It was during this period that the Japanese culture became its own and flourished. Everything from haiku poems to flower arranging to tea ceremonies was created during this time. By the end of the Edo period a gentleman was expected to know “medicine, poetry, the tea ceremony, music, the hand drum, the noh dance, etiquette, the appreciation of craft work, arithmetic and calculation . . . not to mention literary composition, reading and writing.” (Hidetoshi)

During the time of Great Peace the samurai became the new noble men of Japan. No longer needed as warriors, many were given government jobs to help ease them into normal lives. As consequence, the men became some of the more educated citizens. That being said, the pay they received for working for the government was terrible. Most samurai had to pick up 2nd jobs; many of them become traveling schoolteachers.

The stage was now ripe for an explosion of learning. We had farmers that needed to learn math, we had samurai that needed second jobs, and a place for it all to happen, the local shrine or Buddhist temple. Since there were no school buildings, most lessons happened at the shrines and temples that dotted the land. This encouraged more people to gather together for religious, educational, social functions. Over the next century the Japanese people would have the highest literacy rate of all the nations and become one of the most educated.

During this time, the people began to make sangaku, which is basically an artistically made wooden tablet containing a geometric problem and most of the time the solution. These tablets would adorn the temples and shrines showing off the newest knowledge learned. However, these tablets also had a deeper meaning. These sangaku became a way of thanking the gods and spirits for the new knowledge.

Many of the sangaku that have been found focused on finding lengths, areas of various shapes, and even volumes. The sangaku found below is one example of finding a length. The problem asks to find the diameter of the north circle inside of the fan. The problem is setup so that the entire area of the fan is a third of a circle and you can assume you know the diameter of the south circle. The answer ends up being (sqrt(3072) + 62 )/193 times the diameter of the south circle.


Sadly, wasan (Japanese Mathematics) was one of the few things that didn’t survive the Japanese Renaissance, which is why many of the records of wasan and sangaku are only now being discovered. At the end of the Edo period a new government was formed that outlaws wasan from being taught. It turns out that wasan lacked Calculus but more importantly, was different than the rest of the world. With the opening of the boarders, the government needed to adopt Western Mathematics to be able to communicate with all the new trade partners that were being re-established. To that end, a law was created that outlawed wasan and Western math was forced in the schools. Anyone that still taught wasan had his teaching license stripped and imprisoned.

Reference:

Hidetoshi, F., & Rothman, T. (2008). Sacred Mathematics. Princeton, New Jersey: Princeton University Press.

What Does Being Correct Mean?

In class, we were discussing the Parallel Postulate by Euclid. Basically it says that if you draw a straight line on top of two other lines so that they intersect, and if the angles on the same side of the first line are less than 2 right angles (180o), the two lines will intersect at some point on the same side.

Image: 6054, via Wikimedia Commons.

It’s weird learning about proving something that feels so elementary that I assumed it was just true by definition. I mean I can just look at the picture and it certainly looks like it should be correct just by careful inspection. But I guess that doesn’t really prove it without a shadow of doubt. What if what I was looking at was 179.999o and I just said they would never touch even though they would intersect given enough space. Granted, I would assume it was 180o so I would be correct based on the assumption being true.

When I look at this problem, I can’t help but reflect on the lessons, experiences, and “truths” that have instilled within me from previous mentors and teachers. It becomes very hard to try and think about other approaches or ideas other than “duh that true”.

What allowed me to think about this Postulate was learning about how other people through out history thought about the Parallel Postulate and created their own “new math”; their own pseudogeometry; their own imaginary geometry. Here I am unable to think “outside the lines”, but these other people created whole new systems from looking at the problem from a different angle. I have no problems creating weird parallels with my jokes and puns but can’t seem to do the same thing with math. (Yes, I love bad puns).

Poincare and Lobachevski were both people that worked in this pseudogeometry, which is now called hyperbolic geometry. (The former or “normal” geometry is considered “Euclidean Geometry”). In hyperbolic geometry it’s possible to have lines that would normally intersect in Euclidian space be considered parallel and non-intersecting in hyperbolic space. I think looking at the picture below will really help. I know it wasn’t until I built a hyperbolic plane by hand that it really sunk in for me. ( Make your own at http://www.math.tamu.edu/~frank.sottile/research/subject/stories/hyperbolic_football/index.html )

A hyperbolic triangle. Public domain, via Wikimedia Commons.

Reflecting on the on hyperbolic plane I began to try to remember a time when what the instructor was teaching conflicted with something I already knew. As I thought I remembered something an art teacher told me about vanishing points. So imagine you’re standing on some railroad tracks that stretch straight forward for miles. As you look down the tracks, as you would if you were actually a train, at some point the individual components would become one whole line. Instead of seeing the left rail, the right rail, and everything else you would see a railroad track. At that point, the left and right rails have effectively become one, unable to tell them apart. Now what would happen if a train went down those rails that look like they became one? The train becomes smaller, or at least, it looks like the train is shrinking. At the time I could only think about how the teacher lost her mind. It wasn’t until I looked down a straight road that I realized how right she was.

After thinking about how perspective is everything I began to wonder what other things are different than they appear? I asked a friend, and she mentioned she actually had to unlearn some thing to be able to Fence (as in the sport) correctly. She told me that she had to change the way she extended her arm in order to be able to obtain the longest reach possible.

It turns out that a straight line with your arm is not the best way to have the longest reach. In all my learning, I had been taught that you to get the longest linear distance with line segments are to put each line segment end to end along the same axis. But in fencing, doing just that with your arm is not the longest. Why is fencing different?

When hold the sword in your hand, it seems that your muscles tighten to hold the load and your arm up. By tightening your muscles, you shorten your reach by as much as 2 inches for some people. When your muscles are relaxed, the joints can loosen allowing more space between the bones, which lengthens your arm. So by relaxing your arm a bit so it’s not parallel with the ground, your sword can reach just a little bit further.

Is math wrong when it comes to the physics of people and fencing? Absolutely not! In my case, it’s the model that the math was used on that was wrong. I assumed the arm was a rigid object with hinges at the shoulder, elbow, and wrist. Since I had modeled the arm in this fashion any math done to the model would never take into account the possibility of expansion of the hinges. Assumptions are the downfall of many people.

Proof of the Pythagorean theorem

History of the Pythagorean theorem

The Pythagorean theorem is one of the greatest scientific discovery of the human, and it is also one of the basic elementary geometry theorems. There are also many other names to call this theorem, like Shang-Gao theorem, Bai-Niu theorem and so on. Someone maybe will ask that what is the Pythagorean theorem. According to Wikipedia, the Pythagorean theorem “is a relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.[1]” This theorem has a very long history. Almost all ancient civilizations (Greece, China, Egypt, Babylon, India, etc.) have studied this theorem. In the West, this theorem was called Pythagorean theorem. According to legend, Pythagoras, an ancient Greek mathematician and philosopher, was the first person to discover this theorem in 550 BC. Unfortunately Pythagoras’ method of proving this theorem had been lost and we could not see how he proved now. But another famous Greek mathematician, Euclid (330 BC – 275 BC), gave us a good proof in his book called Euclid’s Elements. But Pythagoras was not the first person who discovered this theorem around the world. Ancient China discovered this theorem much earlier than him. So there is another name for the Pythagorean theorem in China, the Gou-Gu theorem. Zhong Jing is the first book about mathematics in China. And in the beginning of this book, there was a conversation between Zhong Gong and Shang Gao. They were talking about the way to solve the triangle problem. From this conversation, we could know that they already found out the Pythagorean theorem around 1100 BC. They found this theorem 500 years earlier than Pythagorean.

Proof of the Pythagorean theorem

Usually in a right triangle, we need to find the length of the third side when we already know the length of other two sides. For such problems, we can directly use the formula to calculate. In many problems, we need this theorem to solve many complex questions. And then, I will introduce two basic method to prove the Pythagorean theorem.

1) Proof by Zhao Shuang

In China, Zhang Shuang was the first person who gave us the earliest proof of the Pythagorean theorem. Zhao Shuang created a picture of “Pythagorean Round Square”, and used method of symbolic-graphic combination gave us a detailed proof of the Pythagorean theorem.

Assume a, b are two Right-angle side (b > a) and c is Hypotenuse. Then each area of a right triangle is equal to ab/2.

ZhaoShuangRtTriangleGraph

[Fig.1] Proof by Zhao Shuang

∵ RtΔDAH ≌ RtΔABE,

∴ ∠HDA = ∠EAB.

∵ ∠HAD + ∠HAD = 90º,

∴ ∠EAB + ∠HAD = 90º,

∴ ABCD is a square with side c, and the area of ABCD is equal to c2.

∵ EF = FG =GH =HE = DG―DH , ∠HEF = 90º.

∴ EFGH is also a square, and the area of ABCD is equal to (b-a)2.

∴ 4 *(1/2)(DG*DH)+(DG-DH)2=AD2

∴ DH2+DG2=AD2

2) Proof by Euclid

Just like we said before, Euclid gave us a good proof in his Euclid’s Elements. He also used method of symbolic-graphic combination.

In the first, we draw three squares and the side of each square are a, b, c. And then, let points H、C、B in a straight line. Next we draw two lines between F、B and C、D and draw a line parallel to BD and CE from A. This line will perpendicularly intersect BC and DE at K and L.

[Fig.2] Proof by Euclid

[Fig.2] Proof by Euclid

∵ BF = BA,BC = BD,

∠FBC = ∠ABD,

∴ ΔFBC ≌ ΔABD

∵ The area of ΔFBC is equal to (1/2)*FG2 and the area of ΔABD is half of the area of BDLK.

∴ The area of BDLK is equal to FG2. And then we can find the area of KLCE is equal to AH2 with the same method.

∵ The area of BDEC = The area of BDLK + The area of KLCE.

∴ FG2+AH2=BD2

Conclusion

The Pythagorean theorem’s development has exerted a significant impact on mathematics. And this theorem gave us an idea to solve geometric problems with Algebraic thinking. It is also a great example about symbolic-graphic combination. This idea is very important for solving mathematical problems. By the Pythagorean theorem, we can derive a number of other true propositions and theorems, which will greatly facilitate our understanding of geometry problems, but it also has driven the development of mathematics.

Reference

[1] http://en.wikipedia.org/wiki/Pythagorean_theorem

[Fig.1] https://upload.wikimedia.org/wikipedia/commons/e/e0/ZhaoShuangRtTriangleGraph.PNG

[Fig.2]https://upload.wikimedia.org/wikipedia/commons/5/59/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem2.svg

Parallel Lines

I learned the parallel postulate in middle school. The best known equivalent of the postulate is attributed to Scottish mathematician John Playfair, and it says that “in a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.”

The reason that I have a special impression on this postulate may be probably due to a popular metaphor in my middle school period. That metaphor related the parallel lines with the mutual feelings between girls and boys: when a girl and a boy cannot stay together, or they do not develop a mutual affection, we say that they are like two parallel lines. No matter what the two parallel lines “do”, they cannot have an interaction. Similarly, for the two unlucky people, no matter what they do, they can never fall in love with each other. I have to say this metaphor describes a tragic situation and sometimes I do not feel satisfied with the “tragic” destinies of the two parallel lines. Fortunately, as my mathematical knowledge grows, I do find that in some other branches of geometry, the seemingly unbreakable law in Euclidean geometry no longer holds. Among the new branches are hyperbolic geometry and elliptic geometry, which will be the main topic of this blog.

Before we talk about non-Euclidean geometry, let me have a brief introduction to the differences between non-Euclidean geometry and Euclidean geometry. The fundamental difference between them lies in the parallel postulate. We already stated a widely adopted equivalent of parallel postulate in the beginning of this article. For two thousand years after Euclid’s work was published, many mathematicians either tried to prove this “fifth postulate” (in Euclid’s Element) or tried to show that it’s not necessarily true. Actually, even in Euclid’s own book, this parallel postulate was left unproved; Also, unlike the first four postulates, the fifth postulate — the “tragic” parallel postulate, was not being used to prove his following theorems in the book. A breakthrough in this topic came out in the 18th century. A Russian mathematician,  Nikolai Lobachevsky, developed the hyperbolic geometry. His most famous contributions are in two aspects: he convincingly showed that Euclid’s fifth postulate cannot be proved, and he presented hyperbolic geometry to the world.

Multiple parallel lines in hyperbolic geometry. Image: Vladimir0987, via Wikimedia Commons.

In the original parallel postulate, we said for any given line R and point P, there is exactly one line through P that does not intersect R; i.e., parallel to R. In hyperbolic geometry there are at least two distinct lines through P which do not intersect R, rendering the parallel postulate invalid. Hyperbolic geometry may be against common sense at first glance, because usually, our recognition about the shape of a space is limited to Euclidean space. However, hyperbolic geometric space does exist, one example is the saddle space with a constant negative Gaussian Curvature. Hyperbolic space is possible in dimensions that are larger than or equal to two. It is curved — the reason why it differs from Euclidean planes — and is characterized by a constant negative curvature. Euclidean spaces are always with zero curvature. To make it more vivid in my own words (which very likely will not be so rigorous), if we observe a small region in the hyperbolic plane, it looks like just a concave plane. And when you draw a triangle in this concave plane, the sum of its inner angles is always less than 180 degrees. This is also a proved theorem in hyperbolic geometry.

In elliptic geometry, we have the following conclusion: “Given a line L and a point p outside L, there exists no line parallel to L passing through p, and all lines in elliptic geometry intersect.” This means we can never find any parallel lines in elliptic geometry. This kind of geometry together with hyperbolic geometry, perfectly form a counter example of the parallel postulate’s assumption “there is one and only one parallel line…”: in elliptic geometry, there is more than one parallel line, and in hyperbolic geometry, there are none. Examples of elliptic geometry are more common in our real life than hyperbolic geometry. One example is the surface of Earth. A line in such a space becomes a great circle (a circle centered at earth’s core). When you draw a line through point P and if P is away from line (great circle) L, the new line you get will be a new great circle, and it will always have two intersections with great circle L, because any two great circles on the surface of sphere will have two intersections.

Here we have three pictures visualizing the relationship between Euclid’s geometry, hyperbolic geometry and elliptic geometry.

Image: Joshuabowman and Pbroks, via Wikimedia Commons.

The establishment of non-Euclidean geometry is the outcome of many generations’ collective endeavors. For example, classical era’s scholar Proclus commented some attempts to prove the postulate, esp. Those attempts tried to deduce it from the previous four postulates; Arab mathematician Ibn al-Haytham in the 10th century, tried to prove the theorem by contradiction; in the Age of Enlightenment Italian mathematician Giordano Vitale and Girolamo Saccheri both contributed new approaches to this problem although they finally failed; Gauss and Nikolai Lobachevsky (we already mentioned him above) also joined the sequence — the latter finally finished this task by establishing a new geometric branch. This mansion was built over such a long time and I am fortunate to feel part of its grandeur and beauty.

So for those suitors who complain their misfortune that their dream lovers and they are like two parallel lines, I think you are too pessimistic. You can imagine yourself being in a elliptic geometric space. Then as long as you try your best, you will always have an intersection with the other line. I am not sure whether this will convince those guys and give them confidence. For me, I am now feeling happy and believe that everything is possible in our real world, just like that everything is possible in mathematics. The story about seemingly very simple parallel lines do make me feel the power and beauty of mathematics.

References:

  1. http://en.wikipedia.org/wiki/Non-Euclidean_geometry
  2. http://en.wikipedia.org/wiki/Elliptic_geometry
  3. http://en.wikipedia.org/wiki/Parallel_postulate
  4. http://en.wikipedia.org/wiki/Hyperbolic_geometry
  5. H. S. M. Coxeter(1942) Non-Euclidean Geometry, University of Toronto Press, reissued 1998 by Mathematical Association of AmericaISBN 0-88385-522-4.
  6. Hazewinkel, Michiel, ed. (2001), “Elliptic geometry”Encyclopedia of MathematicsSpringerISBN978-1-55608-010-4
  7. Weisstein, Eric W.“Hyperbolic Geometry”MathWorld.

Euclid’s five postulates in Descartes’ Coordinate System

  1. Introduction

As we learned about the Euclidean geometry and its five basic axioms in class, some terms like “straight line”, “circle”, and “right angle” kept jumping in my mind. I thought I had a picture of them. for example, a straight line is as straight as the rope with a ball attached and hang in the air, and a right angle is shown like a corner of a rectangular table. However, as a math major student, such a simple cognition of them is not enough, I hope to have some more mathematical concept to express them.

  1. The Cartesian coordinate system

2.1 The invention of Cartesian coordinates

In the 17th century, René Descartes (Latinized name: Cartesius), a well-known mathematician and philosopher to today’s people all around the world, published his work La Géométrie , in which he made a breakthrough. More concretely, Descartes uses two straight lines that are perpendicular to each other as axes x, y, and uses these axes to measure the positions of any points in a plane.

2.2 The rule of representing a point in Cartesian coordinates

One point in Cartesian coordinates has two parameters: one is the x parameter, the other is the y parameter. To measure the x parameter, we need to draw a straight line y’ parallel to the y axis(we will discuss the definition of parallel in Cartesian coordinates later) that through the point, and then set the x parameter of that point as the number of the intersection of y’ and x axis, for its y parameter, draw a line x’ parallel to the x axis through the point and take the number on the intersection of x’ and y-axis as this point’s y parameter.

2.3 To express a straight line in Cartesian coordinates

A straight line in Euclidean geometry is a straight object with negligible width and depth. So, it is an idealization of such objects in Euclidean geometry. However, in Cartesian coordinates, a line has a strict definition, a straight line is the set of points that satisfies a certain equation. And the line equation usually can be written as:

A*x + B*y + C = 0,

The A, B, and C are the coefficients of x, y, and constant. Moreover, the -A/B is the slope of the straight line, -C/B is the y-intercept of this line, which means the intersection of the y-axes and the line.

So, all above is how we express a straight line in Cartesian coordinates.

  1. To express the five postulates in the Cartesian coordinate system

1.”To draw a straight line from any point to any point.”

In Cartesian coordinates, to express a line we only need one point and a direction. Suppose we have two points a=(A, C) and b=(B, D). By doing a subtraction of the two points, we can get a vector (B – A, D – C). We only need this vector to provides a direction, which is (B – A)/(D – C). So this unique straight line can be expressed as

(x – A)*(B – A)/(D – C) = y – C;

2.”To produce [extend] a finite straight line continuously in a straight line.”

Any line in Cartesian coordinates is a straight line(infinite). It can be limited by a range for x or y, such as

A*x + B*y + C = 0,( a < x < b or c < y < d)

  • “To describe a circlewith any centerand distance [radius].”

To describe a circle in the Cartesian system, we only need the center’s x and y coordinates (x0, y0), and a distance as the radius r, it is

(x – x0)2 + (y – y0)2 = r2,

So, above is a typical circle in the Cartesian coordinates.

A right angle in the Cartesian system is always equal to the angle between x and y axes, for x, y axes in the Cartesian system is perpendicular to each others.

  • “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”

For the parallel postulate, it is far more easier to be expressed in Cartesian coordinates, suppose we already know a line as

A*x + B*y + C = 0,

And we have a point (x1, y1) out of the line, have the point and a direction of the other line, the slope is – A/B, and the other line can be described as

-A/B*(x – x1) = y – y1;

And it is easy to know these two lines are parallel, because they have same slope and do not share one point. And by the property of Cartesian coordinates, this is the only line that parallel with the first one.

  1. Conclusion

In Euclidean geometry, some concept are hard to imagine or describe, while Cartesian coordinate make it possible and easy to express, such a great combination of geometry and algebra!

The Power of Construction

Eighth grade Geometry was one of my favorite classes. I thoroughly enjoyed the material, as it was unlike anything I had seen in my math education so far. Personally, I felt that Geometry helped combat the classic refrain of nearly every elementary school child of “When am I going to use this”. Geometry had the unique ability to take all of the abstract ideas taught thus far and ground them in the physical world. It was also the first time we were required to write a logical proof, which has been such an important skill to have and understand. However, my favorite part of the class was the constructions.

I clearly remember being excited about buying my very first compass. I knew that with my compass and stainless steel ruler I used as a straightedge, I was ready for anything Geometry could throw at me. In class, we constructed all of the basics: perpendicular bisectors, bisected angles, various regular polygons, and so on. The pentagram construction was my favorite. That knowledge of that construction, combined with chalk, some string, a yardstick and the driveway made for a very interesting conversation with my parents. Construction is a great hands-on approach to math but is rarely seen past the context of 8th grade geometry.

After 8th grade, I never gave it any thought. It wasn’t until my History of Math course that I realized how much power straightedge-and-compass construction really had.

So let us go back to the basics. A straightedge is a ruler without any graduation. It can be used in straightedge-compass constructions to connect two points on a given plane, and extend lines on a given plane. The compass is used to draw circles (that is, a set of all points equidistant from another point), and be able to ‘measure’ a given line segment, and construct that same length elsewhere. So what can you do with this?

As it turns out, quite a bit! For starters, you can do basic arithmetic. Adding just becomes combining two line segments on the same line, and subtraction is the reverse. Multiplication has a geometric representation as similar triangles. If you have a triangle whose base is 1 unit and a hypotenuse of length a, if you draw a similar triangle whose base is length b, the resulting hypotenuse is length ab.

Trisecting a segment. Image: Goldencako, via Wikimedia Commons.

I find this incredible! What took my math educators 8 years to get around to, the Ancient Greeks did right off the bat. They made mathematics immediately tangible and constructible, instead of relying on the esoteric notion of numbers and Algebra. But construction doesn’t stop there. As I’ve mentioned you can create perpendicular lines, create regular polygons, bisect angeles, bisect segments, trisect segments, and more. Given a unit length, you can even construct whole-number measurements of that length, and even some irrationals like square roots. Take this example:

Construction of a square. Image: Aldoaldoz, via Wikimedia Commons.

To construct a regular unit square (all sizes equal to a given unit of measure and all angles are right angles), it’s a simple matter of constructing perpendicular bisectors of segments and measuring with a compass. But can we construct a square that is exactly twice as much area? If the area of a unit square is 11 = 1then the area of a doubled square must be 2. The sides of the square must then be 2 = s2 ; s = 2. To the uninitiated, this might seem like an impossible task! How on earth using a straightedge and a compass can you construct an irrational number? Euclid, however, found a way. Create two lines that are perpendicular to each other. Use the compass to measure out a unit length along each of the two lines, starting from the intersection. This has given you two sides of a right triangle, each with a length of 1 unit. If you connect them an form a hypotenuse, a2+b2= c2; 1 + 1 = c2; c = (1+1) = 2. With this new length as a measurement for your compass, a square with side length 2 is entirely possible.

Euclid did have one problem them though, and that was cube roots. We have shown that doubling the square is possible, but what about doubling the cube? The construction is analogous to the double the square, just with an added dimension. Therefore, if a cube has a volume of 1 cube unit, it’s double should have an volume of 2 cubed units. V = s s s = s3; s = 3V

No matter how hard Euclid tried, he could not construct a cube root. His limitations didn’t stop there. Most famously, he was unable to construct a square the with the area of a circle. This is known as “Squaring the circle.” Again, if we have a circle with the radius of 1 unit, it will have an area of A = r2 =. So to make a square have an area of , we simply have to construct a side with length . Square roots are no problem, it’s just the hypotenuse of a right triangle with sides that sum to . But…how do we sum to with constructions? It turns out, it is impossible. The number is not a “constructible number, ” as they are known, but a “transcendental number.” This wasn’t proved until 1883 by Ferdinand von Lindemann.

While I would never give up the power of algebra and the tools it provides, Euclidean geometry holds a special place in my heart for its sheer physicality. The ability to construct basic arithmetic, regular polygons, and even the odd irrational number grounds math in a way that I think is delayed for far too long in standard Western education. But at least they get around to it.

I end, as always, with wise words from Randall Munroe.

Image: xkcd by Randall Munroe.

Sources

http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html

http://www.infoplease.com/encyclopedia/world/greek-architecture-ancient-greek-construction-methods.html

http://www.mathopenref.com/constructions.html

“The Copernicus of Geometry”

A young Nikolai Lobachevksy. Image: Lev Kriukov (father), via Wikimedia Commons.

On December 1st, 1792 one man, who would create a revolution in geometry, was born. Actually, a lot of people were born on December 1st, 1792. I can’t name any others, but I’m 99% sure that more than one person was born on that day. I’m not a betting man, but if I was I’d even gamble that more than three were born that day. But I don’t really care about them (no offense to them of course, I’m sure they were fine people). I only care about Nikolai Lobachevsky, the man who would take geometry from the ideas of Euclid, throw those ideas away (he didn’t do that), and change the rules and our perception of shapes, angles, and all things geometric.

Since Euclid’s Elements, circa 300 BC, geometry had been looked at in Euclidean way. Euclid’s axioms and postulates were how it was, and mathematicians had to work within those confines. One similarity is how humanity thought that everything revolved around the Earth, the very human, egotistic geocentric model of the cosmos. In Elements, Euclid’s mathematical magnum opus, which may be the most influential treatise of all time, Euclid creates axioms, propositions, and proofs giving an overview of Euclid’s ideas on number theory and geometry. One of the most important axioms within Elements is Euclid’s parallel postulate.

The parallel postulate states that if a line segments intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. For many, many years, around two thousand, various mathematicians attempted to prove this postulate. With no success, again and again mathematicians would never see the day the postulate was proven, nor were their names engraved in history as the person to prove such an important postulate. However, ideas of negating the postulate altogether came into fashion in the early 19th century. Lobachevsky took this task to hand and worked his way towards an entirely new view of geometry.

Lobachevsky decided to abandon the idea of the parallel postulate, negating its meaning trying to see if there was a possibility for geometry that did not follow the rules Euclid put into place. Lobachevsky worked around the idea that there exist two lines parallel to a given line through a given point not on the line. In 1829, Lobachevsky published a paper in the Kazan Messenger on his new, non-Euclidean geometry, doing this before anyone else had. Unfortunately due to the small nature of the paper, as well as the fact that it was Russian, his work went largely unnoticed. While others like the famous János Bolyai later discovered this new non-Euclidean geometry completely separate from Lobachevsky, they discovered it years after Lobachevsky did. This new geometry became known as hyperbolic geometry, a geometry that Pringles would sponsor if people didn’t hate math and for some reason math had sponsors. A new form of geometry was born, and Lobachevsky discovered his own personal heliocentric cosmos.

Lobachevsky had many other findings. He discovered the angle of parallelism in hyperbolic geometry, the computation for the roots of a polynomial, and the “Lobachevsky criterion for convergence of an infinite series.” When it comes to his life, it unfortunately wasn’t as great as his discoveries. The combination of his radical new theories, findings that were found same time others discovered them (this can be seen with the Graeffe’s method, which is the computation of the roots of a polynomial that I previously mentioned, and Peter Dirichlet’s definition of a function), and being Russian led him to quite the sad ending. Left without the ability to walk and blind with no job due to his quickly deteriorating health, his life ended in poverty. He had lost his son he loved the most to tuberculosis, came from a poor family, died a poor man, and worked hard all his life without much humor or relaxation. He is quite the Russian stereotype. If I were to make a movie about Russia, he would be the person who symbolizes the Russian winter.

Luckily for Lobachevsky, and moreover mathematics as a whole, his legacy and ideas in his works have lived on. Much work has been done in hyperbolic geometry since his time, as well as the extension of non-Euclidean geometry to Riemannian geometry. Taking what we consider as fact and not only negating it but also proving there is more to it, in this sense going from Euclidean to non-Euclidean geometry, is a revolutionary task that not many people in the history of, well, the universe, have done. It’s like that one Arcade Fire song, they just tell us lies.

On February 24th, 1856, a lot of people died. Like, a lot of people. I don’t know how many people, but I assume there were quite a few. When you think of how many people die each day, it’s slightly horrifying. On that day Nikolai Lobachevsky died, a poor man with no vision and not much left to live for. However on February 24th, 1856, many people were born. And even today, even more people were born. And who knows, maybe the next Nikolai Lobachevsky was born today.

Sources and Further Reading:

http://en.wikipedia.org/wiki/Nikolai_Lobachevsky

http://en.wikipedia.org/wiki/Euclid%27s_Elements

http://www.math.brown.edu/~rkenyon/papers/cannon.pdf

http://www.britannica.com/EBchecked/topic/345382/Nikolay-Ivanovich-Lobachevsky

http://www.regentsprep.org/regents/math/geometry/gg1/Euclidean.htm

http://www.encyclopedia.com/topic/Nikolai_Ivanovich_Lobachevsky.aspx

http://en.wikipedia.org/wiki/Hyperbolic_geometry

http://www.cut-the-knot.org/triangle/pythpar/Drama.shtml

http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.html