Tag Archives: hyperbolic geometry

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.

Advertisements

“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

János Bolyai

A portrait, allegedly* of János Bolyai, by Mór Adler. Image: Pataki Márta and Szajci, via Wikimedia Commons.

One of my favorite things that we’ve been able to learn about this semester has been the different mathematicians that we’ve studied. It fascinates me to hear of their interaction with each other and how they affected one another’s work. As much as I love the math, the historical aspect is something that I had never heard and something that I love learning about. One of those mathematicians that I wanted to know more about was János Bolyai. Of him Gauss would say, “I regard this young geometer Bolyai as a genius of the first order.” Coming from someone like Gauss, that is quite the compliment. I wanted to know what made Gauss say that about such a young mathematician.

János Bolyai was born in Kolozsvár (which is now the city of Cluj in Romania) to Zsuzsanna Benkö and Farkas Bolyai, who was also a great mathematician, physicist and chemist at the Calvinist College. Like many fathers, Farkas wanted his son to follow in his footsteps and perhaps to achieve more than even Farkas himself had achieved in the field. So, he raised János with that goal in mind. However, Farkas was a firm believer that a strong body would lead to a strong mind, so in János’s younger years, most of the attention was spent developing his physical body. (O’Connor & Robertson, 2004)

János quickly became a child prodigy. According to Barna Szénássy in History of Mathematics in Hungary until the 20th Century, “… when he was four he could distinguish certain geometrical figures, knew about the sine function, and could identify the best known constellations. By the time he was five [he] had learnt, practically by himself, to read. He was well above the average at learning languages and music. At the age of seven he took up playing the violin and made such good progress that he was soon playing difficult concert pieces.” (Szenassy, 1992). Bolyai’s childhood and adolescence were fascinating. His father wanted to send him to live with Gauss as a student in order to accelerate his mathematical education, but Gauss would not agree to it. Because the Bolyai family didn’t have the financial assets to send János to an expensive university, they made the decision to send him to the Royal Academy of Engineering at Vienna to study military engineering. He truly was a “jack of all trades.” He finished the seven year engineering program in just four years, became an excellent sportsman and even performed as a violinist in Vienna. He was in the military for eleven years, where he became known as the greatest swordsman and dancer in the Austro-Hungarian Imperial Army. (O’Connor & Robertson, 2004) It wasn’t until 1820 that he began intense study on Euclid’s parallel postulate and the development of hyperbolic geometry. One of János Bolyai’s most recognized quotations comes from a letter that he wrote to his father when he said that he had, “created a new, another world out of nothing.”

The story is well-known of the publication of Bolyai’s work on hyperbolic geometry. During János’s military service, his father read the mathematical work that his son had sent him previously and then went to where János was stationed. Farkas then encouraged his son to publish his work. János later said, “Had my father not happened to urge or even force me at Marosvásárhely, on my way to duty in Lemberg, to immediately put things to paper, possibly the contents of the Appendix would never have seen the light of day.” When Farkas sent a copy of his son’s work to his old friend, Gauss, Gauss responded by saying, “To praise it would amount to praising myself. For the entire content of the work … coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years.”

Bolyai’s work on the parallel axiom led to the development of what would be known as a “pseudosphere,” which is an object that extends infinitely, but has a finite volume. This object was created by Beltrami many years later, but now is seen as an embodiment of hyperbolic geometry.

The story of János Bolyai ends as a sad one. He did not manage his money very well, gave very little care or attention to the family estate he had inherited, and finally left his wife and children. Years after his work on hyperbolic geometry, he found the works of another geometer named Lobachevsky, who he thought was fictional; a cover that Gauss had created in order to steal his work on hyperbolic geometry. He quit working on mathematics entirely and focused on “a theory of all knowledge.” (O’Connor & Robertson, 2004) Although he may not have felt like he received the credit that he deserved for his work, János Bolyai was indeed, as Gauss called him, “a genius of the first order.” He gave the world of mathematics a new way of understanding the concept of parallelism and the way in which mathematics relates to our natural world.

*Editor’s note: The portrait here, which also appears on postage stamps honoring János Bolyai, has long been associated with the mathematician but is not authentic. For more information, see “The Real Face of János Bolyai” by Tamás Dénes.

Works Cited

O’Connor, J., & Robertson, E. (2004, March). János Bolyai. Retrieved from MacTutor History of Mathematics: http://www-history.mcs.st-and.ac.uk/Biographies/Bolyai.html

Szenassy, B. (1992). History of Mathematics in Hungary until the 20th Century. New York: Springer-Verlag Berlin Heidelberg.

Circle Limit III

The angles of a triangle must add up to 180°. This is a simple fact that you were probably taught fairly early in your math career. It’s been known for millennia and is pretty simple to prove: for a right triangle, assume we have two parallel lines, one line perpendicular to them, and a fourth line between one of the intersections and an arbitrary non-intersection point on the opposite line as shown below.
triangle-1
This makes a triangle with one right angle, C, and two acute angles, A and B. We also need to consider angle D, the complementary angle to A. We know that A+D has to be 90° since they sum together to make a right angle, so the measure of angle D must be 90° – A. Since D and B are alternate interior angles with respect to the parallel lines and the red transverse line (remember all those awful congruence theorems you learned in your high school geometry class?) they have to be congruent angles. This means that the measure of angle B has to be 90°-A as well. So if we sum up the angles inside the triangle, A + B + C = A + 90° – A + 90° = 180° + A – A = 180°. The proofs for acute and obtuse triangles are similar, but a bit more complicated so we won’t go through them. The point is, we proved it! Triangles have to have 180°, right? Wrong.
The proof we used—and indeed all proofs that triangles must have 180° inside them—relies in some way on an infamous postulate used by Euclid around 300 BCE that says (more or less) that given a line and a point not on that line there is exactly one line through the point that does not intersect the original line. This postulate, though reasonable sounding, foiled mathematicians for thousands of years. Despite attempt after attempt to prove this postulate, no one was ever able to succeed. In fact, it was eventually proven that there IS no proof of this persnickety postulate. The angry mathematicians, having been foiled by this simple-yet-unprovable statement, began to consider what would happen if, indeed, it were not true. What would happen if, for example, there were an infinite number of lines through the point that didn’t intersect the original line? This line of questioning led to the discovery of hyperbolic geometry: a world where there are infinitely many parallels to a line through a given point off the line.
One of the many interesting aspects of hyperbolic geometry is that triangles don’t have to have 180°—In fact, they must have less than 180° (otherwise they could be a triangle in spherical or euclidean geometry). These triangles can still tessellate a plane though! In one particular representation of hyperbolic space, called a Poincaré disk, this tessellation would look like the image below.
The Poincaré disk is a way to show the hyperbolic plane on a circle. The idea is that straight lines are represented as curves from one side of the circle to another with the intention of preserving angles without necessarily preserving lengths. These curves must be circles that intersect the boundary of, or must be diameters of, the disk. The result is that each triangle in the picture above is the same size! From the large-looking central triangles to the itsy bitsy ones on the edge, each triangle would have exactly the same area in a hyperbolic space.
M.C. Escher was a Dutch artist whose graphics are widely known for their otherworldly bizarre mathematics. Stairs that led up to themselves and water that flowed in a ring are just two examples of his pieces, enacted with an almost formulaic mathematical exactness. He is well known in scientific communities for the diagramesque works of art.
You may be asking what this little Dutch artist has to do with our discussion of “curved” triangles. Well, Escher had become somewhat famous for using tessellations in his work. Creating shapes, especially in the shape of animals, which would tessellate all the way across the pieces, forming a lattice of cells that had only to be filled with a clever image. In the early 1950s, he became curious about finding different ways to “draw” infinity on a page. A letter from a friend came to him with some of these Poincaré tilings in the hyperbolic plane and became enamored with them. The images in the letter were a type of tiling denoted by {p,q} that was a tiling of p-gons with q of them meeting at each vertex. These images of hyperbolic tilings inspired Escher to create his Circle Limit series in 1959 and 1960. Circle Limit III was inspired in particular by the {8,3} tiling—4 octagons meeting at every vertex, and is a beautiful reimagining of the tiling with fish in place of the triangles.

Circle Limit III by M.C. Escher. His other work, including the other Circle Limits, can be found at http://www.mcescher.com.

caption

Circle Limit III with the {8,3} tiling overlaid on it. Image by Doug Dunham.

Escher’s works seem to represent the very nature of the hyperbolic plane that we have talked about. After all, in a world where there are an infinite number of parallel lines, why couldn’t I draw infinite fishes on a page?
Anyone wanting to know more can Google hyperbolic geometry, parallel postulate, M.C. Escher, or triangle group.

Making a Pseudosphere

When I learned about models of the hyperbolic “plane” I was intrigued mostly by the Pseudosphere, and how it represents a surface of constant negative curvature, this grabbed my attention so much that I knew I had to have one. And thus began my epic quest to make a rotation of a tractrix about its asymptote.
One thing to note about the tractrix is that it is a curve that you can create through rather simple methods. All you need is to have a string, paper, a pencil, and an object of small but noticeable mass. The tractrix is created by tying one end of your string to the mass. Place the weighted object on the paper as far away as it can go so that your taut string makes a right angle with the edge of your paper. Holding your end of the string in your hand at the edge of your paper, move your string along the edge of the paper so that your hand follows a straight path. As you move the string, the weight will begin to move. Now mark the location of your mass every centimeter or so. This curve is called the tractrix.
There are many methods to create this shape, one of the more popular modern methods for making these sorts of shapes is on a 3D printer. Since I don’t have a 3D printer, I decided that I would have to make this Pseudosphere by hand which is obviously much cooler. The method I chose was a lathe, which is a tool that turns a chunk of wood at high speeds, allowing the woodworker to remove parts of the wood, yielding radial symmetry about the axis of rotation.
The first thing I had to do was determine how large I wanted my Pseudosphere to be. To do that, I looked up the equation for a Pseudosphere, which is defined parametrically.

Parametric equations for a pseudosphere.

Parametric equations for a pseudosphere.

The constant value “a” can be changed at will, so I decided to make “a” be 3 which would give me a main diameter of 6. At a length of 10 inches, five on each side, I would have sufficient width at the ends to maintain structural integrity. The units I decided to use were inches, due to the fact that I am already familiar with them, and my measuring tools are all in inches.

One-inch blocks of wood glued together before lathing.

One-inch blocks of wood glued together before lathing.

For my choice of wood, I had two options. The first was take a big log and try to cut it down to the desired size. Due to the fact that it would take a while to cut a six inch diameter log that is ten inches long down to the proper size, and since large logs often have splits in them, I chose another option. That was taking a series of one inch thick squares of wood and gluing them together in a pattern roughly resembling the shape of the Pseudosphere.
Using a trace of the tractrix, my father and I determined the placement of the blocks and began the process of gluing pieces together in the right configuration. The gluing process took two days to complete, because I didn’t want my project flying to pieces for a lack of patience on my part.  Essentially, the glue had to set up to full strength to withstand the turning.

An intermediate pseudosphere.

An intermediate pseudosphere.

Once the pieces were all glued together, I had the momentous task of taking squares, and rounding them out to a closer approximation of a Pseudosphere. The process wasn’t easy, throughout the whole project I had a sore hand from having to hold onto the tool through all the jerking motions made by the heavy wood slamming into the metal cutting tool.
After having removed all of the corners, I took calipers, and used them to determine the diameter that the Pseudosphere should have at every inch. Cutting down to those points, I began slowly, and carefully working inwards, trying to get the right curvature. Because the natural tendency of all of these tools is to cut a straight line across a surface, I had to pay special attention to the tool’s path, so that I could get a constantly changing curvature throughout the whole length of the Pseudosphere. There were some tense moments when the knife didn’t behave in the way I was intending, and I cut too deeply, but the wonders of foresight helped, because I had, at the beginning given myself a 1/8 inch cushion in my measurements, so that I could sand, and perform the finishing touches properly.

The finished pseudosphere.

The finished pseudosphere.

The sanding was a fun process, the high friction caused a heat buildup in the sandpaper, making it necessary to use a glove. Once we had it all sanded we applied a finishing oil called “Tung Oil” to give the wood a darker, richer color. It also smells pretty good.
In the end I was left with a wonderful mathematical object. There were some questions from people, such as, why I didn’t make the Pseudosphere infinitely long. My response is twofold. One I would have needed infinite wood. Despite the fact that a Pseudosphere has finite volume, the cylinder I would have needed to start with would have needed to be of infinite volume. Also structural stability had a role to play there, I can only make something so thin and rotate it at high speeds without it suffering a physical breakdown.
I do however hope you like my project, and remember, our lives are only as interesting as we decide to make them.

-Travis

Sources
http://mathworld.wolfram.com/Tractricoid.html

Image credits
(I made them myself, don’t tell anyone)