Tag Archives: Non Euclidean Geometry

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.

Straight or Curved?

When you think of a straight line, I’m sure you think of using a ruler or going between two points on a flat surface, but how do you think of a straight line in space? Straight lines are typically thought of as the shortest distance between two points. Take a piece of paper, make two dots, then use a ruler or another piece of paper and connect the dots with the least amount of drawing. This is called a straight line to most people and they don’t know any different. What they don’t realize is that everywhere in the universe is on a curved or elliptical plane. Newton states that the presence of mass causes curvature based on gravity, but Einstein had theories as well. In Newton’s theory, gravity makes particles leave their straight paths causing them to come together and not follow a straight line. In Einstein’s theory of general relativity, gravity is a distortion of space-time. Particles still follow the straightest possible paths in that space-time. But because space-time is now distorted, even on those straightest paths, particles accelerate as if they were under the influence of what Newton called the gravitational force, (Physics, 2015) Basically, Newton’s theory says in space when you have two objects close together the gravitational pull they exert on each other causes curvature, making a straight line actually be curved, but when you’re drawing a straight line you don’t imagine that the line would end up back where you started, even though that is exactly the “straight line” planets take in their orbits.

Just like drawing on a sphere, start on one side; continue the line and you end up right back where you started; this is what Einstein’s theory was stating. A straight line can really be curved- the shortest distance doesn’t always have to be straight in the sense that most people look at. Einstein viewed the world from a bigger picture and saw that outside forces reacting to each other could change the way straight lines are perceived. It somewhat boggles the mind, but when you think about it for a bit, it all seems to fall into place and makes sense.

When we discussed the parallel postulate in class it seemed as if it could be true in every situation, until you thought about it and applied it in the non-Euclidean space, which is one type of 2-dimensional geometry. In Euclidean geometry it is very cut and dry on how lines look and can or cannot intersect, but as you look into hyperbolic geometry or elliptic geometry things start to look a bit different. In hyperbolic geometry your parallel lines seem to curve away from each other and in elliptic geometry they curve towards each other.

Image: Joshuabowman and Pbroks13, via Wikimedia Commons.

Image: 6054, via Wikimedia Commons.

Image: 6054, via Wikimedia Commons.

When non-Euclidean geometry first came to fruition, Lobachevsky helped show how lines could curve and still be straight by negating the parallel postulate. The parallel postulate states that if a line segment 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. α+β<180°, therefore those two line will intersect at some point, (wikipedia, 2015). This is all fine on a flat surface. I know that it’s a bit complicated to think that there really are not many flat surfaces though.

Think of the bigger picture: look down on Earth as if you were on the moon. No matter which way you look the path curves around. When you walk to your car it seems as if you are walking on a flat surface and the further in you look at the ground it appears to seem as if there’s undeniable proof that you are on a flat surface. What most people don’t see is that it’s not flat.

Imagine you’re a ladybug; you are living a very happy life. You have friends and loved ones that are impossible to replace and your home is amazing, but you need to go on a trek to get food every day. You have to go many inches across the treacherous terrain to find the best aphids to bring home for dinner. You start your journey each morning and it seems like you cross many flat surfaces with only a few bumps along the road and you’re successful every day. Now jump back out and look at this journey from a human perspective. This ladybug thought the journey was long and flat in areas where you see that it was short and somewhat curved slightly in spots. Crossing an almost flat leaf to the lady bug looks differently to you. You see the slight curve that it had to take and realize that to the ladybug the leaf is somewhat like the earth.

It’s a bit mind boggling to realize that depending on the scale you look from it changes the ideas of lines and what straight is drastically. It makes you wonder if there is any one way to describe straight lines or if you have to look at it differently within the parameters you set. The idea that space is curved has changed how many have approached the mysteries of the universe. It has brought about many discoveries like gravitational lensing.

Gravitational lensing takes a look at the whole picture of space and looks at the way light bends from a star to show other stars and planets that you can’t see. A gravitational lens looks at a distribution of matter, such as a cluster of galaxies, between a distant source and an observer, which is capable of bending the light from the source, as it travels towards the observer. This is only one way that a curved, but straight line, in space can show things we couldn’t see before. Which has also helped in the exploration of space with probes and satellites, we can’t just plan to launch satellites in a straight line off of the planet and hope that they don’t hit something or they don’t go off of course. It’s the same with missions further into outer space, you have to take in account the curvature of space and the gravitational pull of everything that the ship or craft could come close to, making straight lines that you need to go along curved. The gravitational pull could also be used to make the journey faster and more energy efficient if you can use the pull of the object you fly by to somewhat slingshot yourself around and back onto the direction you intended to go. These discoveries are wonderful and have further advanced our knowledge of space and the way we view lines all around us.

Works Cited

Physics, M. P. (2015). Elementary Einstein General Relativity . Retrieved from Einstein online: http://www.einstein-online.info/elementary/generalRT/GeomGravity

wikipedia. (2015, March 7). Non-Euclidean geometry. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Non-Euclidean_geometry#External_links

Math: Is It All In Our Head?

After years of math classes, the crazy truth is finally coming out.  It is all just in our heads.  No way! How can that be? There’s an interesting debate in the world of math.  Are math principles the creation of humanity, or are they universal truths that humans discovered? There are compelling arguments on both sides of the debate and both sides have several different sub-levels of thought.  In this article, I will discuss them both generally.

The realists maintain that mathematical principles would exist even without people.  Humans discovered the principles and brought them into practical use and any intelligent human being could also discover the same principles.  This argument is supported by the fact that many cultures have discovered mathematical principles independent of one another.  Also, mathematical concepts, such as the Fibonacci sequence and some fractals, occur in nature which would suggest that they exist even without people.   Some realists, like the Pythagoreans, believe that the world was created by numbers.  The realist point of view can lead to an almost supernatural view of mathematics.

The challenge with mathematical realism is that there is no physical domain where math entities exist.  We cannot draw a perfect circle or even a line.  We can conceptualize these things in our mind and we can prove them in theory; however, we cannot actually manipulate math entities in the physical world.  Many math concepts exist only in the context of our understanding about them and conceptualizing them.

Another view is the anti-realists.  They maintain that math is the creation of humans in order to make sense of the world.  They recognize that math is an amazing, complex system and that it works as modeled by science.  However, some argue that scientific principles could be explained without math.  One anti-realist, Hartry Field, demonstrated this by explaining Newton Mechanics without referencing numbers or functions.  He explained that, in his opinion, math is fictional and is true only in the context of the story in which it is being told.

So, is it all in our heads?  A fiction that was created to explain properties in our world?  In reality we may never be able to settle the debate and it may not matter.  Math works.  That is the beauty of it.    In his article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, Eugene Wigner observes that

the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.  This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.

Albert Einstein, 1921. Image: Public domain, via Wikimedia Commons.

Albert Einstein, 1921. Image: Public domain, via Wikimedia Commons.

Perhaps the best thing to commend mathematics as being real, is that it works.  Time and time again, it works.  Its principles, laws and theorems, applied over and over, in different settings produce accurate results and predictions.  Einstein commented in a 1921 address titled Geometry and Experience, “It is mathematics which affords the exact natural sciences a certain measure of security, to which without mathematics they could not attain.”  He explored the question of how math, a product of our mind can be so applicable to the concrete world.  He asked, “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirable appropriate to the object of reality?”  Einstein answers this question with the statement, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”  He looks specifically at the field of geometry and the need humans have to learn about the relationships of real things to one another.  Even though the axioms of geometry are based on “free creations of the human mind”,  he says, “Solid bodies are related, with respect to their possible dispositions, as are bodies on Euclidean geometry of three dimensions.  Then the propositions of Euclid contain affirmations as to the relations of practically-rigid bodies.”  The abstract principles, when applied to “real” world situations prove to be accurate.  Einstein continues to explain how the theory of relativity rests on the concepts of Euclidian and non-Euclidian geometry.  He challenges the mind to conceptualize a universe which is “finite, yet unbounded”.  In the end, it is this ability to use conceptualized principles and apply them to our world that makes mathematics work.  So yes, mathematics may be all in our head and it may be a huge puzzle created by humanity, but it is effective, useful, and even beautiful.

Sources

Einstein, Albert. Geometry and Experience. http://www-history.mcs.st-andrews.ac.uk/Extras/Einstein_geometry.html

Wigner, Eugene. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Wikipedia. Philosophy of Mathematics. http://en.wikipedia.org/wiki/Philosophy_of_mathematics

Non-Euclidean Geometry in Pattern Making

Original Dress Inspiration pic

The “inspiration pic” for the costume.

Not so long ago, my sister came home from work quite frustrated.  In her line of work she is responsible for costume productions for Hale Centre Theatre in West Valley.  For the upcoming production, the costume designer wanted to recreate a dress she found while doing her research.  The research image was passed off to one of the pattern makers, whose attempt to craft the pattern followed none of the principles of aesthetics, pattern making, or the basic principles of geometry, Euclidean, and non-Euclidean.

After she related this portion of her story, I asked my sister how it should have been done.  So she began explaining some of the concepts in pattern making, and how to create different effects in clothing using what I later learned to be principles of hyperbolic and spherical/elliptic geometries.

The image in [Fig 1a] is a technical sketch of the original pattern created to emulate the skirt from the research image, as can be seen, each section has a triangle attached at the side.  These triangles, when they were sewn in, gave the skirt an effect not of a smooth flowing silk dress, but rather appeared to be “the fins of a rocket ship preparing for launch.”  Upon seeing a mockup of this design, one of the other pattern makers refused to cut out the mockup,  as it “would not look good on a human body.”

Image: Kristy Draper.

Image: Kristy Draper.

My sister began to explain to me how these concepts work on the human body.  To explain these, since costuming is a visual art, I will need to present some pictures of what I am describing.

When we make a skirt flow, we create a type of parabola or hyperbola in 3D that surrounds the lower half of the body, and so any increase in the diameter of the parabola needs to be taken into account with respect to the whole shape.  Skirts can have greater hem circumferences than a circle with the same radius as the skirt length, this means that sometimes a simple parabola doesn’t adequately describe the situation at hand.

In this particular project the goal was to give the skirt a hem circumference greater than that given by a full circle. This is basically intended to make it more flowy and elegant.  In order to accomplish this, while still yielding an even curve to the skirt, the pattern maker needs to perform an action known as “slash and spread.” (See figure 1b.)

The “slash and spread” technique is used to create a fuller skirt based off of another basic skirt pattern.  It is done by taking a pattern piece, and drawing a certain number of lines down it parallel to the front center seam, or back center, depending on what part of the skirt we are working on. The pattern piece is then cut along those lines, the top point of each cut is still connected at a point on the waistline, or in this case a design line, and the bottom edges are separated, each by an equivalent amount.  After completing this step, the pattern is traced onto a new sheet of pattern paper, and all of the connecting lines are smoothed out using a tool called a french curve, which is a drawing tool based off of sections of the “euler curve”.  The smoothing is done in such a way that all four corners are locally right angles, even if they curve afterwards, they must be right angles to be able to fit on the shape of the human body. (This rule does not include design lines, only structural ones.)  (See fig 1b.)

After applying this technique to the skirt, it was able to hang well, no rocket fins, and it actually had a fuller skirt as can be seen in the drawing (compare length l to length k). Each section was made to fill much more space than just adding triangles to the edge could.

The finished dress. Image: Kristy Draper.

The finished dress. Image: Kristy Draper.

I found it interesting that this discipline, which sometimes is considered to be limited in its use of mathematics, actually uses many complicated methods that are reminiscent of non-euclidean geometry, such as its use of parallel lines that appear curved on paper, but on the complex shape of the human body appear straight.  It is also striking that an innate understanding of these concepts are required to create something that looks good.  It was interesting that even though I am not a trained pattern maker, I was able to understand these concepts as well as I did.  In our discussions my sister and I learned that our disciplines have a lot more similarities than is often thought: this is great for all those kids who say that they will never need geometry after they leave school, they don’t realize that they are wearing it.

Sources:

Personal Interview with Kristy Draper, Costume Shop Manager, Hale Centre Theatre.

Armstrong, Helen Joseph. Patternmaking for Fashion Design. 4th ed. Upper Saddle River, N.J.: Pearson Prentice Hall, 2006. Print.

Image credits Kristy Draper: Technical Drawings, Finished Dress

“Inspiration Pic”    http://www.pinterest.com/pin/339529259379580951/

Transition from Euclidean to Non-Euclidean Geometry

Euclidean geometry is the geometry that everyone learns and uses throughout Middle School and High School. In general, geometry is the study of figures, such as points, lines and circles in space. Euclidean geometry is specifically any geometry in which all of Euclid’s postulates and axioms hold. Axioms and postulates are the beginning of reasoning, they are simple statements that are believed to be true without proof. Assuming Euclid’s axioms and postulates found in his book Elements, the rest of Euclid’s classical geometry could be deduced. However, Euclid’s fifth postulate, the parallel postulate, was disconcerting because it was lengthy compared to the rest and not necessarily self evident. Many other ancient mathematicians were dissatisfied with Euclid’s fifth postulate. They thought that it was presumptuous and tried to prove it using lesser axioms or replace it altogether with something they thought to be more intuitive. But their proofs always included an assumption equivalent to the parallel postulate, so for centuries the postulate was assumed to be true.

Centuries passed and the postulate remained unproven; however, development to understand Euclid’s postulate continued into the eighteenth century. Perhaps the most well-known equivalent to the parallel postulate is Playfair’s Axiom, which states “through any point in the plane, there is at most one straight line parallel to a given straight line.” Arguably one of the most influential mathematicians, Carl Friedrich Gauss became interested in proving Euclid’s fifth postulate. After attempting to prove the postulate, he instead took Playfair’s Axiom and altered it, creating a completely new postulate. Gauss’ new postulate stated “Through a given point not on a line, there are at least two lines parallel to the given line through the given point.” With this Gauss had unearthed a completely new space that today is called hyperbolic geometry. However, he chose not to publish any of his results, wishing not to get caught up in any political strife. The work was later published  by Johann Bolyai and Nikolay Lobachevesky, who both had academic ties to Gauss.

Shortly after this discovery another type of Non-Euclidean geometry was discovered by Gauss’ student Georg Friedrich Bernhard Riemann. Riemann looked at what would happen when you altered Playfiar’s Axiom in the opposite direction than Gauss. Riemann’s alternate postulate is stated as follows, “through a given point not on a line, there exist no lines parallel to the line through the given point.” With this, what is known as elliptical or spherical geometry was discovered.

Spherical geometry. Image: Anders Sandberg via Flickr.

Spherical geometry. Image: Anders Sandberg via Flickr.

Spherical geometry provides a somewhat simpler model then hyperbolic geometry. Anyone who has ever attempted to wrap a basketball in paper understands that there are some discrepancies between the two surfaces. For instance, triangles behave quite differently than they do in Euclidean geometry. In hyperbolic space, the angles of a triangle add up to less than 180 degrees, and in spherical space, they add up to more than 180 degrees. Spherical geometry also has other inconsistencies with Euclid’s initial assumptions other than the parallel postulate. In Leonard Mlodinow’s book Euclid’s Window, the author describes how Riemann’s space was inconsistent with other of Euclid’s postulates. For instance, Euclid’s second postulate states that “any line segment can be extended indefinitely in either direction.” In spherical space this is not true; the lines, or great circles, have a limit to their space, namely two pi times the radius of the sphere. Mlodinow describes how Riemann saw this postulate as “only necessary to guarantee that the lines had no bounds, which is true of the great circles.” Also, Euclid’s first postulate became less clear, “Given any two points, a line segment can be drawn with those points as its endpoints.” This postulate can be used to easily describe whether a point is between two other points. However, on the globe, choosing two points on the equator such as Ecuador and Indonesia it is difficult to say whether a third point, Kenya, is “between” them. The problem is that there are two ways to connect the points, one passing over North America and another passing over Africa.

For much of our day to day lives Euclidean geometry works great, because on a local scale we appear to live on a flat world. I can go to a soccer field and trust that it will take four 90 degree turns to walk around the perimeter, or that the Pythagorean theorem will work to describe the path between opposite corners. But looking at a larger scale, the surface that we live on is spherical and has different properties than the flat plane. It is interesting to see how Gauss and Riemann, going against the grain of conventional mathematics, led to new and vast fields of undiscovered mathematics. To me, this shows how mathematics is just as much an experimental science as physics or engineering. These new discoveries of mathematical spaces made possible Einstein’s physical description of the space in which we live. Mlodinow closes his section on Gauss and Riemann saying, “though thoroughly remodeled, geometry continued to be the window to understanding our universe.” Even though the properties of these new geometries differ from classic Euclidean geometry and may have more or less practical use, they are just as important. From Euclid up until Gauss, mathematics was largely pragmatic, but the discovery of these new geometries highlights how math can be appreciated for its own sake.

References:

Case, William A. Euclidean vs. Non Euclidean Geometries. Web. http://www.radford.edu/~wacase/math%20116%20section%207.4%20new%20v1.pdf

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Mlodinow, Leonard. Euclid’s Window. Touchstone New York, 2001. Print

Weisstein, Eric W. “Non-Euclidean Geometry.” http://mathworld.wolfram.com/Non-EuclideanGeometry.html