Monthly Archives: April 2015

Random walk

George Polya was a Hungarian mathematician who made contributions in many branches of mathematics, among which was probability theory. In this blog we will discuss the “Random Walk” problem in probability. What is interesting is that George Polya actually first theorized this problem by accident.

George Polya was at that time a professor at a university in Zurich, Switzerland. The beautiful landscape there helped him develop a habit: taking afternoon walks in local gardens. One day, when he took his afternoon walks, one strange thing happened: he met a young couple six times. Well, I don’t know how large the garden was and how many paths were there, but this coincidence did surprise our mathematician. He was wondering how could this be possible, considering that he was then taking a random walk. After he mentioned this to his wife, he decided to do some research on the probability regarding random walk. His research actually established a new topic in probability theory.

Now let’s begin our discussion about the random walk problem with the simplest case. Consider one object moving along a straight line. Our assumptions are that this object can only move leftward or rightward, and the distance it moves each time is just one unit. For each time, it has a 50% chance to move leftward and a 50% chance to move rightward. A random walk model is thus created. And now we can ask: if when the object return to the starting point, the movement will end, then what is the probability that the object could return to the starting point? Thanks to George Polya’s work, the answer is 100%. Although some people may think this is unlikely, that’s only because when we think about this problem, we have a pre-assumption that the time is limited, or finite, which is always true in our real life. However, this mathematical model sets the time as endless, so it means when there is enough time and the object keeps moving endlessly, at last it will almost always return to the starting point.

When combining this theoretical conclusion with real life experience, mathematicians created the so-called “Drunken Man Going Home” problem. Assume there is a drunken man. He comes out of a bar and walks randomly along the line connecting his home and the bar. Once he gets home he will stop. Then what is the probability that the drunken man could finally get home? The answer to this funny question is still 100%. The math model of this problem is equivalent to the previous one. Because when time approaches infinite, the different positions of fixed points on the line actually make no differences. In my own words, any finite number in front of infinite vanishes to zero. So in this case, we just move the ending point from the starting point in the first example, to a new fixed point which denotes the drunken man’s home in the second example. The result keeps unchanged.

But mathematical problems always go from simple cases to complex cases. In the original random walk along a straight line, if we change the condition from “along a straight line” to “in a two-dimensional plane”, the complexity of this problem will definitely increase. If we change the condition to “in a three-dimensional space”, the complexity of this problem will increase dramatically. And the answers to this problem also change. The law is that, when the dimension increases, the chances that the object could get back to its starting point decreases, with the assumption that other conditions keep unchanged. For example, for a three-dimensional space, the probability is 34%; for four-dimensional space, the probability is 19.3%; when the dimension reaches eight, the probability that the object can get back is only around 7.3%. This means that there is only a lucky “drunken man”, who could always find a way home; there is no lucky “bird”: in a three-dimensional space, if the bird flies randomly, its chance to get back to its nest is much smaller! (Even if it could fly endlessly.)

Some people may cannot help ask, what can the theorem on random walk be used for. Well, obviously it is not developed to help drunken men feel confident when they need find a way home. It does have very wide applications in various areas. I will broadly list its applications in three different areas.

In economics, the theorem on random walks generates a hypothesis called “random walk hypothesis”. With the help of this hypothesis, economists could construct math models to research factors affecting shares price and the change of shares prices. It is quite understandable because randomness is q characteristic of the stock market.

In physics, the famous Brownian motion can apply the theorem on random walk to achieve the purpose to simplify the system. Since molecules move randomly to every direction, if we make an assumption that the molecules will only move to a finite number N directions, then the problem can be converted to a random walk problem in finite dimensional space. We could estimate the case in real physical world through our approximation in theoretical random walk models.

In genetics, a random walk could be used to explain the change of genetic frequency — a phenomenon that finally leads to genetic drift. Because gene mutation is always random, when we view the original gene pool as a point in a multi-dimensional space, then each gene mutation will generate a new gene pool, which can be viewed as a new point that forms through the original point’s random walk. After countless random walks, we could use the computer to draw pictures that show the trajectory of these abstracted points. The trend and some important characteristics of a gene drift could be described by this.

Randomness is everywhere in nature. At first glance, randomness connotes lack of order, unpredictable and uncontrollable. However, after mathematician’s great job, we do gain many laws and theorems on randomness. Random walk now has very wide applications in various disciplines, and we must attribute this partially to the coincidence happened to George Polya.

Reference:

  1. http://www.math.wichita.edu/history/men/polya.html
  2. http://www.guokr.com/article/53059/
  3. http://en.wikipedia.org/wiki/Random_walk

Reflections on Zeno’s Paradox —— A Problem about Geometric Series, or Not?

Our knowledge of mathematics develops along with the long history of human civilization. Ancient Greece is usually considered as the cradle of western civilization and the birthplace of mathematics. Here I will discuss the famous Zeno’s Paradox, an intellectual legacy we inherited form those great thinkers in ancient Greece, whose philosophical thinking has been energetic and attractive since ancient times; Then I will have a brief introduction about the “solving” of the paradox using geometric series; In the end I will show that, in some sense, the paradox has not been fully unraveled, by reference to another problem proposed by contemporary scholars. I believe the charm of mathematics will be presented after these efforts.

The ancient Greek philosopher Zeno once created quite a few paradoxes to show his skepticism about some common phenomena. He thought plurality and change were not a universal truth, and in particular, motion was only our illusion. Among his paradoxes that survived today, most of them have equivalent math models. So I will pick up one of them, “Achilles and the Tortoise”, to represent his logic.

The problem is like this: Achilles, the most famous Achaean warrior in Homer’s Iliad, the “swift-footed” hero, is chasing a tortoise. Suppose the initial distance between them is 100-meters, and Achilles’ speed is 10 m/s while the tortoise’s speed is 1m/s. After the chasing begins, Achilles will spend 10 seconds to finish a first the 100-meters. Then he will be at the spot where the tortoise was, at 10 seconds ago; In this period (10 seconds), the tortoise also proceeds 10 meters. Then, to finish the second distance, 10 meters, Achilles spends 1 second, while in the same period, the tortoise proceeds 1 meter; Then it goes on, every time Achilles reaches the tortoise’s previous spot, he still needs to chase more because in that period the tortoise proceeds to another further spot. Hence, Zeno concludes, in this case, Achilles will never overrun the lucky tortoise, which is a very bizarre conclusion against our common sense.

This paradox raised in history of great interest. Many scholars tried to give an answer or explanation, including Aristotle, Archimedes, Thomas Aquinas, etc. The joint efforts of philosophers and mathematicians did not succeed immediately. Without the help of rigorous mathematical tools, their solutions cannot resist questioning from skepticism. To make it more clear, philosophical thinking alone could hardly solve this problem; even if it accomplished so, to convince others to believe this will be no less difficult. Immanuel Kant in his Critique of Pure Reason mentioned that rationality is not omnipotent. It has its own structure of a priori knowledge, and after itself combined with a posterior experience, it becomes useful knowledge, which guides our cognition. However, due to the nature of human’s longing for perfection, eternal, and universality (I would like to add “infinite” here), we are inclined to abuse our rationality and expands it to areas that it in fact does not apply. This is to say, human rationality arises from very specific experience, and is applicable there; but due to our preference, we create some concepts (like “perfection”, “eternal” and “universality” I mentioned above), which is non-existent in real life and also beyond rationality’s realm. But we are so confident and accustomed to our rationality that we apply it to those concepts generated by ourselves, without noticing it is not applicable there. After the abuse, confusion subsequently follows.

I really admire Kant’s genius in his noticing that a critique of human reasoning itself is very much needed. And I would use his theory to help form my personal understanding about this problem. But I will leave it here and deal with it later, after the introduction of the rigorous mathematical proving with respect to this problem.

Thanks to the invention of calculus and the epsilon-delta language, we now have the rigorous mathematical tool to deal with problems about infinity. A brief solution is to use geometric series. With respect to the “Achilles and Tortoise” problem we mentioned above, the time that Achilles needed to catch up with the tortoise can be represented as:

series

This means Achilles could overrun the tortoise after approximately 11.11 seconds. Thus, the sum of a series with infinite terms, are quite possibly finite, which may be beyond our predecessors’ understanding. But, does this problem stops here? Some modern scholars believes not. Why, because we are not sure what is Zeno’s true meaning. This is to say, the result of the formula may not answer Zeno’s question. Let me here give an example, which is called Thomson’s Lamp: suppose there is such a lamp with a toggle switch. After you start the game, it’s switched one after 1 minute, then switched off after half minute, then on after fourth minute, then off after eighth minute, and so goes on. The sum all the time we spend in the game is 2 minutes, according to the same method about sum of geometric series above. Then, one question follows: After exactly two minutes, is the lamp on or off?

This time we find it’s also very difficult to answer this variation of Zeno’s paradox, even if we know geometric series. And because of this, I believe to use geometric series could give a result, but could not solve the problem about the process, which may be Zeno’s real point. And Kant’s argument gives me guidance in understanding this paradox. Also, there is a scholar making this point more explicitly: according to Pat Corvini, this paradox arises from “a subtle but fatal switch between the physical and abstract”. When we expand our mathematical abstractions to the physical world, even it’s applicable almost everywhere, with respect to some concepts, it’s quite unimaginable and confusing. This time, we may still need mathematics as well as philosophy, to finally solve this paradox.

References:

Binmore, K. G. & Voorhoeve, A. (2003). Defending Transitivity against Zenois Paradox, Philosophy and Public Affairs, Vol.31(3), pp.272-279

John, L. (2003). Key Contemporary Concepts from Abjection to Zeno’s Paradox, Ebrary, Inc.

Wikipedia, Zeno’s paradoxes, http://en.wikipedia.org/wiki/Zeno%27s_paradoxes

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

Two Pi or not Two Pi

The date I’m writing this blog post is March 14th, 2015. If you live in a very particular part of the world, you might represent that date as 3/14/15, which in turn might make you excited to see the first 5 digits of π all nicely lined up in a row. The rest of the world might be confused why you’re making such a big deal out of 14/03/15, and the Western engineers are just biding their time until it’s 3/14/16. The arbitrariness of π day not-withstanding, I’m here to talk about how π, and any days relating to it, pales in comparison to Tau (and any days relating to it!)

Still looks delicious, however. Image: Public domain, via Wikimedia Commons.

But first, some history: π, as we all should know, is the ratio between the diameter of a circle and its circumference. This is 3.141592…  for each and every circle EVER. Which is pretty cool! This usually marked an important discovery for each. Most early cultures didn’t get it quite right, but made their best rational number approximation. In ~1700 BCE, a Babylonian clay tablet uses a constant represented as 25/8, or 3.1250. Around that same time period, an Egyptian papyrus scroll approximates π as (16/9)^2, or 3.1605. One of the most successful techniques of approximating π during this time period is often attributed to Archimedes, where he calculates the perimeter of an inscribed N-gon. Archimedes used an 96-sided polygon, and calculated π to be somewhere between 23371<< 227 (roughly 13.1408 and 3.1429). At the same time, a famous Ancient Chinese mathematician, Liu Hui did the same algorithm on an 3,072 sided polygon, approximating π to be 157/50 3.1416. However, Liu Hui developed a similar, faster algorithm after noting that successive inscribed polygons formed a geometric sequence with a factor of four. Some 200 years later, another Chinese mathematician used this algorithm on a 12,288-sided polygon, calculating the π approximation out to be 31464625<100<31464625 +169625, which translates to roughly 3.141592920.

But enough about the history of π. What I couldn’t find, in all my research, is why ancient peoples were obsessed with the ratio of the circumference and diameter, rather than the circumference and the radius!? After all, a circle is literally defined by its radius as a distance from its origin. Why go through all that unnecessary doubling! I guess ancients needed their line segments to touch something ‘tangible’ on both sides, but thanks to them, let’s take a look at what we have to deal with:

The number π is such an entrenched constant that we developed a unit of measure to use it: radians (which are, of course, dimensionless). If we wanted to measure a full circle it is 2π radians. Wait, what? We have got this glorious constant that people constructed 12,000 sided polygons to calculate, and it only gets us halfway around the circle? So now π/2, π/4, π/8 don’t actually mean half of a circle, or a quarter of a circle, but half of a half of a circle, and quarter of a half of a circle. Enter Tau.

Why not use Tau to express the ratio of the circumference and the radius, that ever pivotal piece of circular information. Because the radius is exactly half the diameter, you can clearly see how Tau is 2π, or 6.28(ish). Why the letter Tau, though? Well my theory is that it is similar enough to Pi that folks don’t feel too threatened by it’s emergence. So what does Tau get us?

For starters, how many radians is a full circle using Tau? Just, Tau! And how about half a circle: Tau / 2! There is no need for a quick mental check of divide by two conversion (…or, wait, was it multiply by two? Which way are we going again? See how this is confusing!), what you want and what you’re looking for are simply in the constant you use.

See how much easier tau is? Image: Michael Hartl, via Wikimedia Commons.

“But Ryan,” you exclaim, “What about ei? Without π, how will Euler be identified!” And to that I say: never fear. While it is true that ei=-1, if you substitute in Tau for Euler’s Identity, you’ll find that cos() + isin() = 1 (Euler’s Formula substituting=) reduces down to a very tasty 1. Identity saved.

And the list goes on! A sine wave is still a sine wave, using instead of π. Using a constant derived from the actual construction of a circle, rather than a near arbitrary doubling of the radius that has been passed down from Ancient peoples and entrenched into our present day mathematical dogma, just seems to make more sense.

And best yet, our deliciou Pi(e) day of March 14th is only delayed a few short months until June 28th,which can be represented as 6/28 which as someone pointed out to me happen to be perfect numbers. This switch practically sells itself, to be honest…though we’ll have to come  up with a dessert called “Tau.”

Of course, there are alternative solutions…

Image: xkcd by Randall Munroe.

Sources:

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

http://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/

The Importance of ‘Nothing’

I’m a programmer. When I ask people their impression of what I do, the usual response is a long string of ones and zeros, said in a robot voice. Before I first started my Computer Science degree, I probably would have said the same thing. After my first semester, I would scoff at such binary answers, and feel powerful knowing I know how to write code. Halfway through my degree, I discovered that when you get down to brass tacks, zeros and ones are really all that comes down to. Finally, here at the end of my degree, I’m really happy that I don’t have to work in raw ones and zeros.

And it has always tickled my fancy that there is no Roman numeral representation for the number zero. I usually just pull this out for fun trivia, but after discovering in class that the Egyptians and Babylonians also struggled with the concept, I thought it might warrant a little extra research.

In this day and age, with our modern schooling, it seems as if zero is trivial. It literally means nothing, after all. It might have a few cool properties. For example, zero added to any number will result in the number as one example…but you can get the same behavior by just multiplying by one! For a computer scientist, zero is a boolean value. Zeroes also have a very friendly feel to them. If you see a lot of zeros at the end of a number, you know that number is a nice round one. And we like round numbers.

But being able to use zero is HUGE! Without it, we would either have an ill-defined positional notation for our numbers, or have to resort to an additive system like Roman numerals.  The lovely round number of 100,000 so cleanly represented here (with a little help from a comma) would require 100 M’s in a row using present day Roman Numerals. Even ancient cultures that used a positional notation would just use contextual clues to figure out if 216 meant 2016 or 2160 or what have you. Babylonians started to help with this problem by making two tiny stylus tick marks. So now, 2106 became 21”6. Interestingly enough, there was never any tick marks at the end of numerals, only in the middle. This leads scholars to believe that these tick marks were not an idea of zero; simply punctuation, much like our helpful little comma from before.

Zero is special in that it has two roles. It can be used for positional notation as we have just seen, but that was just as easily solved with punctation. Zero is also, of course, a number in and of itself, which brings on a whole barrel of troubles. Historically, numbers were thought of much more concretely. People used them to solve ‘real’ problems rather than abstract ones. It is a pretty far jump from for a farmer to go from five horses he owns, to five “things” in existence, to an abstract idea of ‘five’. If the farmer is solving the problem of how many more horses he needs, it is going to be “zero more horses.”

For this reason, perhaps it was lucky for earlier civilizations to miss out on zero. Working with zero can get you into a lot of trouble. There are cases of some of the brightest mathematicians of their time struggling with the concept of zero. And Indian mathematician has this to say about division:

“A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.” – Brahmagupta

You can tell he is reaching when he suggests a number divided by zero is N/0.

The first known zero, found in Cambodia. Image: Amir Aczel.

What’s even more mysterious is how there isn’t some clearly defined point in history where zeros are firmly established. There are some hints and teases in the nautical readings of Greeks and odd punctuation marks in Egypt, but nothing concrete. The earliest known writing of zero is famously from a stone tablet found deep in Cambodia, where it has the date of 605 in sanskrit, with a small dot to denote the zero between the six and five.

A clean rendering of the oldest known numeral using zero. Image: Pakse, via Wikimedia Commons.

It seems odd that such a powerful and tricky number wouldn’t have a more auspicious start. Instead, somewhere, someone in India put a dot on a tablet…and the world was changed forever.

I just hope something like this doesn’t happen:

Zach Wiener, SMBC-Comics 08/29/2012.

-Fin-

Sources:

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

http://www.sciencealert.com/search-for-the-world-s-first-zero-leads-to-the-home-of-angkor-wat/

http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM