Author Archives: rwsargent

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

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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