Discrete Ideas » Fun

Elegance in motion

The Count — Thu, 27 May 2010 04:41:54 +0000

Every once in awhile, I come across a video (usually on Youtube, isn’t that where everthing ends up) that quite elegantly illustrates something math-related. I haven’t been collecting them long, but I thought I’d share a couple of them with you all. Please enjoy them as much as I have.

Autotuning is a relatively recent fad in the music industry which has everything to do with the mathematics of sound. How else can they take someone’s speaking voice and make them sing!?

This is a quite brilliant video showing how mathematics is present in the shapes of nature. I find it amazing that such “irrational” numbers come from naturally occurring phenomena.

There you have it. I hope you enjoyed those videos. I have a few more, but I think I’ll save them for later. You probably have to get back to work, or some equally non-Math-related thing. Sorry about that; I’ll be back to distract again later.

The Count

Pi for lunch?

The Count — Sun, 14 Mar 2010 19:18:23 +0000

Happy Pi Day! Yes, it’s that time of year again when the month and day (in the American form of date representation) for those legendary 3 digits 3/14, also known as the beginning of the mathematical term Pi1. Being The Count, however, I’m certainly not satisfied with just one Pi Day each year, or just matching 3 digits of Pi for my festivities. No, I must venture forth to find other Math-related dates to share the joy that is the geeky holiday.

Now, Pi Day comes every year but only in America, as the European version of date display comes Day then Month. Unfortunately, this leaves Europe with no way to put forth 3.14 (note, they use a period (.) instead of a slash (/)), as alas, there are only 12 months. For the most part, Europe seems to be out of luck. After searching several dozen math constants2, and I can’t find any that start with a number from 1-30, and have 2 digits after the decimal that form a number less than 13. Viswanath’s Constant3 comes closest, but it looks like we’ll just have to exclude Europeans from our celbratory antics… No pie for you!

Not so fast! Let’s try not to leave them out of all the fun. I present below some alternatives to Pi day, some of which can even be translated to Day before Month!

Date	Reason	Name	Celebration
3/14/15	Pi	Super Pi Day	Eat nothing but pie all day. Luckily, there are many, many varieties4
11/23/58	Fibonacci Sequence	Fibonacci Day	Every hour, give a gift that costs as much as the last two gifts, starting with 2 $1 gifts. (20 points to whomever figures how much the last gift costs)
1/6/18	Phi	Golden Ratio Day	Do unto others only what you’d want to do to you, 1.618 as much!
2/7/18	e	Euler’s Number Day	Do something that people used to believe was impossible.5

We’ll have to wait a bit for these extra-geeky days to arrive, but at least we have today! So, those of you who can celebrate Pi day, run out to your Marie Callander’s or Coco’s, or even the local Denny’s, grab a slice, and enjoy! Save some room for Super Pi Day in a few years though, ok?

The Count

The Gift of Years

The Count — Sat, 30 Jan 2010 21:30:26 +0000

What?! Another birthday so soon!? No, you haven’t lost 6 months of your life, and The Count isn’t 41… yet! Due to the overwhelming popularity of my Birthday article, it became clear to me that even those people that weren’t born during the awesome year of 1969 would like to be able to see their own age in huge-numbered detail.

After many hours of toil, The Count has made available to you, his loyal reader, a page designed to give you all the joy he felt in seeing his age in Earth years converted to galactically-large numbers of other units. Simply click the link below, and enjoy!

The Count’s Age Converter

The Count

Discretely simple

The Count — Sun, 25 Oct 2009 14:00:43 +0000

I’ve said it before, and I’ll say it again – I like math. Now, don’t get me wrong; I don’t run around solving Fermat’s Last Theorem for fun on Sunday nights, or anything like that. Perhaps it would be more clear to say I enjoy math, for many reasons, not the least of which is, when you prove something in math, it’s true every time!

Since merely saying, “I can’t find a contradictory example” is never accepted as “it’s always true”, mathematical proofs are often filled with a multitude of complex concepts and references to other also very complex proofs. It’s very refreshing, I think, to find a few that are so simple and elegant, they make you wonder if there aren’t very simple proofs for those other theorems, that we just haven’t discovered yet. Here I present two of my favorite math proofs. I hope you’ll enjoy them as much as I do.

It’s well known that the sum of the integers from 1 to N is N*(N+1)/2, and this can easily be verified for any number you care to choose. However, it’s the proof for this statement that’s makes it so I can say that it’s true for ALL positive integers. To find the sum for an unknown N, see the image below.

First, write the numbers from 1 to N across the top, then write the same numbers underneath those, only in reverse order. This gives us N columns of numbers which we can now sum individually; as it happens, each column totals N+1. The sum of all these N columns must be N*(N+1) and, since we used all the numbers 1 to N twice each in the rows, we need only divide that sum by 2 to find the solution to our problem… N*(N+1)/2. This elegant little exercise proves that the sum will always work out to that formula. How cool is that?

The second proof is a little more esoteric. While it may not be entirely as useful in your life as the previous proof (if you found that one so), it’s another example of just how simple and elegant some math proofs can be (though most aren’t). Look at the image on the right. The blue shape is a semicircle where A-B is a diameter. What’s not entirely intuitive and/or noticeable from the picture is that all 3 of the angles (marked 1, 2, and 3) are 90-degree or right angles. In fact, the angle formed by the segments between any point on the semicircle and the 2 diametric points must be a right angle. Useful information? Maybe not, but let’s look at how we know it’s always true.

In the picture on the left, we take a look at a single arbitrary example of the angle. By treating it in a completely generic fashion, what we do will be applicable to all possible angles. Note that an additional segment has been drawn from the point D to the center of the diameter (and of the entire circle, were it shown) at point C. The angle we’re trying to prove is a right angle is the sum of the two angles a and ß. Now, since AC and CD are both radii of the semicircle, they must have the same length, and their corresponding angles in the smaller triangle ACD must be equivalent (we’ll call that a). The same thing goes for the BCD triangle, only the angle there is probably different than the ACD pair, so we’ll call it ß. Now, since ABD is a triangle too, its angles must total 180 degrees, yet from our previous statements, it must also total 2*a + 2*ß. Basic algebra will reveal that (a+ß) must equal 90 degrees. Simple and elegant, just as I promised.

Well, these are just two of my favorite math proofs. While neither is exactly mind-shattering, I hope they show that it is possible to prove something mathematically without obscure references to complex transformational theorems and multitudinous graphs and charts. The next time you see a numeric pattern or shortcut, make a try at proving it. You may just come up with a short method that’s eluded us all.

The Count

Litter By Numbers

The Count — Fri, 09 Oct 2009 04:31:20 +0000

About two years ago, I got interested in Geocaching1. I call it “organized littering”. Essentially, people have taken the time to hide caches (usually tupperware containers full of bric-a-brac) all over the world. They then log the lat/long coordinates of their stash, and enter them along with a description on the GeoCaching site2. The rest of us use that site to find caches near where we’ll be, and off we go using our portable GPS units to find these little pockets of fun all over the world.

Our family was vacationing in the Pocono Mountains when we starting ‘caching. I had researched several caches in the area, so one day my father, my wife, my son, and I all headed out into the wilderness. We spent all day traipsing around and seriously enjoyed ourselves getting dirty. The locations we found were a little remote and, since it was summer, the foliage was dense enough to block the view of much. When I decided to write about it, I got to thinking how cool it would be to look for caches in exotic locations. So here they are, some of the coolest locations in the Northern Hemisphere to go looking for litter.

Location	Lat	Long
Pyramids of Egypt	N 29° 58′ 34.00″	E 31° 07′ 52.00″	link
Stonehenge	N 51° 10′ 43.00″	W 01° 49′ 52.85″	link
Mall of America	N 44° 51′ 13.64″	W 93° 14′ 32.43″	link
Waimea Canyon, Kaua’i	N 22° 02′ 55.00″	W 159° 39′ 29.49″	link
Goat Island, Niagra Falls	N 43° 04′ 50.15″	W 79° 04′ 07.92″	link
Eiffel Tower	N 48° 51′ 21.46″	E 02° 17′ 27.75″	link
Checkpoint Charlie, Berlin	N 52° 30′ 23.33″	E 13° 23′ 24.69″	link
Tarifa, Spain	N 36° 00′ 35.10″	W 05° 36′ 24.01″	link

Now, I know you’re just going to run out and fly to all these great places just to go find the closest cache, right? Ok, maybe not, but I hope I peaked your interest in Geocaching, it’s a great way to learn about global coordinate systems and geography. The next time you have a free Saturday afternoon or want a reason to take a hike, give it a try.

The Count