Modeling Nature

You know those movies with all of the realistic scenery and special effects?  What you are seeing on the screen may or may not be real, and the effects are often made possible by applying fractal techniques.

Fractals are the answer to creating irregular geometric shapes such as mountain ranges and coastlines.  That is because, like these shapes, a fractal is self-similar.  Self-similar means that the appearance is the same no matter how far you zoom in or out.  Take a tree, for instance.  If you cut off a branch and stick it in the ground, the branch looks like a tree by itself.  Take a branch off that branch and stick it in the ground, and you have what looks like a tree.  The smaller part looks like the whole.

Trees, snowflakes, mountain ranges, coastlines, blood vessels, EKG readouts, and many other things have fractal properties and can be modeled using fractal techniques.

To find out more, check out NOVA’s special.

Hunting the Hidden Dimension

The Big Reveal

Okay, the answer from yesterday’s post is…

5050

How did he get that?  Well, he did not start by adding 1 + 2 + 3 + …  Instead, he found a pattern.

Try writing 1 to 100 on one line and then 100 to 1 on the next.  Add up the numbers directly up or down from each other.

1 + 100, 2 + 99, 3 + 98, 4 + 97, 5 + 96, …

Each sum is 101, and you get 100 of these sums which comes out to 10100.

You need only half this amount, as you added each number in twice.

10100 / 2 = 5050

How about you?  Did you get the answer?  Whether you did or did not, what did you try?

Mental Sum

Take a moment and add up the numbers 1, 2, 3, …, 98, 99, 100.  Now, there’s a catch before you try.  Do it in your head, and you have 10 minutes.

If you did it, congratulations!  Please don’t share the result, though.  If you did not do it, reflect on the fact that Carl Friedrich Gauss was able to do so at the young age of 10 years old.  Apparently, his teacher had a hard time with keeping the lad busy, and she sent him off with this problem to get him out of her hair for a bit.  I can just imagine her expression when he came back in just a few minutes with the correct answer!

I will decline from explaining for today.  Tune in tomorrow for how Gauss tackled the problem.  In the meantime, see if you can find not only the answer, but the technique.

Striking Goldbach

Goldbach’s Conjecture was first proposed in a weak form by Christian Goldbach in the 18th century and then in a stronger form by Leonhard Euler.  It says that every odd number can be broken up into a sum of, at most, 3 prime numbers.

35 = 19 + 13 + 3

21 = 3 + 7 + 11

It has not been proven so far, but the young Terence Tao seems to be making progress.

Sacrificial Solid

A dodecahedron is a solid, 3-D figure made up of 12 regular pentagons.

It is said that when the dodecahedron was discovered, the Pythagoreans sacrificed 100 oxen in celebration.  (1)

Mathematical Graffiti

Speaking of Sir William Rowan Hamilton and quaternions, the solution to how to multiply quaternions came to Hamilton one day while he was out on a walk with his wife.  Worried that he would forget, Hamilton immediately carved the answer in the Brougham Bridge.  That piece of mathematical graffiti went unpunished and in fact is commemorated by a plaque today on that same bridge.  (3)

When are we ever gonna use this?

Sometimes, mathematical discoveries come about because people are trying to solve real-world application problems.  Other times, mathematical discoveries come about because people have incredible imagination and insight, but then those discoveries are seemingly useless, for a while.

Quaternions are something that fall into the latter category.  These members of a noncommutative divisional algebra were first discovered by Sir William Rowan Hamilton in 1843.  They were fun for theoretical mathematicians but did not seem to have a practical use until 1985, when computer scientists realized that they can be used in digital animation.  Quaternions are perfect for work involving orienting and rotating objects in 3-D space.  (4)

Prime Time

Prime numbers are integers greater than 1 that just have factors of 1 and themselves.  Some examples are 2, 3, 5, 7, 11, and so forth.  Composite numbers are the ones in between, excluding 1.  (One is neither prime nor composite.)  Composites have more factors than just 1 and themselves.

Mathematicians, especially number theorists, love talking about prime numbers and looking for patterns in them.  One such mathematician, Bernhard Riemann, hypothesized that there is a hidden pattern to the way that prime numbers fall on the number line.  The hypothesis holds true for the first billion cases, but this does not stand as rigorous proof.  A proof has to show, without a doubt, that it is true for all cases.

You think that you can prove it?  If so, then you will be the next winner of \$1 million, for the Riemann Hypothesis is one of the Clay Mathematics Institute’s Millennium Prize Problems.  (4)

It’s Magic

Take a minute and try to put the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 into the grid so that the sum is the same for each row, column, and large diagonal.

If you have succeeded, then you have filled out something called a magic square.  (I will refrain from posting the solution so that inquiring minds do not accidentally see it.  Just Google “magic square” or “Lo Shu diagram” if you need an answer key.)

These squares have intrigued people through the ages.  Some thought that the completed squares held mystical properties and proceeded to carry them around as talismans against evil.  Others, like Benjamin Franklin, found them so interesting that they spent much time on finding bigger ones.  Ben Franklin even delved into creating magic circles!

Zero Zeroes

Imagine a time without zero.  Hard to do, huh?  Actually, zero was not recognized as a number until 628 AD with Brahmagupta.  He wrote out rules for using zero, which have been edited since then.  People as early as the Babylonians and indigenous Central American peoples used symbols for missing numbers, but that was not the same as recognizing zero on its own.   (1)