Deceptive and Misleading Mathematical Patterns

In the Mathematical Palette, I have mentioned about mathematics as a science of patterns.   I have highlighted that some mathematical patterns are obvious, some can be solved mathematically, and some are a bit counter-intuitive.

Venus Transit (2004)

In reality, we have improved our way of living by recognizing and generalizing patterns. For example, we are able to predict the weather through the data we have collected all over the years. We look for patterns from the data and use probability to announce that there will be rain showers and thunderstorms for the next three days and feel pretty sure about it. Through patterns, we have even predicted the movement of planets. We know that the next transit of Venus  is  in 2117 (too bad if you didn’t see it on June 5).  That is how beautiful and powerful patterns are. » Read more

Understanding the Fermat’s Last Theorem

The Fermat’s Last Theorem is one of the hardest problems in the history of mathematics.  The problem was written by Pierre de Fermat in 1637, and it was only solved more than 300 years later —  in 1995 by Professor Andrew Wiles.

But what is exactly the Fermat’s Last Theorem?

The Fermat’s Last Theorem is an extension of the Pythagorean Theorem.  Recall that the Pythagorean Theorem states that given a right triangle whose side lengths are x, y and hypotenuse z, x^2 + y^2 = z^2 is satisfied. For example, a right triangle with side lengths 2, and 3 has hypotenuse \sqrt{13}.

There are some interesting things that we can see if we examine the side lengths of right triangles.  For instance, if we let the triples (x,y,z) be the side lengths of a right triangle, where z is the hypotenuse, we can find triples such that all lengths are integers. The triples  (3, 4, 5), (5,12,13), (8,15,17) are integer triples, and they satisfy the Pythagorean Theorem. These triples are called Pythagorean Triples. It is not also difficult to see that there are infinitely many Pythagorean Triples (Can you see why?). » Read more