Fibonacci Fun

The Fibonacci sequence has been mentioned in a previous post, but let’s look at one of the many cool patterns hidden in the numbers.

Just to refresh your memory about the Fibonacci sequence, it appears as follows:  1, 1, 2, 3, 5, 8, 13, 21, …  Each new term in the sequence is found by adding the last two terms.

Now that you have the sequence in your mind, let’s look at a pattern.  Find a number and multiply it with the number following the next.  For instance, if you take 3, multiply it with 8.  The result will always be 1 more or 1 less than the square of the number in between.  So, 3 x 8 = 24 while 5 squared is 25, a difference of 1.  The product is 1 less than the square.  How about we take 5 with 13.  The 8 squared is 64 while 5 x 13 = 65.  Again, there’s a difference of 1.  The product is 1 more than the square.  Whether the result is 1 more or 1 less will alternate down the sequence.  Pretty cool, huh?

The alternating pattern is made all the cooler when you stop to realize that the ratios of successive numbers alternate higher and lower, always approaching phi (the Golden Ratio).

Jail Bird Mathematician

Evariste Galois was an early bloomer.  He was so early, that he developed group theory (a major advancement in abstract algebra) while he was in his teens.  At the same time that he was developing all kinds of new mathematics, he stayed heavily involved in politics during a tumultous period of history in France.  He did spend some time in jail for his political involvement, which gave him plenty of free time to work.  Tragically, we cannot see what may have become of Galois as an adult mathematician.  At the age of 20, he was killed in a duel over a woman.