Pythagoras and Fibonacci

A

few weeks ago, I was walking through the local B&N, and found A Slice of Pi1. In it, Liz Strachman outlines some of the short mathematical tidbits she would employ as a means to keep her students' attention and interest during her years as a teacher.

One of those was an exercise she refers to as "Fibonacci Meets Pythagoras:"

Take any four sequential Fibonacci Numbers ... 2 3 5 8.
Multiply the outside numbers,
Multiply the inside numbers  and double,
, .
16 and 30 are the legs of a right-angled triangle.
Using the Theorem of Pythagoras, we can calculate the hypotenuse:
...
Notice that 34 is also a Fibonacci number further along the line.

As I explained in an earlier post, a "Fibonacci Number" is a number in the series where each number is the sum of the previous two:

...

This is pretty easy to see and understand, and, with a little bit of substitution, leads us to a formula for finding that fourth sequential number using the first two:

If  and  then  or

A Pythagorean Triplet is a set of three numbers which can be used as the legs and hypotenuse of a right triangle. Numbers where .  To express Ms. Strachman's example algebraically:



I'm not sure if I can prove that will always be a Fibonacci Number, but that definitely appears to be the pattern (notice also that the ordinal of the resulting Fibonacci Number is the sum of the ordinals of the first and fourth Fibonacci Numbers used in the equation):

Fn (a)Fn (b)Fn (c)Fn (d)Resulting NumberResulting Fn
Fn1 (1)Fn2 (1)Fn3 (2)Fn4 (3)5Fn5
Fn2 (1)Fn3 (2)Fn4 (3)Fn5 (5)13Fn7
Fn3 (2)Fn4 (3)Fn5 (5)Fn6 (8)34Fn9
Fn4 (3)Fn5 (5)Fn6 (8)Fn7 (13)89Fn11
Fn5 (5)Fn6 (8)Fn7 (13)Fn8 (21)233Fn13
Fn6 (8)Fn7 (13)Fn8 (21)Fn9 (34)610Fn15
Fn7 (13)Fn8 (21)Fn9 (34)Fn10 (55)1597Fn17
Fn8 (21)Fn9 (34)Fn10 (55)Fn11 (89)4181Fn19
Fn9 (34)Fn10 (55)Fn11 (89)Fn12 (144)10946Fn21
Fn10 (55)Fn11 (89)Fn12 (144)Fn13 (233)28657Fn23

I am positive that it will always be a whole number. And (as you could probably guess), even if a & b weren't Fibonacci Numbers, the formula would still generate a Pythagorean Triplet.

I wouldn't be so hyperbolic as to call that "magic," but it's definitely cool.

  1. Strachan, Liz. A slice of Pi: all the math you forgot to remember from school. New York: Fall River Press, 2010. Print. []
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