The Golden Mean occurs traditionally as the ratio of long side to short in a "golden rectangle."
The golden rectangle has the following property: If a square is cut off from one side of the rectangle ... | ![]() |
... the rectangle that remains ... | ![]() |
... has the same ratio of sides as the original rectangle. | ![]() |
If we call this ratio x, then a rectangle with sides 1 and x will have the correct ratio. The rectangle remaining after cutting away a square will have sides x-1 and 1. | ![]() |
Setting the two ratios equal leads to the equation x2-x-1 = 0. | ![]() |
We can solve this equation for the Golden Mean: we find onepositive root, x= (1+)/2 = 1.618033...This rectangle seems to have nothing to do with a pentagon.Nevertheless,
Finally, to round off the argument, note that
So the pentagon argument gives us a geometric proof of the irrationality of the Golden Mean. See The most irrational number (this column for July, 1999) for more lore about this amazing number.