visual3Visual Explanations in Mathematics
3. The link to the Golden Mean
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,
- In a regular pentagon, the ratio of diagonal to side is thesame as the ratio of long side to short in the golden rectangle, namelythe Golden Mean 1.618033...(Link to Proof)
Finally, to round off the argument, note that
- A number is rational if and only if it is commensurable with 1.(Link to Proof).
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.