Visual 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
Setting the two ratios equal leads to the equation
x2-x-1 = 0.
We can solve this equation for the Golden Mean: we find one
positive root, x= (1+)/2 = 1.618033...
This rectangle seems to have nothing to do with a pentagon.
Finally, to round off the argument, note that
- In a regular pentagon, the ratio of diagonal to side is the
same as the ratio of long side to short in the golden rectangle, namely
the Golden Mean 1.618033...(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.
number is rational if and only if it is
commensurable with 1.(Link to Proof).
© copyright 2000, American Mathematical Society.