The relatively formal and orthodox way that math is usually taught in school usually leaves students knowing how to “Solve for X,” but often isn’t as effective at teaching how to use math to solve problems.
“Golden Ratio” rectangles provide a useful example here. A “Golden Ratio” rectangle has been known to artists for thousands of years as being especially aesthetically pleasing. It has the property that if you cut it lengthwise so as to leave a square, the remaining rectangle has the same proportion as the original.
This doesn’t seem to describe how to go about building one — until you rephrase the definition in the language of mathematics. If we call the width X and the height Y (and assume it is placed so X is the longer dimension), we can describe this relation in terms of X and Y, and make an algebraic equation out of it.
Since the original rectangle is X long and Y high, we know that if we cut off a YxY square from one end, the resulting rectangle will have the same dimensions as the original.
So, X/Y (the original ratio) = Y/(X-Y), since Y is now the longer end.
From here, algebraic manipulations give us what we’re looking for — a fixed numerical ratio between X and Y…
X/Y = Y/(X-Y)
Set Y to 1 to obtain the unit ratio:
X/1 = 1/(X-1)
Simplify: X = 1/(x-1)
Cross-multiply to get X(X-1)=1
Distribute and regroup: X^2 – x – 1 = 0
This is a quadratic equation, and is easily solved with high-school algebra. One root of the equation is negative, which doesn’t make sense, but the other is positive: x=[sqrt(5)-1]/2, or roughly 1.618.
This can easily be checked: if you remove a 1 x 1 square from a 1.618 x 1.0 rectangle, you’re left with a 1 x 0.618 rectangle, which has the same aspect ratio (it’s the same shape, just smaller.)
Once you learn to “ask questions” using math like this, you can often find the answer you’re looking for. The art is phrasing the question.
This post is dedicated to the memory of my grandfather, Millard C. Carr, who would have been 100 years old today, and who was partly responsible for my love of technology and engineering, and pretty much wholly responsible for my love of aviation and classic propliners. We miss you, Granddad.