A Euclidean Construction of a Golden Rectangle in Which All Circles Used Have Radius One or Two

There is more than one way to construct a golden rectangle using the Euclidean rules, but all the ones I have seen before use circles with irrational radii. This construction, which I believe to be new, does not use that shortcut, which helps explain its length. The cost of avoiding circles of irrational radius is decreased efficiency, as measured by the number of steps required for the entire construction.

In the diagram below, the distance between points A and B is set at one. All of the green circles have this radius, while the magenta circles have a radius exactly twice as long.


To make following the construction from the diagram above easier, I named the points in alphabetical order, as they appear, as the construction proceeds. The yellow rectangle is the resulting golden rectangle. The blue right triangle is what I used to get a segment with a length equal to the square root of five, which is a necessary step, given that this irrational number is part of the numerical definition of the exact value of the golden ratio (one-half of the sum of one and the square root of five). In order to make the hypotenuse have a length equal to the square root of five, by the Pythagorean Theorem, the two legs of this triangle have lengths of one and two.

The Golden Ratio: Working from a Definition to Find a Value


I found the image above through the Wikipedia article on the golden ratio. After using what appears above to define the golden ratio, the article then reveals its exact and approximate values. Later, the writers of the article do show the calculations involved in doing this, but they seem unnecessarily complicated. I’m going to try to simplify the process here, and might later edit/simplify this Wikipedia article to make it more understandable.

So, first, “a + b is to a as a is to b” need to be written as a fraction, which is easy enough: (a + b)/a = a/b. The value of this fraction, a/b, is, by definition, φ, the golden ratio. As an equation, this can be written a/b = φ.

Next, apply cross-multiplication to (a + b)/a = a/b, and it becomes (a + b)(b) = (a)(a), which simplifies to ab + b² = a².

Also, since a/b = φ, this means that a = φb (via the multiplication property). Next,  ab + b² = a² is rewritten, with φb substituted for each a. The result of this substitution is (φb)(b) + b² = (φb)², which then becomes φb² + b² = φ²b². To simplify this, b² may be cancelled (via the division property), producing φ + 1 = φ². This may then be rearranged (via the subtraction and symmetric properties) to φ² – φ – 1 = 0. Two values of  φ can then be found via the quadratic formula, and they are {1 ± sqrt[1 – (4)(1)(-1)]}/2 = [1 ± sqrt(5)]/2. Use “+,” and calculate a decimal approximation for this irrational number, and you get ~1.618, which is the golden ratio. Use “-” instead, and you get a negative number (approximately -0.618), which can be rejected on the grounds that a ratio of two lengths must be positive, since all lengths, themselves, are positive.

Also, I’m changing my mind regarding changing Wikipedia, on this subject. The two versions of the calculation (the one now on Wikipedia, and mine) don’t match, but both are mathematically valid — and, while my version makes more intuitive sense to me, that doesn’t mean it would make more sense to others, and Wikipedia isn’t there for me alone. Until I actually wrote the calculation out, I thought my version would be simpler, but I cannot claim that now.

[Later addition: see the first comment below for a way, suggested by a reader of this blog, to simplify the calculation, as I wrote it above. I’m not going to take credit for his improvement, of course — that would violate mathematical etiquette!]