The Golden Mean (or golden ratio) is a famous number that has been know from ancient times. For example some ancient buildings were built with this ratio used for its proportions.
Here is an algebraic conditions that defines the golden ratio.
So, it satisfies the equation x-1 = 1/x. It is not hard to show that the golden mean is one half of the sum of one and the square root of 5. Since the square root of 5 is about 2.236, and one plus the square root of 5 is 3.236, the golden ratio is approximately 1.618.It is positive and subtracting 1 from it gives a number that equals its reciprocal.
The geometric construction that I have seen most often golden ratio uses a square A, B, C, D. Find the midpoint of the side AB and call it M. Make the circle with center M that passes through C (and D). Let E be the point where this circle meets the ray AB. Then the ratio AE / AB equals the ratio AB / BE. It is easy from this that that AE/AB is the golden ratio. Click here to see a picture (coming soon).
A recent and surprising construction is one given by George Odom. George constructs an equilateral triangle ABC with vertices on a circle. Let M be the midpoint of AB and N be the midpoint of AC. Then the ray MN meets the circle at a point P. The ratio MP/MN is the golden ratio. Click here to see a picture (coming soon).
Another pleasant surprise construction is given by Gabriel Bosia. Gabriel starts with the famous pythagorean triangle with sides 3, 4, and 5. Construct the circle of diameter 5 with center at the midpoint of the hypotenuse.
Suppose that BC is the side of length 3 and AC is the side of length 4. Let M be the the midpoint of side AC. Construct the perpendicular bisector of AC. It goes through the center of the circle and meets the circle in two points, one of them 4 units from the circle and the other 1 units from M. (Ptolomy). This second point, call D. The segment CD has length square root 5. Extend along the ray CD one more unit to E. Then the midpoint N of CE gives the golden ratio. Click here to see a picture (coming soon).
Three-Four-Five Triangle and the Golden Mean
The figure above, shows how one may use the famous three-four-five triangle inscribed in a circle to construct the Golden Ratio. Of course this is not the most efficient construction in terms of the number of steps required, but it is interesting. It was discovered by Gabriel Bosia. Find out more about constructions for the golden mean
Here are some very nice geometrical things.
