Please note: These papers were prepared for the Greek Science course taught at Tufts University by Prof. Gregory Crane in the spring of 1995. The Perseus Project does not and has not edited these student papers. We assume no responsibility over the content of these papers: we present them as is as a part of the course, not as documents in the Perseus Digital Library. We do not have contact information for the authors. Please keep that in mind while reading these papers.

The Golden Section

If you don't think math is fun, be gone. For everybody else on the web who enjoys math, here is a beautiful and interesting element of geometry tracing as far back as the Greeks, and maybe even further. Note: I'm only human, and a student at that, so please look kindly upon my humble attempt to verbalize beauty that was never meant for words.

WARNING: BORING MATH AHEAD! ONLY THE STOUT OF HEART DARE BRAVE THIS WORLD.

What is The Golden Section? The golden section is a line segment divided into two parts. Point C is positioned such that the ratio of the short half to the long half is equal to the ratio of the long half to the whole.
Symbolically:

A------C---B where CB / AB = AC / AB, or AB^2 = BC x AC

This was the "mean and extreme ratio," according to the Greeks, while it has been known in more recent times as the "divine proportion" and "golden section." There is some confusion over the origin of the later term, the "Golden Section". It is attributed to Johannes Kepler in one source, who mentions it as being one of the most beautiful constructions in the mathematical world. There is evidence that something similar was used by the Greeks, as Eudoxus is credited with applying Plato's methods to the "section."{Things as I have found them are ambiguous.}

From a more esthetically, it is one of the most beautiful quarks of the mathematical universe because of its combination of elegance and simplicity.

We'll get into the good stuff in a minute, but first a little history and background. The pentagon, dodecahedron(a solid made of 12 regular pentagons), and the golden section are intimately intertwined in mathematics that is more than simply elementary. Yet, despite this, the earliest evidence of the golden section is found in an Etruscan dodecahedron from before 500 BC, and similar dodecahedrons have been found elsewhere in the world. It is unlikely that these people understood the mathematics of this object, but it's presence raises the question of how far basic mathematical knowledge had progressed before its formal introduction by the Greeks. The first mathematical occurrence of the golden section and it's associated figures is found in the school of thinkers founded by Pythagoras. The Pythagoreans, as they are known, adopted the pentagram as the symbol of health in their brotherhood, and it eventually came to be the distinguishing badge of their school. Unfortunately, little of their actual mathematics survives, but it is highly likely that they were the ones who derived the construction of the pentagon and decagon from the golden section. Eudoxus can be given credit for several theorems about the ratios found in the golden section. Euclid then used the basic properties discovered by his predecessors to construct regular pentagons, decagons, and dodecahedrons from the ratio of means and extremes.

The golden section then sank away into the background as an unimportant curiosity until the work of Leonard Fibonacci of Pisa. One of the problems Fibonacci dealt with was the reproduction of rabbits. The series that resulted is now commonly know as the Fibonacci series and remains a very interesting phenomena because it is so commonly echoed in nature. Where is the connection to the golden section? The ratio of successive elements in the Fibonacci sequence approaches the mean-extreme ratio , which happensto be the most irrational of all irrational numbers.

Now down to the good stuff.

To divide a line segment(AB) into lengths of the mean-extreme ratio:

Let the length of AB be x. Thus the length of AZ must be 1/2 x. From this, it is not hard for one to figure the length of ZB, ZY, AY, and finally AC. The ratio of AC/AB can then be found to be , and the ratio of CB/AC is found to be . They don't look equal at first, but they are. Go ahead and check them.

How does this relate to triangles and pentagons?

If an isosceles triangle is now constructed, such that the base AD is the same length as AB and sides AC and CD are congruent, and then segment DB is constructed, the resulting triangles are found to have some interesting properties. With some knowledge of Euclidean geometry, a pencil and several sheets of paper, it can be proven that CD is congruent to BD. It then becomes apparent that in isosceles triangles BDC and DAB, the base angles are exactly twice as large as third angle. Thus, the angles in triangle DAB are 72deg. and 36deg., or in radians, 2[[pi]]/5 and [[pi]]/5.

Now, every high school student who has had basic trigonometry has had a table of cosines and sines drilled into them. Cosine of [[pi]]/4. (1/2)^1/2. Sine of [[pi]]/6. 1/2. Tan of [[pi]]/2. 0. Cosine of [[pi]]/5. This is the one teachers avoid because it is more complicated than others. If segment AB has a length of one, the circle centered at A becomes a unit circle. Since CDB is an isosceles triangle, then a line bisecting angle CDB is a perpendicular bisector of segment CB. Then, the cosine of [[pi]]/5 is only AC+1/2 CB, and the sine of [[pi]]/5 can be found by using the Pythagorean theorem.

Cool? Yes No