There are certain things, says Aristotle, that suffer no alteration - save magnitude - as they grow.  Thus if we add to a square an  L-shaped portion, shaped like a carpenter's square, the resulting figure remains a square; and the portion which we have added, with this singular result, is called in Greek a `gnomon'.

a) A rectangle having sides one and square root of two, respectively, is a gnomon of itself!
b) A rectangle whose sides are in proportion to the golden ratio has a square on its longer side as gnomon.
c) Let ABC be an isoceles triangle having apical angle of 36 degrees
and base angles, hence, of  72 degrees.  Then bisecting one base angle subdivides the larger triangle into two smaller isoceles triangles, one of which is similar to ABC and the other of which is a gnomon.
 
 
 

2) The following image is obtained by starting with the a rectangle whose sides are in the golden ratio, then adding a gnomon - a square in this case - to build a bigger golden rectangle. Starting with the smallest rectangle, sweep out an arc by connecting  a chosen corner of this rectangle to the corresponding corner of the next larger rectangle, then continue doing so until you get to the largest rectangle.  Does the curve you swept out bear any resemblance to the spiral below? What about the nautilus shell below?
 

All figures formed by gnomons will have these characteristics.

Next let's look more closely at the golden ratio.
The Greek's were intrigued by the golden ratio because they regarded golden rectangles as being the most pleasing to observe. They built the Parthenon with this in mind.

Artists throughout history, most notably Leonardo da Vinci, have been intrigued with the golden ratio.
For example, if one draws a rectangle bounding the Mona Lisa's face, one will find its sides to be in the golden ratio.

If you want to know the meaning of the image below, you'll need to read Sigmund Freud's biography of Leonardo Da Vinci.
In Leonardo's portrait of an Old Man, he partitions the face into rectangles, many of which bear the golden ratio.

3) Using what you know about the construction of Golden rectangles by adding squares, and also simply the fact that phi is about 0.618, see how many significant ways you can divide the Leonardo painting below.

Now we move to another Leonardo, Leonardo of Pisa, also known as Fibonacci.
In his Liber Abaci, c. 1202, he posed the problem of  counting rabbits, under some basic assumptions.  One is given a pair of rabbits. After one month it reproduces another pair.  Each succesive month, every new pair becomes an adult pair and itself reproduces a baby pair, while each mature pair continues to reproduce likewise.  Let F_N , called the Nth Fibonacci number, be the number of adult pairs of rabbits after N months.  So F₀=F₁=1, while F₂=2,  F₃=3,  F₄=5 and so on.

1) Compute F_N for N=1,2,...,12.
2) Explain why the description of how the rabbit pairs are generated can be translated into the mathematical formula:  F_N+1    = F_N + F_N-1
3) Let R_N   denote the ratio F_N+1 /F_N of the numbers of rabbits from two successive months. Compute these numbers for the first twelve months.   Do you see any pattern?
4) In the long run the ratio approaches a fixed number R.  Using the recursion formula for the Fibonacci numbers, show that the number R satisfies the quadratic equation R² - R -1 =0.  Now use the quadratic formula to determine the value of R.