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PENTAGON {5}
A pentagon has five equal sides, and five equal angles.
Check your answers. A pentagon can be constructed from a strip of paper by folding it in a simple knot.
In the figure, using the obvious symmetries of the pentagon, we see that the two blue triangles are similar (the marked angles are equal, so all corresponding angles are equal). Hence: 1/x = d/1, and x = 1/d. Now noting the parallelograms in the figure, from the orange diagonal we have This number
Early Greek geometry was largely based on the properties of straight lines and circles, so it was natural that they were interested in geometric constructions using just straightedge and compass. Here are some basic constructions that one can do: (a) Draw a line through two given points.
We have seen that we can construct the equilateral triangle using straightedge and compass. An obvious challenge for the early Greeks was to construct all the regular polygons in this way.
The regular pentagon can be constructed too, but with a little more difficulty and some insight! Euclid (ca 300 BC) gave a construction, and simpler constructions were provided by Ptolemy (85 – 165 AD) and then Richmond in 1893.
F0 = 21 + 1 = 3, F1 = 22 + 1 = 5, F2 = 24 + 1 = 17, F3 = 28 + 1 = 257, F4 = 216 + 1 = 65537. At this point the history of polygon constructs becomes rather absurd. Richmond gave a simple construction for {17}, Richelot and Schwedenwein constructed {17} in 1832, and one J. Hermes spent ten years constructing {65537}, depositing his manuscript in a large box in the University of Göttingen, where it may still be found!
In constructing the regular pentagon, we can think of taking five points equally spaced around a circle, and then joining successive points using straight line segments. However, instead of choosing successive points, what if we choose every second point? We then obtain a new non-convex regular figure called the pentagram. It has five vertices (on the circle) and five connecting (and intersecting) edges. It is denoted by {5/2}. The meaning of the 5 is clear here; the 2 denotes the number of circuits around the centre, or equivalently, the general number of crossings that must be made to reach the centre from outside the figure. Clearly {5/2} = {5/3}. (But notice that the ‘density’ interpretation of the 2 does not carry over to the 3.) The pentagram is a very old symbol, dating back to around 3500 BC. It has in the past been used as a Christian symbol, but today is more likely to be associated with magic and witchcraft.
We say this figure has a fractal structure. An original large (elaborate) pentagon occurs again and again in the figure on a smaller scale. Here is an easier fractal construction which conveys the same idea. At left, a central pentagon is surrounded by five others. Notice that the outline of this extended pentagon is essentially another pentagon. In the sequence on the right, we start with a base pentagon, and replace it by the figure at left. We then replace each smaller pentagon by the (scaled) figure at left, and so on. The limiting figure obtained (in theory!) by this process is a fractal.
Golden Section : http://mathworld.wolfram.com/GoldenRatio.html Straightedge and compass : Coxeter, H. S. M., Introduction to geometry, Wiley (Edition 2) (1969) Chapter 2. History of the pentagram : http://www.fabrisia.com/pentagram.htm Fractal patterns : http://internal.maths.adelaide.edu.au/people/pscott/fractals/
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Our next convex regular polygon in our sequence is the (regular) pentagon {5}. 
Intrinsic Properties 
The pentagon has five diagonals. Interestingly, the diagonals at each vertex trisect the vertex angle into three equal-sized angles. Can you see why? It is easy! Now suppose the pentagon has side-length 1 and diagonal length d. We seek to find the value of x in the adjacent diagram.
= 1/2( 
5 + 1) = 1.61803 ... . This ratio d : 1 of the diagonal to the side length is the golden ratio, and the diagonals are said to cut each other according to the golden section.
We wish to inscribe a regular pentagon P0P1P2P3P4P0 in a given circle with centre O. 
So we can now construct {n} for n = 3, 4, 5 and 6. For any given {n} we can construct the perpendicular bisectors of two adjacent sides to determine the circumcentre of {n}. Then further we can bisect the central angles to determine {2n}. Hence it is clear that if {n} is constructible with straightedge and compass, then so is {2n}. However, in fact, we cannot even construct {7} with straightedge and compass. The question was completely answered by Gauss (1777 – 1855) (pictured) at the age of 19 (!). Gauss found that a regular n-gon, {n}, can be so constructed precisely when the odd factors of n are distinct Fermat primes: that is, prime numbers of the form 
. Sadly, in terms of our polygon problem, the Fermat primes are pretty thin on the ground: the only known primes of this kind are:
Albrecht Dürer (1471 – 1528), artist and part-time mathematician is probably best known today for the magic square which appears in his picture Melancholia. Albrecht Dürer also published The Painter's Manual in which he illustrates various ways of drawing geometric figures. One section on ‘Tile Patterns Formed by Pentagons’ foreshadows our modern understanding of fractals.
