Step II Gently make an over and under knot rolling the paper round as in the

# Step ii gently make an over and under knot rolling

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Step II : Gently make an over-and-under knot, rolling the paper round as in the diagram. (This is the slightly tricky bit!) Step III : Gently pull the paper so that it tightens and you can crease the folds as shown to make it lie perfectly flat. Now if you hold it up to a bright light, you'll notice you almost have the pentagon shape. Try drawing a five-pointed star in the pentagon without lifting your pen/ pencil. The pentagram is a symmetrical 5-pointed star that fits inside a pentagon. Starting from a pentagon, by joining each vertex to the next-but-one you can draw a pentagram without taking your pen off the paper. Consider the pentagon ABCDE. Draw the pentagram inside it. The pentagram has 5 triangles on the edges of another pentagon FGHKL at its centre. The process can continue infinitely number of times (though may not be feasible to do so practically).
Let's focus on one of the triangles and the central pentagon as shown here. All the orange angles at the vertices of the pentagon are equal. They are called the external angles of the polygon . What size are they? This practical demonstration will give us the answer: * Take a pen and lay it along the bottom right edge pointing right. * Turn it anticlockwise through the orange angle so that it points up to the next vertex. * Move the pen along that side of the pentagon to the next vertex and turn it anticlockwise through the next orange angle. * Repeat moving it along the sides and turning through the rest of the orange angles until it lies back on the bottom edge. * The pen is now back in its starting position, pointing to the right so it has turned through one compete turn. It has also turned through each of the 5 orange angles. * So the sum of the 5 orange angles is one turn or 360°. Each orange angle is therefore 360/5=72°. The green angle is the same size as the orange angle so that the two "base" angles of the blue triangle are both 72°. Since the angles in a triangle sum to 180° the yellow angle is 36° so that 72° + 72° + 36° = 180°. Hence the blue triangle is a golden triangle. The basic geometrical facts we have used here are: The external angles in any polygon sum to 360°. The angles on a straight-line sum to 180°. The angles in a triangle sum to 180°. Pentagon – The US Military Department Building Order of the Eastern Star emblem
The Pythagoreans used the Pentagram as a sign of salutation, its construction supposed to have been a jealously guarded secret. Hippocrates of Chios is reported to have been kicked out of the group for having divulged the construction of the pentagram. The pentagram is also called the Pentalpha, for it can be thought of as constructed of five A's. For more examples on the pentagram visit - masonry/pentagrams_additional.html Pentagons in Nature FIBONACCI SERIES IN NATURE We have seen that the ratio of the two consecutive terms of Fibonacci series gives us the Golden ratio. The Fibonacci series , generated by the rule, f 1 = f 2 = 1 ; f n + 1 = f n + f n " 1 is well known in many different areas of mathematics and science.

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