Step II
: Gently make an overandunder 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 fivepointed star in the pentagon without lifting your pen/
pencil.
The pentagram is a symmetrical 5pointed star that fits inside a pentagon. Starting
from a pentagon, by joining each vertex to the nextbutone 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 straightline 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.
You've reached the end of your free preview.
Want to read all 18 pages?
 Fall '09
 Math, The Da Vinci Code, Golden ratio, Golden Section