Geometrical Constructions 1
7 Regular Polygons, Golden Ratio
Regular Polygons
A polygon is a many-sided shape. A regular polygon is one in which all of the sides and angles
are equal. Some examples are shown below.
n=3
n=6
n=5
n=4
n=7
(non constructible)
n=9
n=8
(non constructible)
…
Only certain regular polygons are "constructible" using the classical Greek tools of the compass
and staightedge. According to Gauss´ theorem, a regular n-gon can be constructed, if and only
if the odd prime factors of n are distinct “Fermat primes”
k
Fk = 22 + 1.
F0 = 2 2 + 1 = 3, F1 = 2 2 + 1 = 5, F2 = 2 2 + 1 = 17, F3 = 2 2 + 1 = 257, F4 = 2 2 + 1 = 65537,
0
1
2
3
4
and it is known, that Fk is composit for 5 ≤ k ≤ 32.
http://mathworld.wolfram.com/FermatNumber.html
http://en.wikipedia.org/wiki/Tilings_of_regular_polygons
http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html
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Geometrical Constructions 1
7 Regular Polygons, Golden Ratio
Approximate Construction of Regular Polygons
Approximate construction of
regular heptagon:
C
1)
Draw a diameter AB of the circumscribed circle
2)
Construct an equilateral triangle with the base of the
diameter
3)
Divide the diameter into n=7 equal parts
4)
Project the second point of division from the vertex C
of the triangle onto the circle
5)
segment a is the approximate length of the inscribed
regular polygon (heptagon)
About the accuracy of the approximate construction, if the
radius of the circle is 10 cm,
A
1
a
2
3
4
5
6
7
B
an is the length of a side of the inscribed polygon, calculated in
analytical geometry,
t n is the length of a side calculated by trigonometry (the
“exact” value).
n
3
4
5
6
7
8
9
10
11
12
13
an 10 3 10 2 11.75 10. 00 8.69 7.68 6.89 6.23 5.70 5. 26 4. 87
t n 10 3 10 2 11.76 10. 00 8.67 7.65 6.84 6.18 5.64 5. 18 4.79
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Geometrical Constructions 1
7 Regular Polygons, Golden Ratio
Golden Ratio
Divide a segment in such a way that the ratio of the larger part to the smaller is equal to the
ratio of the whole to the larger part .
Golden triangle
Golden rectangle
1
τ
=
τ 1−τ
2
τ + τ − 1= 0
τ =
1-τ
5 −1
2
36°
Construction of τ
1
1
½
τ
τ
½
τ
36°
36°
τ
1
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τ
1-τ
72°
τ
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Geometrical Constructions 1
7 Regular Polygons, Golden Ratio
Golden Kite an Dart
“Sharp” triangle
36°
72°
“Flat” triangle
36°
36°
kite
dart
http://goldennumber.net/penrose.htm
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Geometrical Constructions 1
7 Regular Polygons, Golden Ratio
Golden Spiral
O: center of “dilative rotation” or
“spiral similarity”
o
Nautilus shell
The true spiral is closely
approximated by the artificial spiral
formed by circular quadrants
inscribed in the successive squares.
http://www.geocities.com/jyce3/spiral.htm
http://www.sciencenews.org/articles/20050402/mathtrek.asp
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Geometrical Constructions 1
7 Regular Polygons, Golden Ratio
Constructions on Regular Pentagon
Given: O, A
Given: O, a
Given: A,B
A
‚
ƒ
R/2
ƒ
…
O
R/2
„
O
• „
R
‚
r
r
„
ƒ •
•
r
‚
a
A
r
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M
AB/2 B
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