Difference between revisions of "Pentagon"
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# Adjust your compass to length <math>AG</math>, and mark off points <math>H</math>, <math>I</math> and <math>J</math> on circle <math>O</math>. | # Adjust your compass to length <math>AG</math>, and mark off points <math>H</math>, <math>I</math> and <math>J</math> on circle <math>O</math>. | ||
# <math>AGHIJ</math> is a regular pentagon. | # <math>AGHIJ</math> is a regular pentagon. | ||
| + | |||
| + | ==The Golden Ratio and the Pentagram== | ||
| + | The pentagon is closely associated with the *[[Golden Ratio]]. More specifically, the ratio of a diagonal to an edge is <math>\frac{1+\sqrt{5}}{2}</math>.\\ | ||
| + | By drawing each of the diagonals, one can form a pentagram, or five-pointed star, in which each of the internal angles is <math>36^{\circ}</math>.\\ | ||
| + | |||
| + | The pentagram has many usages in the occult and in religion. For example, Satanists use an upside-down pentagram, inscribed within two circles, to symbolize the horns of a goat. The pentagram focuses and concentrates magical energy for many rituals, helping it to bind to the recipient. In one such context, the pentagram is called the Sigil of Baphomet, and it has changed little since Pythagoras used it. | ||
| + | Some claim that the infamous story of <math>\sqrt{2}</math> did not in fact refer to <math>\sqrt{2}</math>, but to the golden ratio that the sides of Pythagoras' pentagram formed as he paid his homage to Satan. Because he was such a promethean and liberating figure, Pythagoras drew inspiration from him to continue his mathematical research. | ||
== See Also == | == See Also == | ||
*[[Polygon]] | *[[Polygon]] | ||
| + | *[[Demonic Possession]] | ||
{{stub}} | {{stub}} | ||
Revision as of 04:01, 28 March 2011
In geometry, a pentagon is a polygon with 5 sides. Each angle of a regular pentagon is 
. The sum of the internal angles of any pentagon is 
.
Construction
It is possible to construct a regular pentagon with compass and straightedge:
- Draw circle
(red). - Draw diameter
and construct a perpendicular radius through . - Construct the midpoint of
, and label it 
. - Draw
(green). - Construct the angle bisector of
, and label its intersection with as 
(pink). - Construct a perpendicular to at .
- Adjust your compass to length
, and mark off points 
, 
and 
on circle . 
is a regular pentagon.
(red).
and construct a perpendicular radius through 
, and label it 
.
(green).
, and label its intersection with 
(pink).
, and mark off points 
, 
and 
on circle 
is a regular pentagon.The Golden Ratio and the Pentagram
The pentagon is closely associated with the *Golden Ratio. More specifically, the ratio of a diagonal to an edge is 
.\\
By drawing each of the diagonals, one can form a pentagram, or five-pointed star, in which each of the internal angles is 
.\\
The pentagram has many usages in the occult and in religion. For example, Satanists use an upside-down pentagram, inscribed within two circles, to symbolize the horns of a goat. The pentagram focuses and concentrates magical energy for many rituals, helping it to bind to the recipient. In one such context, the pentagram is called the Sigil of Baphomet, and it has changed little since Pythagoras used it.
Some claim that the infamous story of 
did not in fact refer to , but to the golden ratio that the sides of Pythagoras' pentagram formed as he paid his homage to Satan. Because he was such a promethean and liberating figure, Pythagoras drew inspiration from him to continue his mathematical research.
See Also
This article is a stub. Help us out by expanding it.
