Math Mutation 102:  A Brush With Evil

    Let me start by apologizing for the lag since my last podcast.  The
school board campaign has kept me pretty busy, but it's almost over.  If you
are one of my listeners in the Hillsboro school district, don't forget to
vote for me!
    Today I thought it would be fun to discuss pentagrams.  You know the
figure I'm talking about:  basically you draw a pentagon, then draw all the
diagonals connecting the vertices, and you have a pretty star shape.  The
pentagram seems to have had deep symbolic meaning in many cultures since
ancient days, and most popularly brings demonic connations to mind in our
modern culture.  Why do we find this shape so fascinating?
    To start with, I think it's the easiest "interesting" picture to draw.
Think about it:  one lazy way to doodle, especially for the
non-artistically-inclined, is to draw a bunch of dots and connect them.  If
you draw 2 or 3 dots, you get a simple line or triangle.  If you draw 4 dots
and connect them all, you get a quadrilateral with an X in it, which is not
very exciting.  But draw 5 dots with even spacing, connect them-- and you
have a pentagram!  If the spacing is not even, you get some kind of
distorted pentagram, except in the degenerate cases where several of your
dots coincide or fall on a line.  If you're aiming for a star shape,
the pentagram star is also an easy figure to draw, in that you can
draw the complete star in five lines without lifting your pencil, as
I'm sure you learned in grade school.   
    Each intersection between diagonals of a pentagram divides the
line into segments measurable by the "golden ratio", the famed
mathematical constant phi equal to the ratio a/b such that a/b =
(a+b)/a, which is (1+ root 5)/2. You can also see that the center
lines of the pentagram together form a smaller pentagon, in which you
can draw another pentagram.  In fact, if you look carefully, you'll
see that in the 'arms' of the star, you can also inscribe a
diminishing infinite series of smaller pentagons and pentagrams. If
the starting pentagram was inscribed in a circle of radius 1, or had a
circumradius of 1, then the nth iteration of smaller pentagrams has a
circumradius of phi to the -n.  The show notes contain a link to a
nice illustration of this at Wolfram's Math World.   
    Aside from these properties, the close association of the pentagram with
the number 5 is probably another reason why human cultures have continuously
found it fascinating.  As early as 4000 BC it is said to have represented
the "sacred feminine".  To the ancient Hebrews, it represented the
five books of the Torah.  Perhaps because the 5 points can be put in
correspondence to 2 arms, 2 legs, and a head, the top point of an
upward-pointing pentagram can be said to represent the Spirit.  According to
the Skepic's Disctionary, an inverted pentagram may be considered especially
evil because it relegates the spirit to the bottom of the metaphysical heap.  
In ancient Greece, the mathematical cult of Pythagoras used pentagrams as a
symbol of health, and wore them so followers could recognize each other.
The pentagram was a Christian symbol to the Roman emperor Constantine
I, representing the Five Wounds of Christ.  The knight Gawain in medieval
literature inherited this use.  Due to the ability to draw the full
figure without lifting your pencil, the pentagram also became known as
the Endless Knot, a symbol of Truth and protection against demons.  A
pentagram was used to contain the demon Mephistopholes in Goethe's Faust,
which probably helped lead to its adoption by Satanists in more recent
times.  As a result, many Christians consider the pentagram in general
to be demonic, regardless of which way it's pointing.  To Wiccans and
Japanese,  however, the pentagram has no evil connotations, and merely
symbolizes the five mystical elements.   
    Perhaps the most amusing comment I found in articles online is the most
obvious.  As you've probably seen in thousands of cartoons and children's
drawings, the five-pointed pentagram star seems to be a near-universal
symbol for the stars in the sky.  But as the skeptic's dictionary points
out, no star ever looks anything like that, whether to the naked eye
or through a telescope.  Strange, isn't it.  If you have a good
explanation, be sure to send me an email! 
    And this has been your math mutation for today.




References:
<li><a href="http://en.wikipedia.org/wiki/Pentagram">Pentagram at Wikipedia</li>
<li><a href="http://mathworld.wolfram.com/Pentagram.html">Pentagram at
Wolfram's Math World</li>
<li><a href="http://www.skepdic.com/pentagram.html">Pentagram at the
Skeptic's Dictionary</li>
<li><a href="http://www.apologeticsindex.org/p00.html">Pentagram at
Apologetic's Index</li>
<li><a href="http://www.thornr.demon.co.uk/kchrist/pent.html">Another
Pentagram article</li>