Pentagram

The pentagram, also called the five-point star, pentacle, pentalpha, or pentangle, is the star polygon .

It is a pagan religious symbol that is one of the oldest symbols on Earth and is known to have been used as early as 4000 years B.C. It represents the "sacred feminine" or "divine goddess" (Brown 2003, pp. 35-37). However, in modern American pop culture, it more commonly represents devil worship. In the novel The Da Vinci Code, dying Louvre museum curator Jacque Saunière draws a pentagram on his abdomen with his own blood as a clue to identify his murderer (Brown 2003, p. 35).

In the above figure, let the length from one tip to another connected tip be unity, the length from a tip to an adjacent dimple be , the edge lengths of the inner pentagon be , the inradius of the inner pentagon be , the circumradius of the inner pentagon be , the circumradius of the pentagram be , and the additional horizontal and vertical distances be and as labeled. Then the indicated lengths are given by simultaneously solving the system of seven equations

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

to obtain

			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)

This gives the ratio

(15)

where is the golden ratio (Wells 1986, p. 36; Brown 2003, p. 96). The tips of the pentagram are therefore golden triangles.

The area of the filled pentagram with unit tip-to-tip edge length (as above) is given by

			(16)
			(17)

while the area of the tips only (corresponding to the even-odd winding rule) is given by

(18)

A series of embedded pentagrams can be constructed to form a larger pentagram, as illustrated above (Williams 1979, p. 53). If the central pentagram has center (0, 0) and circumradius 1, then the subsequent pentagrams have radii

(19)

and centers

			(20)
			(21)

modulo rotation by , where is the golden ratio.