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Pentagram


Pentagram

The pentagram, also called the five-point star, pentacle, pentalpha, or pentangle, is the star polygon {5/2}.

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).

PentagramLengths

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 a, the edge lengths of the inner pentagon be b, the inradius of the inner pentagon be r, the circumradius of the inner pentagon be R, the circumradius of the pentagram be rho, and the additional horizontal and vertical distances be x and y as labeled. Then the indicated lengths are given by simultaneously solving the system of seven equations

2a+b=1
(1)
r^2+(1/2b)^2=R^2
(2)
r^2+(a+1/2b)^2=rho^2
(3)
(rho-r)^2+(1/2b)^2=a^2
(4)
x^2+(rho+R+y)^2=1
(5)
x^2+y^2=a^2
(6)
x^2+(y+R)^2=rho^2
(7)

to obtain

a=1/2(3-sqrt(5)) approx 0.381966
(8)
b=sqrt(5)-2 approx 0.236068
(9)
r=1/2sqrt(1/5(5-2sqrt(5))) approx 0.16246
(10)
R=sqrt(1/(10)(25-11sqrt(5))) approx 0.200811
(11)
rho=sqrt(1/(10)(5-sqrt(5))) approx 0.525731
(12)
x=1/4(sqrt(5)-1) approx 0.309017
(13)
y=1/2sqrt(1/2(25-11sqrt(5))) approx 0.224514,
(14)

This gives the ratio

 a/b=phi,
(15)

where phi 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

A_(filled)=1/4sqrt(650-290sqrt(5))
(16)
=5/(sqrt(130+58sqrt(5))),
(17)

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

 A_(unfilled)=5/4sqrt(85-38sqrt(5)).
(18)
GoldenPentagram

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

 r_n=phi^(-n)
(19)

and centers

x_n=-1/4(1-phi^(-n))sqrt(50+22sqrt(5))
(20)
y_n=1/2phi(1-phi^(-n))
(21)

modulo rotation by 2pik/5, where phi is the golden ratio.


See also

Dissection, Golden Triangle, Great Dodecahedron, Great Icosahedron, Great Stellated Dodecahedron, Hexagram, Hoehn's Theorem, Kepler-Poinsot Polyhedron, Miquel Five Circles Theorem, Miquel's Pentagram Theorem, Pentagon, Polygram, Small Stellated Dodecahedron, Star Figure, Star of Lakshmi

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References

Brown, D. The Da Vinci Code. New York: Doubleday, 2003.Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, pp. 43-44, 2002.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 122-125, 1990.Pappas, T. "The Pentagon, the Pentagram & the Golden Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 188-189, 1989.Schwartzman, S. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. Washington, DC: Math. Assoc. Amer., 1994.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 211, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 36, 1986.Williams, R. The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, 1979.

Cite this as:

Weisstein, Eric W. "Pentagram." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Pentagram.html

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