(This is part of a series I'm writing for another site. I'm hoping posting it here will inspire me to finish it soon. I'm having trouble fitting some of the modern uses into this and will hopefully post them when they're done.)
Most Pagans are familiar with the pentagram. An outgrowth of this symbol is the septagram, a star with seven points. It is also known as the Faery Star, Elven Star, Elven Star of the Seven Sisters and acute heptagram. There are representations of it found in mathematics, paganism and even the subculture called Otherkin.
A Smattering of Math
Mathematically, the septagram is called the acute heptagram ("hepta-" being Greek for seven.) It belongs in a class of shapes called star polygons. They are usually represented by a fraction composed of prime numbers. In this case, the heptagram is known as a 7/3 star polygon. This simply means seven points are spaced evenly around the circumference of a circle and when drawn, the straight lines are connected at every third point. However, it is theoretically impossible to accurately draw a septagram with only a pencil and a straight edge, unlike the pentagram. Where the interior angles of a pentagram come to 36 degrees (180 degrees divided by 5), the interior angles of a septagram come to about 25.714 (180 divided by 7.)
There are as many reasons the number seven is sacred as there are people. One theory is it may have come about from the number of planets that could be observed in the ancient sky. It could also come from the seven observable stars of the pleiades, also called the Seven Sisters, or those seven stars that every child learns in the night sky called the Big Dipper. Another theory says the moon is responsible. It changes phase every seven days - from new to crescent, half, full and back to new again.
A Smattering of History
One of the oldest representations of the septagram is linked to the seven planets. It comes from the Hellenistic (circa 323 BC to about 146 BC) world This was a time of expansion for the Greeks, whose influence was felt from Italy to Egypt, the Middle East and Asia. During this time, the seven day week was widely adopted and each day was associated with a planet. The planets were placed in a counterclockwise circle in the order of the apparent speed of their travels across the sky, from Moon to Saturn. If a line is drawn from the point representing the Sun, not only does it form a septagram but also shows the days of the week in correct order.
The septagram makes another appearance during the Renaissance as an alchemical symbol. Usually seen reversed, or upside down, it was used to represent the seven recognized metals: lead, tin, iron, gold, copper, mercury and silver. Again, each metal was associated with one of known planets. It was also a depiction of the seven steps of the alchemical process: calcination, sublimation, solution, putrefaction, distillation, coagulation and tincture. Black (or "nigredo") was considered to be the starting point. As Saturn was the slowest moving planet in the sky, lead was considered by many alchemists to be the prima materia, or base material. With each step, impurities were supposed to be removed from the lead until all that was left was gold.
Upcoming posts: Modern uses of the Septagram and explorations into the meanings of the seven points - sun, land, water, magic, wind, moon and connection.
Sources (includes sources for info on some modern use as well):
Seven point star
Weisstein, Eric W. "Star Polygon." From MathWorld--A Wolfram Web Resource.
The Metal - Planet Affinities - The Sevenfold Pattern
The Heptad, 7 from Numbers, Their Occult Power and Mystic Virtues, by W. Wynn Westcott, 
Sublime Lead: The Biography of a 5000 Year Toxic Love Affair, Chap. 6 -
The Honeymoon: Lead and Alchemy: Decoding Chemistry Within the Imagery (pdf file)
an essay on the seven pointed star, by magpie
Blue Star Beliefs
What Does the Star Mean?