Today it all comes to an end. No, not the world, but a 12-year streak in which each year has offered a magical day like today, when the day, month, and year all coincide. Harmonic convergences such as 12/12/12 aren’t to be sniffed at, as most of us won’t see another one, Mayan calendar or no Mayan calendar. But part of what makes this particular day special is the number 12 itself. How special is it? Here’s a guided tour:
The number 12 has religious significance far beyond the 12 days of Christmas. Its role is reinforced by Twelfth night, the 12 Apostles, the 12 feasts of Eastern Orthodoxy and the 12 Tribes of Israel. These 12 tribes are associated with the 12 sons of Jacob, so it bears mention that the Norse god Odin also had 12 sons.
The concept of a dozen was also alive and well in Greek mythology, as evidenced by the 12 principal gods of the Greek pantheon atop Mount Olympus. These gods won’t be listed here because their total number actually exceeded 12, but 12 was apparently the limit at any given time. However, we can list the 12 labors forced upon Hercules by King Eurystheus of Tiryns. Most involved killing some ghastly creature or another. One notable exception was labor number three, as the Cerynian hind was actually a delicate deer, loved by Athena, that Hercules had to stalk for a year before gently carting away.
THE TWELVE LABORS OF HERCULES
One: Kill the Nemean Lion
Two: Kill the Lernean Hydra
Three: Capture the Cerynian Hind
Four: Capture the Erymanthian Boar
Five: Clean the Augean Stables
Six: Kill the Stymphalian Birds
Seven: Capture the Cretan Bull
Eight: Capture the Horses of Diomedes
Nine: Take the Girdle of the Amazon Queen Hippolyte
Ten: Capture the Cattle of Geryon
Eleven: Take the Golden Apples of the Hesperides
Twelve: Capture Cerberus
On the other hand, whereas today it is standard for a jury to consist of 12 people, Ancient Greece had no such limit. The trial of Socrates involved 501 jurors.
One reason for the prominence of the number 12 is that it is evenly divisible by 2, 3, 4, and 6, and is therefore convenient for all sorts of applications, from eggs to donuts to numbers on a clock to months in a year to signs of the zodiac.
And let’s not forget the arts. An alexandrine, found in French literature from Racine to Baudelaire, is a line of 12 syllables. Alexander Pope, alas, was not taken with this construction despite its virtually eponymous form, as in his second line below:
A needless alexandrine ends the song
that, like a wounded snake, drags its slow length along.
Moving to mathematics, there are 12 edges on a cube. One of the most famous two-dimensional depictions of a cube is the Necker cube, a 12-lined optical illusion that first appeared in 1832. Because the cube has no dotted lines, the figure produces (at least) two different interpretations.
There are 12 different shapes that can be made out of five squares joined at the edges. Called pentominoes, these shapes are frequently labeled by the letters they most closely represent. Altogether, the 12 pentominoes account for 5×12 = 60 square units, and in fact it is possible to arrange the 12 pieces so as to make rectangles measuring 6×10, 5×12, 4×15 and 3×20.
There is even a board-game variation in which players alternate placing pentominoes onto an 8×8 grid until someone (i.e. the loser of the game) cannot place a remaining pentomino without overlapping one that has already been played.
Author and futurist Arthur C. Clarke was a big fan of pentominoes, and he filmed a scene for 2001: A Space Odyssey that showed HAL the computer playing the 8×8 pentomino game. Unfortunately for fans of the game, the scene was cut in favor of a different 8×8 game: chess.
Twelve is also the “kissing number” in three dimensions. To see what this is all about, it might help to visit the two-dimensional case, in which the kissing number is defined as the number of circles of radius 1 that can simultaneously touch a given circle. That number is clearly six.
The situation in three dimensions is necessarily more complex. When you put 12 spheres around a central sphere, there is substantial free space, and it is tempting to believe that there might be room for a 13th sphere.
Apparently the possibility of a 13th sphere arose in a conversation between Isaac Newton and Scottish astronomer David Gregory back in 1694. Although the precise nature of their disagreement has been lost to history (most accounts suggest that Newton was on the 12 side and Gregory on 13), there is no record of so much as a gentleman’s wager on the subject and certainly no possibility of anyone collecting, as the question wasn’t fully resolved until 1953.
Speaking of spheres, consider the traditional black pentagon-white hexagon “Telstar” soccer ball design that was the official ball for the 1970 Mexico City World Cup and for several years thereafter. (1970 was the first year in which the World Cup was televised, and the new ball was especially easy to follow on television.)
There are 12 pentagons and 20 hexagons on this ball, but only the 12 is sacred. The Telstar ball is actually a special case of a more general concept called a “Buckyball,” named for Buckminster Fuller of geodesic dome fame.
It turns out that many different sphere-like structures can be made from a combination of pentagons and hexagons, but whereas there is no limit on the number of hexagons, the number of pentagons must always be 12.
In particular, it is possible for the number of hexagons to equal zero, in which case you’re left with the standard 12-faced “Platonic solid,” the dodecahedron. There is also a dodecahedron whose faces are all rhombi. Rhombic dodecahedra, as they are called, can, like cubes, be fitted together to fill three-dimensional space, in the same sense that rhombi — or squares, but not pentagons — can tile the plane. But a desk calendar can be made out of either form of dodecahedron, with each face getting a different month.
Having returned to the calendar, our tour is now complete.
Derrick Niederman is a math professor at the College of Charleston.