Math Chat: Symmetries and The Millennium

Old Symmetries Challenge

What are the symmetries of a cube in space? (What about a hypercube in 4-dimensional space?)

Jan Smit counted up all 24 symmetries. You can rotate the cube 90 degrees (a quarter turn), 180 degrees, and 270 degrees about a vertical axis, and similarly about two horizontal axes, for a total of 9 symmetries. You can rotate the cube 180 degrees about a horizontal diagonal axis and five other similar axes for a total of 6 more. You can rotate the cube 120 degrees and 240 degrees about any of the four long diagonals from one corner (where three faces meet) to the opposite corner for a total of 8 more. Finally you can leave the cube alone, for a grand total of 9 + 6 + 8 + 1 = 24. Unfortunately, this method does not work in higher dimensions, where a symmetry is not simply a rotation about an axis.

Another approach focuses on where the top ends up, as one of the six faces of the cube. If the top stays on top, there are four possibilities: rotation through 0 degrees (leaving it alone), 90 degrees, 180 degrees, and 270 degrees. Similarly if the top goes to any other face, there are four possibilities. The grand total is 6 x 4 = 24 symmetries. Mathematicians call these six collections of four symmetries "cosets."

Just as the square has four sides and the cube has six square faces, the hypercube in 4D has 8 cubical faces. If you leave the top one on top, there are the same 24 possibilities as above. Similarly if the top goes to any other face, there are 24 possibilities for a grand total of 8 x 24 = 192 symmetries.

In summary, in 3D there are 6 x 4 = 22 x (3x2) = 22 x 3! symmetries, and in 4D there are 8 x 6 x 4 = 23 x (4x3x2) = 23 x 4! symmetries. In general, in n dimensions there are 2n-1 x n! symmetries.

Millennium

D.D. Trent asks for the authoritative Math Chat word on when the new millennium begins.

Our answer is January 1, 2001. The calendar was established to make the year AD 1 the first year after Jesus' birth, with the previous year 1 BC and no year zero. Hence 2000 is the 2000th year, the last year of the second millennium, and 2001 begins the third millennium.

The calendar actually had Jesus's birth too late by about 6 years; the exact date is still unknown. At his Web site, Michael Donner argues that the one certain date is the lunar eclipse of March 12 and 13, 4 BC, tied by historian Flavius Josephus to Herod's death, which certainly followed Jesus's birth. Therefore, as of March 13, 1997, we can be sure that it has been 2000 years since Jesus' birth and celebrate the arrival of the third millennium.

New Challenge

Assuming the third millennium arrives on January 1, 2001, where on Earth should the celebration begin?

* Send answers and new questions to:

Math Chat, BSC, Williams College

Williamstown, MA 01267

or by e-mail to:

Frank.Morgan@williams.edu

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