## Roots of Unity

Mathematics: learning it, doing it, celebrating it.

# The Most Mathematically Perfect Day of the Year

|

The Paley graph of order 9 is a perfect graph, making it an appropriate object of veneration and study on June 28, a perfect day. Image: David Eppstein, via Wikimedia commons.

Whether you write it 6/28 or 28/6, today is a perfect day. A perfect number is a number that is the sum of its factors besides itself, and 6 (1+2+3) and 28 (1+2+4+7+14) are the first two perfect numbers. Hence, June 28 is a perfect day. Perfect numbers are few and far between, so don't hold out for a celebration in which the year is perfect as well. The last time that happened was in 496, and the next time is over 6,000 years away, in 8128.

Mathematical date enthusiasts will note that today is also Tau Day, if you didn’t get enough circle constant celebration in March. Tau is the ratio of a circle’s circumference to its radius, and it is arguably a better constant than pi, the ratio of circumference to diameter. But if you choose to celebrate perfect numbers today instead of circle constants, you can transcend the transcendental number battle.

What is the appropriate way to observe this perfect day? First things first: the parfait is the appropriate food, its name coming from the French for perfect. Suggested activities include bowling a perfect game (or as close as you can get) and rooting for a perfect game at your favorite baseball stadium. You could also sing organum with friends to celebrate the musical intervals of perfect fifths and octaves. (You get bonus points if any of you have perfect pitch.) And a couple hours studying your favorite language's grammar may help you perfect the perfect tense. Physics is lousy with perfection: perfectly frictionless surfaces, perfectly rigid bodies, and perfectly elastic collisions abound. There are perfect fluids and perfect gases. Unfortunately, the labs are all out of them. You’ll have to stick with the exercises in your nearest physics textbook to experience any of those perfect objects.

If you’d prefer to keep the celebration mathematical, you could of course go looking for the next perfect number. (Maybe you could find an odd one this time?) But given the amount of computational power required to find Mersenne primes, you might be better off with some other branches of math. Graph theory has perfect graphs, algebra has perfect groups, and topology has perfect sets and perfect spaces (disappointingly not the subject of this Avett Brothers song). To make things more confusing interesting, it also has perfectly normal spaces, which are sometimes called perfect spaces, even though they're not the same as the other perfect spaces! There's certainly enough there to keep you occupied for a while on this perfect day. To wind down, you can try to draw perfect circles with Vi Hart.

No matter how you choose to celebrate, please avoid any involvement in perfect crimes or perfect storms.

The views expressed are those of the author and are not necessarily those of Scientific American.