Here at Uncommon Descent, we never really celebrated Pi Day (March 14) this year because other stuff intervened. But pi is a really important irrational number:

Pi has been calculated to over one trillion digits beyond its decimal point. As an irrational and transcendental number, it will continue infinitely without repetition or pattern. While only a handful of digits are needed for typical calculations, Pi’s infinite nature makes it a fun challenge to memorize, and to computationally calculate more and more digits. More.

Indeed,

In Carl Sagan’s novel Contact, the main character (Ellie Arroway) is told by an alien that certain megastructures in the universe were created by an unknown advanced intelligence that left messages embedded inside transcendental numbers. To check this, Arroway writes a program that computes the digits of π in several bases, and eventually finds that the base 11 representation of π contains a sequence of ones and zeros that, when properly aligned on a page, produce a circular pattern. More.

But some now say that’s wrong. At Science News, Emily Conover argues,

A longtime fixture of high school math classes, pi has inspired books, art (SN Online: 5/4/06) and enthusiasts who memorize it to tens of thousands of decimal places (SN: 4/7/12, p. 12). But some contend that replacing pi with a different mathematical constant could make trigonometry and other math subjects easier to learn. These critics — including myself — advocate for an arguably more elegant number equal to 2π: 6.28318…. Sometimes known as tau, or the symbol τ, the quantity is equal to a circle’s circumference divided by its radius, not its diameter.

This idea is not new. In 2001, mathematician Bob Palais of the University of Utah in Salt Lake City published an article in the Mathematical Intelligencer titled “ π is wrong!” The topic gained more attention in 2010 with The Tau Manifesto, posted online by author and educator Michael Hartl. But the debate tends to reignite every year on March 14, which is celebrated as Pi Day for its digits: 3/14.

The simplest way to see the failure of pi is to consider angles, which in mathematics are typically measured in radians. Pi is the number of radians in half a circle, not a whole circle. That makes things confusing: For example, the angle at the tip of a slice of pizza — an eighth of a pie — isn’t π/8, but π/4. In contrast, using tau, the pizza-slice angle is simply τ/8. Put another way, tau is the number of radians in a full circle.More.

We recommend digging the big coffee urn out of the back of the closet for this one. Also, cookies and crackers.

*See also:* Is celeb number pi normal?

Pi: How did mathematics come to be woven into the fabric of reality?

At PBS: Puzzle of mathematics is more complex than we sometimes think

and

Eugene Wigner: Nobel Prize Winner Promotes ID, Ccirca 1960

The dream lives:

Hartl has a solid point. It is also interesting to note that even though pi has a certain beauty to it when put to music,,,

,,, Tau (2pi) has a deeper beauty to it when put to music:

Pi is really convenient for finding arc angles by radii, which, for me at least, are far more convenient to work with than diameters.

Six of one, half dozen of the other. If it works better for students, do it.

Over here we celebrate “better pi approximation day” on the 22nd of July. If we were to change to tau we would have to celebrate “worse tau approximation day” on the 25th of April.

I’m sticking with pi. 🙂

I agree with local minimum, with this caveat: since 2pi = tau, pi = tau/2. Now we write 2pi everyplace we would write tau, but if tau was the standard, we would have to write tau/2 everywhere we write pi, such as in the formula for the area of the circle. Multiplying by 2 is computationally and algebraically cleaner than dividing by 2, so I vote for sticking with pi, and understanding the important role 2pi plays also, without worrying about 2pi having another name.

When teaching radians, I haven’t found it very difficult to teach students that pi = 180°, and thus fractions of pi are fractions of a semicircle.

And if we changed to tau, we would have no reason to have pie contests on Pi day.

My engineering sensibility leads me to prefer pi/4, which is the ratio of area within a circle, divided by the area of a square circumscribing the circle. It’s also the ratio of the circumference of a circle divided by the perimeter of that same square. This is one of the few examples where distance and area measurements coincide.

Why does this please my engineering sense? It’s a measure of efficiency, a cost function of sorts, of using polar coordinates in a world that lives in a Cartesian grid. It’s the efficiency of putting a round peg in a square hole: approximately 0.7854, a solid C+.

The next time you are on an airplane and see croplands of former of circles set within squares, in order to enable a rotating irrigation pipe fastened to the water source at the center, think pi/4 and 78.54% land use and 78.54% efficiency in fencing.

If we didn’t value private property so much that we chop it into easily trackable squares and rectangles, we might arrange these crop circles into hexagons. Then irrigated crop circles would have an efficiency of 82.70% by area and 95.51% in perimeter. Bees have figured this out in their honeycomb designs, where their larvae can’t tell the difference between circles and hexagon shapes of their nursery tubes, and they enjoy A+ level efficiency in wax material use compared to cylinders with a circular cross section. Since there wouldn’t be mechanized farming by humans in the world without private property, the reduction in efficiency from 82.70% to 78.54% in area, and 95.51% to 78.54%, is the cost of the Fall.

By the way, the efficiency of putting a square peg in a round hole is 63.67%, both in area and perimeter. Maintaining a circular lifestyle in a square world is more efficient than the other way round.

The lesson is that in a fallen world, private property trumps utopia every time, so much so that it even makes makes small-farm utopians more successful.

Yes, the formula A = 0.785 • r^2 is useful. And yes, nesting circles in a hexagonal grid would be more efficient, although there would be a compensating loss of other features, such as the ability to locate points on a rectangular grid.

jdk @7,

True, an orthogonal grid would be inconvenient, which is why we’ve not done it, I suppose. But a non-orthogonal grid with two axes parallel to 4 of the 6 sides of the hexagon would work for many practical uses. Not all cities have orthogonal streets and avenues and we manage.

For tiling applications, the hexagon is the optimum point for maximizing area while minimizing perimeter.

Together, hexagons and pentagons can also be combined into 3-D geodesic domes.

Yes, hexagons are a good shape! 🙂

I once made a hexagonal chess game, with three colors and thus three bishops, and the queen could travel along three axes at 120° to each other. Also, lots of board games, especially those war board games, are on a hexagonal grid.

jdk @ 9,

Wow. You sound like a very interesting person!

Dean

Thanks, Dean.

I’ve loved math since I was a kid, taught high school math for 35 years, and have read a lot about the history and philosophy of math.

Here’s a fact about hexagons that I think is great: If you plot the six sixth roots of 1 in the complex plane, and connect them sequentially, you get a regular hexagon. This statement generalizes to all regular polygons.