## Why can’t Pi and Tau be friends?

π day used to be a straightforward holiday for people who love mathematics. Who could possibly dislike a holiday to celebrate a number that has been celebrated across cultures and throughout history? The Egyptians and Babylonians approximated the ratio of the circumference of a circle to its diameter. The same in Ancient India, China, and Greece. The symbol π itself is newer, but is still over 300 years old.

Only recently have Tauists come along and started their arguments that Pi is wrong. Since then, the Pi and Tau adherents seem to have turned our circles into boxing rings, with Pi on one side and Tau in the other.

The Tauists start it off with a series of quick jabs.  Tau is fundamentally simpler. Radius is more fundamental than diameter, the fundamental constant should be based on radius. Radians finally make sense, it goes against intuition to have 2 Pi radians in a circle. Pi is off by a factor of 2. It is unnatural and confusing. It is wrong!

The Pious punch back that Pi has been the constant for thousands of years, that it’s what everyone knows, that nobody naturally draws a line going halfway across a circle. Tauists demonstrate selective bias to prove their point. The area of 1/4 of a unit circle is Pi/4, or Tau/8. Tau is off by a factor of 2. It is unnatural and confusing. It is wrong!

But, in the midst of all this bickering, we lose the fundamental reason why Pi and Tau are so cool. When we get held up in the details, we forget how magical it is that the ratio of a diameter or radius of a circle to its circumference is somehow not something that can be expressed as a ratio at all. That somehow the observed perfection of a circle is due to something truly transcendental. That a circle is itself the perfect shape, minimizing surface area to volume, exhibiting the same curvature everywhere around itself until you get back to exactly where you started.

Tau and Pi aren’t that different. They’re both irrational, they’re both transcendental, they solve all of the same problems. They are, in fact, the same number give or take a factor of 2. Maybe we can just teach everyone both and let people choose for themselves whichever one is easiest for them?

If we celebrate both, that’s twice as many days to get people excited about mathematics. It’s twice as many days to eat pie and appreciate how round it is.

I mean, Tau is just 2*Pi. Er, Pi is just Tau/2. Fine. Tau and Pi differ by just a factor of 2. Sorry. Pi and Tau differ by just a factor of 2. Really, why are we arguing?

So this π day, lets agree to shake hands, stop fighting, and eat some Tau-co Pi. It’s delicious and the fundamental constant that we know and love makes just as beautiful circles whether it’s 3.14… or 6.28…

Thanks guys. Happy Pi day!

## Non-Euclidean Chess: Part 1

Most traditional grid-based board games are played in Euclidean space. This is straightforward and easy to understand but topologically boring.

Why not spruce up the classic grid-based game of chess by modifying it to be played on a non-Euclidean board?

The simplest way to transform a standard board into a non-Euclidean board is to connect the sides of the board together. Connecting the squares on adjacent sides to each other creates a board with an inherently spherical topology. In contrast, connecting the squares on opposite sides can lead to a board with a toroidal structure. A twist in the way the squares are connected can lead to a board with the topology of a klein bottle.

Connecting opposite sides of a chess board (in the diagram above, red sides connect to blue sides) leads to a board with a toroidal structure.

While these macro structures are interesting, they are not clearly well suited for chess. In particular, none of these boards have edges. Thus, it is impossible for a Pawn to reach the edge of the board and become a Queen. Actually, for all we know chess has always been played on boards with these macrostructures, and we just “walled off” a section that was locally equivalent to a subsection of a Euclidean space.

How can we create a non-Euclidean board that does not lose the edge constraints of the original Chess board? One way is to connect random squares to each other by the edges. We will refer to these kinds of connections as “chutes” in homage to a classic board game with these kinds of connections – Chutes and Ladders. Chutes will be shown as blue lines in the diagrams below.

The side of the tile going into the “top” of a chute connects to the side of the tile coming out of the “bottom” of that chute.

For example, the green “pawn”, above, moves forward one square to the red position. Notice that the pawn is now moving sideways, and can be Queen-ed on its next move. This variant of non-Euclidean chess is best played with pieces with some obvious indication of directionality on them, so that you don’t lose track of their traveling directions.

In order for the board to be connected, we map the tiles that come out of the tops and bottoms of each chute and ladder to each other.

The green piece above ends up where the red piece is when it moves forward one space.

At this point, it is pretty clear how to travel across the sides of the tiles, but what it means to travel diagonally, like a bishop, is less well defined. We will use a “slide-left” metric* to define diagonal moves.

Consider any diagonal move. It can be represented as moving forward one space in one of the four cardinal directions and then sliding to the left one space repeatedly until the destination is reached.

By describing each diagonal move as a series of horizontal and vertical moves, the diagonal movement of pieces through chutes becomes clear.

In the figure above, the green piece begins a diagonal move by moving 1 forward through a chute, then sliding left. The diagonal move can continue until it hits the edge of the board and can no longer complete a “forward one, left one” series.

Note that knights still get to choose whether they move over two and up one or vice versa, and so may easily have more valid destinations than normal.

It is notable that this sort of non-Euclidean chess board is rather unlikely to be fair if you use randomly chosen chutes (perhaps an advantage if you want to handicap players of uneven ability). Can you design a chess board with a single chute such that White as a mate in one?

There are many other fascinating games that you can try playing along similar lines. In one variant of this game, players take turns moving, then adding a new chute to the board. The second post in this series will explore playing chess on a hyperbolic checkerboard (an idea first mentioned to me by Roice Nelson). Are there any other interesting boards that you can imagine playing chess on?

* As David Dalrymple and other people with a better ability to not confuse terminology than me might note: “slide-left” is not a “metric”. It is a homomorphism from the free group on {(1,0),(1,1),(0,1),(-1,1)} to the free group on {(1,0),(0,1)} which preserves actions on the square lattice.

## Ancient Greek Geometry

is a remarkably entertaining game of making compass and straight-edge constructions made by Nico Disseldorp. The first few challenges are easy to construct, but later… pentagons!

http://sciencevsmagic.net/geo/

The author also has some cute mathematical non-game web applications on his main page.

## The Bridges Math Art 2013

is the premier conference on the intersection between Math and Art. This year, it was held in Enschede, The Netherlands. I presented a fun workshop paper on Flipbook Polyhedra and exhibited Seeing Stars in the art exhibition.

George Hart just released this fantastic video of the Bridges Art Exhibition. You can see my pieces briefly at the very beginning and spinning at 1:17. You can also see lots of other awesome artwork from the Bridges Art Exhibition, including spotlights on the four prize-winning pieces.