Quick summary (so you know where this is going)
Try to form triangles which enclose more of the goals than your opponent. The triangles must be formed using the well defined lines.
If you add up the angles in a triangle, you get Pi radians, hence the name.
What you need.
You'll need a hexagonal grid. Pi paper is even better. A good size for the playing field is hexagon shaped, with six hexagons to a side.
For that sized board, you will need two sets of 15 or so playing pieces that can be easily distinguished from each other. You will also need 5 other pieces, which are best if they can be tipped over to point at either player. An Icehouse set works well for this, so I'll be calling the players red and blue. I used the small green pyramids for the goals.
The playing field.
The game is based on particular lines between the centers of the hexagons. You can think of these as either six lines going through each one, or twelve radiating from each center. They are dived into two groups.
The well defined Lines
One group follows the 'normal' hex-to-hex moves. Going from the center of each hex, draw a line to the center of each of the six adjacent hexes. This should give you three bisecting lines for each, or six radiating lines. To follow these lines, you can just move from hex to hex, as long as you don't turn.
The other group follows the 'hex diagonals.' From the center of each hex, go out and bisect each angle, following the line between two hexes, and coming out on the other side to hit the center of another hex. Of course it continues going edge-hex-edge-hex until you hit the boundary of the playing field. This should once again give you three bisecting lines or six radiating lines, depending on persective.
These six (or twelve) lines are the ones that are used in the game.
Game concepts.
Triangles
A triangle is defined by three points connected by three lines. The points are the hexes, specified by the playing pieces. The lines are the well defined lines discussed in the playing field section above. In order for three pieces, A, B, and C, to form a legal triangle for game play, A and B must be on one of those well defined lines, B and C on a different line, and C and A on yet a different line. While any three points make a 'triangle,' only certain sets of three points make a triangle in the game of Pi.
Some examples of triangles,
and one of size precedence.
Triangles have size. The size of a legal triangle is defined as the number of hexes it encloses. In particular these are: The three hexes under it's pieces, (so the smallest triangle has a size of three, although it isn't very useful) hexes whose centers are passed through by the triangle's sides, and any hexes fully enclosed by those sides.
If all three of the pieces forming a legal triangle belong to the same player, the triangle is said to belong to that player, and the area the triangle encloses is enclosed by that player.
Setting up the board:
Place three red pieces so that they surround the center hex, forming an equilateral triangle of size 4.
One player is chosen as the placer. He places the goal pieces on the board anywhere he chooses, except for the corners of the hex board - it would be impossible to enclose those spaces.
The other player is the chooser. After the placer sets out the goals, the chooser decides whether he wants to play red or blue. Blue plays first.
Playing
Players take turns putting new pieces of their own color onto the board. Whenever a player places a new piece on the board, it must form one or more new triangles with existing player pieces (not goals) already on the board. It's okay to divide exiting triangles into smaller ones: the new piece has to be part of a triangle, but it doesn't have to increase the amount of area that is enclosed.
Special rule for the red player
You may only play pieces that form triangles with your own, red, pieces.
Enclosing goals
The object of the game is enclose more goals (using your triangles) than your opponent. You control a goal when it rests on hex that one of your triangles counts for it's size.
Big triangles can enclose multiple goals. But if a goal is enclosed by multiple triangles (and most will be) the smaller triangle wins. If two triangles of the same size enclose a goal, the player who got it first keeps it. Once a size-4 triangle encloses a goal, there is no way to steal it.
The 'first player' rule is one reason why using goal pieces that can be tipped over is handy.
Ending the game
The game is over when all of the goals are enclosed by any player. The player with the majority of enclosed goals wins.
Should a situation arise where both players agree that it will be impossible for either of them to enclose a goal, that goal is tied, it counts as enclosed for ending the game, but belongs to neither player. If both players have an equal number of goals at the end of the game, the game is a draw.
Tip
Play a few games. The outcome of the game is heavily dependant on the starting arrangement. Too many goals near the edge seems to favor blue, while having them all near the middle can lead to a swift victory for red.
Example game
I've drawn in the controlling triangles. I've also done my best to color in the rest of the controlled territory, but it isn't important to the game, and I may have missed something.
Red wins, chooser took blue. Blue was actually one move from A and D for most of the game, and two moves from C. E was pretty much assumed it would go to blue, but red used it as a distraction tactic to build up his options for playing on D. D was pretty close to being tied, and likely to go to blue, but red was actually playing to loose D near the end (though it took realistic threats to maintain blue's attention) so he could take A with the last move(28), ending the game before blue could complete his otherwise guaranteed size-4 triangle.
Pi Paper
This is something I made to make following the lines a little easier. The hexes work fine, but you might want to try this instead.
If you draw all the legal lines out, the playing field will look like a tiling of 30-60-90 triangles. I find it quite helpful to mark the centers of the hexes. I also find it helpful to draw the lines as three orthogonal grids, each rotated 60 degrees from each other. I use red, green and blue. This is the what I call Pi paper. Actually, the board may look better in red, blue and yellow on black, but I'd rather spare my printer the solid black.
The Pi board.
Print twice, rotate one by Pi radians, and attach.
Full Pi paper.
In case you need it.