Poincare’s Square and Unbounded Gomoku

Poincare’s hyperbolic disk model.

Henri Poincar’e was masterful in presenting scientific concepts and ideas in an accessible way. To explain that the Universe might be bounded and yet infinite, he imagined that the temperature of space decreased from the centre to the periphery in such a way that everything contracted with the distance from the centre. As travellers moved outward from the centre, everything got smaller in such a way that it would take an infinite time to reach the boundary.

Poincar’e described a beautiful geometric model with some fascinating properties. He envisioned a circular disk in the Euclidean plane, where distances were distorted to give it geometric properties quite different from those of Euclid’s Elements. For the Poincar’e disk, the metric is

$\displaystyle \mathrm{d}s^2 = \frac{4(\mathrm{d}x^2 + \mathrm{d}y^2)}{(1-(x^2+y^2))^2} = \frac{4(\mathrm{d}r^2 + r^2\mathrm{d}\theta^2)}{(1-r^2)^2}$

Basically, distances shrink with distance ${r=\sqrt{x^2+y^2}}$ from the origin. A diagram of Poincare’s disk is shown in the figure above, showing a selection of geodesics. Each of the (distorted) triangles in the figure has the same area in this metric.

From Circle to Square

We will consider a metric similar in some ways to that of the disk, but with points confined within a square centered at the origin. The metric of the Poincare Square is

$\displaystyle \mathrm{d}s^2 = \frac{\mathrm{d}x^2}{(1-x^2)^2} + \frac{\mathrm{d}y^2}{(1-y^2)^2}$

Let us consider the distance from the origin ${O}$ to the point ${P}$ at ${(x,0)}$ :

$\displaystyle s(O,P) = \int \mathrm{d}s = \int_0^x \frac{\mathrm{d}x}{(1-x^2)} = \log\sqrt{\frac{1+x}{1-x}} \,.$

Clearly, the distance from the origin to the boundary at ${(1,0)}$ is infinite. We can invert this expression to obtain ${x}$ as a function of distance ${s}$ :

$\displaystyle x = \tanh s \,.$

Geodesics

The general equations for geodesics may be written

$\displaystyle \frac{\mathrm{d}^2 x^\lambda}{\mathrm{d}\tau^2} + {\Gamma^\lambda}_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}\tau}\frac{\mathrm{d}x^\nu}{\mathrm{d}\tau} = 0$

where ${{\Gamma^\lambda}_{\mu\nu}}$ are the Christoffel symbols, involving derivatives of the metric coefficients ${g_{\mu\nu}}$ . They can be computed manually, but this is tedious and error-prone. Software to compute them automatically is available in association with the textbook on relativity by Hartle (2003). The Riemann tensor, Ricci tensor and curvature can also be computed in this way.

For the Poincar’e square, the Riemann tensor vanishes identically and the curvature of the space is zero. There are only two non-vanishing Christoffel symbols,

$\displaystyle {\Gamma^1}_{11} = \frac{2x}{1-x^2} \qquad\mbox{and} \qquad {\Gamma^1}_{22} = \frac{2y}{1-y^2} \,,$

so the geodesic equations are uncoupled from each other:

$\displaystyle \frac{\mathrm{d}^2 x}{\mathrm{d}\tau^2} + \frac{2x}{1-x^2} \left(\frac{\mathrm{d}x}{\mathrm{d}\tau} \right)^2 = 0 \qquad\qquad \frac{\mathrm{d}^2 y}{\mathrm{d}\tau^2} + \frac{2y}{1-y^2} \left(\frac{\mathrm{d}y}{\mathrm{d}\tau} \right)^2 = 0 \,.$

These equations are nonlinear, but they can be solved analytically, with the help of Mathematica. For initial conditions ${x(0) = x_0}$ and ${\mathrm{d}x/\mathrm{d}t (0)= u_0}$ , the solution is

$\displaystyle x(t) = \frac {1 - \exp\left\{ - \frac{2 u_0}{1-x_0^2} \left[t - \frac{1 - x_0^2}{u_0}\log\left( -\sqrt{\frac{1-x_0}{1+x_0}}\, \right) \right] \right\} } {1 + \exp\left\{ - \frac{2 u_0}{1-x_0^2} \left[t - \frac{1 - x_0^2}{u_0}\log\left( -\sqrt{\frac{1-x_0}{1+x_0}}\, \right) \right] \right\} }$

The solution for ${y(t)}$ is identical in form, with initial conditions ${y(0) = y_0}$ and ${\mathrm{d}y/\mathrm{d}t (0)= v_0}$ (these solutions were confirmed by substituting them back into the equations).

Left: geodesics through the origin ${(0,0)}$ of the Poincar’e square. Right: geodesics through the point ${(-0.5,0.3)}$ of the Poincar’e square.

In the figure (left panel) we show a set of 100 geodesics passing through the origin. They are all “straight lines” in the Poincar’e metric. With two exceptions, they all extend from ${(-1,-1)}$ to ${(+1,+1)}$ or else from ${(-1,+1)}$ to ${(+1,-1)}$ . The two exceptions are the lines corresponding to the ${x}$ -axis and the ${y}$ -axis.

In the figure (right panel) we show 100 geodesics passing through a general point — in fact, the point ${(-0.5,0.3)}$ . Once again, they all extend from a corner to the opposite corner, with the exception of two geodesics, one parallel to the ${x}$ -axis, the other to the ${y}$ -axis.

Noughts and Crosses Unbounded

Children love to play noughts and crosses — aka tic-tac-toe — but soon lose interest: as long as both players use optimal strategies, the result is a draw, so the game has no deep theoretical interest. The first player has nine options. There are eight cells remaining for the second move, and so on. So there are at most ${9!}$ or about a third of a million games. However, many of these are equivalent. For the first move there are essentially three choices. You might like to estimate the total number of essentially unique games.

Gomoku is a more interesting game of this sort. Players alternately mark a cell with black and white stones or noughts and crosses. The goal is to get five stones or marks in a row, horizontal, vertical or diagonal. The game can be played on a Go board or with pencil and paper. There are typically 15 by 15 cells but, in principle, the playing field can be unlimited.

Left: an infinite grid for Gomoku or infinite noughts and crosses. Right: Infinite chessboard; all squares have equal area.

Is there any way to fit an infinite number of cells on a finite board? In the figure above we show a grid of lines parallel to the axes and equidistant from each other in the Poincar’e Square metric. The left panel shows an “infinite gomoku board”. In the right hand panel is an infinite chess board. Despite appearances, all the squares are of equal area.

Sources

${\bullet}$ Hartle, James B., 2021: Gravity: an Introduction to Einstein’s General Relativity. Republished 1st Edition, Cambridge University Press. 602pp. ISBN: 9-781-3165-1754-3

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