What is the maximum "alive density" of cells in Conway's Game of Life when played on a torus?

Solution 1:

In the specific case of a pattern that is fixed under the rules of the Game of Life, the maximum density is 1/2, as shown by Noam Elkies in The still-Life density problem and its generalizations.

Elkies also presents (on page 22) a simple example of a period 6 oscillator with maximum density 3/4. Here are all its phases: \begin{array}{|c|c|c|c|c|c|c|c|} \hline 1&1&0&0&0&0&0&0\\ \hline 1&1&0&0&0&0&0&0\\ \hline \end{array}

\begin{array}{|c|c|c|c|c|c|c|c|} \hline 0&0&1&0&0&0&0&1\\ \hline 0&0&1&0&0&0&0&1\\ \hline \end{array}

\begin{array}{|c|c|c|c|c|c|c|c|} \hline 1&1&1&1&0&0&1&1\\ \hline 1&1&1&1&0&0&1&1\\ \hline \end{array}

\begin{array}{|c|c|c|c|c|c|c|c|} \hline 0&0&0&0&1&1&0&0\\ \hline 0&0&0&0&1&1&0&0\\ \hline \end{array}

\begin{array}{|c|c|c|c|c|c|c|c|} \hline 0&0&0&1&0&0&1&0\\ \hline 0&0&0&1&0&0&1&0\\ \hline \end{array}

\begin{array}{|c|c|c|c|c|c|c|c|} \hline 0&0&1&1&1&1&1&1\\ \hline 0&0&1&1&1&1&1&1\\ \hline \end{array}

Solution 2:

I think it'll be hard to beat

\begin{array}{|c|c|} \hline 1&1\\ \hline 0&0\\ \hline \end{array}