Shown above is the wick stretcher. If you see just a heading, then your browser probably doesn't support Java. Other Life patterns can be found in the illustrated catalog. The Java source code is available, and the algorithm is described below.
Patterns are downloaded as gif-encoded images. The patterns were originally converted to gif files for display in the illustrated catalog. This turned out to be a very concise representation, so downloading is not a bottleneck even for the patterns that look really big. Once you see the gif, you have the pattern. The Java API provides methods for downloading images and extracting pixels, and this novel application may be of interest to those looking for an example of the PixelGrabber class.
The primary data structure is a list of live cell coordinates in row-major order. That is, cells are ordered top to bottom, and then left to right for cells on the same row. A cell is stored as an int, with coordinates packed into bit fields. The sign bit is unused. Bits 18-30 store the y coordinate, bits 5-17 store the x coordinate, and bits 0-4 are reserved for accumulating neighborhood information. Note that sorting these int values in increasing order is equivalent to sorting the cells in row major order.
The astute reader may object that the universe size is arbitrarily limited to 8192x8192. That's true, but this is big enough for all the patterns in the archive. More importantly, the size is tied to number representation rather than array allocation, so we could make the universe big enough to satisfy just about anyone if we moved up to 64 bit words.
Now, given this representation, it is easy, for example, to compute the set of all neighbors to the east of live cells just by adding a displacement. This operation also preserves row major order. A list of cells and their neighbors can also be merged while preserving row major order in the same manner that sorted lists are merged for merge sort. The only difference is that if cells have identical coordinates, we combine them by adding the contents of bits 0-4. The method for this can be found in LifeGenerate.combineLists(). If we initialize the 0-4 bit field of each live cell to 2, then after a sequence of such merges, we obtain twice the total count of cells in every non-empty 3x3 box in the Life universe.
Finally, we subtract 1 from the count obtained for each box centered at a live cell. It is easy to see that now the neighborhood list contains enough information to compute the next generation. Bit 0 encodes whether the cell is live or dead, while bits 1-4 contain the number of neighbors. A final pass using a lookup table computes the next generation, discarding dead cells, while initializing live cells for the next generation.