In Conway's Game of Life, and other cellular automata with the same neighborhood structure, you can determine the values of the cells in any 2x × 2x square of cells from their values x steps earlier in a bigger 4x × 4x square. If we view this as a time-space diagram, with two of the dimensions as the spatial dimensions of the Life grid and the third dimension as time, we can view the bigger starting square and the smaller resulting square as the bottom and top of a sort of stepped pyramid formed by stacking concentric squares. The values of the cells in each layer of the pyramid are determined by the ones in the layer below. The squares shrink inward as they rise upward, because the other cells outside the pyramid need more information (outside the base of the pyramid) to determine their values. The image below shows this construction for x = 1, 2, and 4, with each time-space state of a cell represented as a cube.
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