Multidimensional additive rules

The 2D analog of rule 90 yields the patterns shown below. The colors of cells are given essentially by Mod[Multinomial[t, x, y], 2]. In d dimensions (2d)^DigitCount[t, 2, 1] cells are black at step t. The fractal dimension of the (d + 1)-dimensional structure formed from all black cells is Log[2, 1 + 2d].

The 2D analog of rule 150 yields the patterns below; the fractal dimension of the structure in this case is Log[2, (1 + Sqrt[1 + 4/d]) d].