Building a pyramid out of bricks

Suppose you are constructing a pyramid out of blocks like the ancient Egyptians did. Each new block is placed centered over the point where the corners of four blocks of the level below meet. What is the ultimate shape of the pyramid as a function of the dimensions of the bricks of which it is composed?

Let the length along the edge from base to apex be called the side length. If cubes are used, the ratio of the side length to the base length is sqrt(6):2., which is taller than an equilateral square pyramid. If flatter blocks are used, with height sqrt(0.5) times width or length, then each face is an equilateral triangle. If one builds an upside-down pyramid beneath it as a foundation, then it's a half-buried regular octahedron.

If the Egyptians made equilateral triangle faces, then they must have known the Pythagorean Theorem, as well as a practical method of extracting square roots.

There is one more interesting square pyramid: the one formed by connecting the corners of a face of a cube to the center. It has side length sqrt(3)/2 and is related to the rhombic dodecahedron. It may be constructed by stacking half-cubes.

What material should it be constructed out of to stand the test of time? Poured concrete, perhaps reinforced, is the choice in this day and age (forget blocks: just pour each layer directly) but it would be very cool to see one constructed of giant blocks of clear artificial quartz. Diamond may catch fire, maybe sapphire would also work: is it aluminum oxide? There might not be enough aluminum.