The height of the 1x1x1 LEGO brick is 6/5 times its width. Is there any particular reason for these proportions? And especially why not making a cubic brick (it would have made SNOT building easier)?
In the 60s, Lego did produce a separate system for architectural modelling called Modulex which used a 1:1 ratio 5mm cube as its basic brick. It wasn't successful and was discontinued in the late 60s.
I believe the 6:5 was chosen so that studs could fit into the geometry. Related: why the plate is 1/3 of the brick's height. By adding two plates to the height of the brick vertically you can now match the centres of two horizontal units (6+2+2):(5+5)
Thinking Lego started with 2×4 bricks of 1.25 inches. As they made model tractors not from bricks, they made the height of bricks to be equal to the difference between the small and large wheels' centres to build tractors. So (√10 - √2) ÷ 2, but needed to add the size of a stud. Or could just be that 2×4 studs fit the size of a large wheel. In fact 2×4 fits large wheel, but 2×2 overlaps the tyre on small wheel. Slope, 1:1, 100%, 45°, but they have a vertical edge. Guess that = 1/16" & needed to take studs underneath, but not as much as a slimbrick as sloping up from it.
Plates need enough height for studs underneath.
I have some bricks (not compatible with Lego) whose height equals the width of one stud pieces, and whose height is equal to that of 2 plates. I guess a 1/2 Lego plate wouldn't look right for a flat roof, which is their main use?