Doodles

The other week, in bed and still awake, I took to wondering how you'd design a building with only horizontal, vertical and diagonal faces so it would be close to a hemisphere. (Such a shape would minimize the ratio of surface area to volume, making insulation easier.)

I came up with this thirteen-story design, with the help of Moira's graph paper.

I made it so that each storey, except for those with the same area as the one below, would be surrounded by a vestibule ring area that wouldn't get the central heating but would still be windowed off, with diagonal glass ceilings. There'd also be a glass pyramid on top of the penthouse.

This is how it looks from above:

The extra contours indicate the diagonal covering. 

Then it occured to me to design a five-story mansion so that a semi-hemisphere would face south for the sun, while northward there'd be a steep diagonal. (There'd still be the vestibules.) Here, north is on the right.

This is the vertical view, with north up:

It all got started from thinking about how if you have a circle centered at (0,0) with five units radius, it would touch (+/-3,+/-4) and (+/-4,+/-3), eight points in addition to (+/-5,0) and (0,+/-5). Similarly, a concentric circle with thirteen units radius would touch (+/-5,+/-12) and (+/-12, +/-5) as well as (+/-13,0) and (0,+/-13). In addition, a sphere centred at (0,0,0) would touch points at (3,4,12) or whatever order and whatever positive or negative you want!

Inside an Aspie mind...