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A sphere’s minimized surface area per volume also means that temperature control is more efficient: there’s less surface area for heat to escape in the winter, or enter in the summer. So for the same (or similar) wall material cost & ongoing energy cost, you get more interior space.

Then the question is: can you actually use the extra interior space?

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This analysis doesn't hold up when you consider that apartment buildings pack multiple cuboid spaces into a single larger cuboid building, sharing floors and ceilings with each other to lower heating and cooling costs.

This becomes obvious when you take it to the extreme. A cube of 10 × 10 × 10 meters can be divided into 1000 units of 1 cubic meter each, and has an external surface area equal to 6 × 10 x 10 = 600 square meters, or 0.6 square meters surface exposed per unit.

A solitary sphere with a volume of 1 cubic meter has a radius of 0.62 cm and a surface area of 4.836 square meters, which is over 8 times worse, even ignoring the fact that it's harder to utilize the space in the sphere.

Yes, the surface area ratio is only applicable to whole buildings. But, regardless of how the interior is divided, the whole space has to be heated. (Empty apartments notwithstanding, since that would change the shared-internal-walls calculations.) The savings you get from multi-unit buildings ultimately comes from the resource-sharing of the multiple tenants, rather than the shape of the whole building.

Now that I think about it, maybe the spherical or cylindrical shape is actually better suited to an apartment building. The inner units _would_ have typical rectangular angles. For units along the curved surface, some walls will be straight and some will be curved.

Plus, the curvature would be “wider” for a larger building, which would make it a little easier to design or arrange furniture.

The inner units also would have zero windows.

Even if that wouldn’t make them so undesirable as to be basically useless, that probably would be in conflict with building codes in large parts of the world.

the best you could do is a fairly small sphere with kitchens, bathrooms and storage spaces in the inside.

Yeah, the wastage of the "average" home bugs me, but there's got to be lower hanging fruit than somehow living in spheres. Even just a cylinder would give you some great improvement (and it would be mostly rectangles, with normal windows), but even better than that? On the other side of the wall I am typing this at, there's a heat engine removing heat from a freezing box, but then it has to drop all that heat into the space I am sitting in . . so then another box has to try and get the heat outside. Oh! But wait! There's another box that HEATS UP - a lot! - so I can put food in it. Another one dries my clothes, and makes hot water, both of them putting heat back in. Jiminy beeswax. Not even going to touch the way water is brutally wasted in the typical setup.

I have often thought of this. It may turn out that those inefficiencies are fairly small - the devices themselves small, well insulated, etc. One would have to do the maths. But I imagine that cooling all the air with AC (and letting it leak whenever you open a door or window) and then heating an outdoor pool is huge

I think we should start with dc power. Each house should have a 12/24 volt low voltage high efficiency power supply to eliminate the need for individual converters everywhere

48V would be a better choice. It is increasingly popular for off-grid applications because the higher voltage reduces heat and power loss, letting you use thinner wires, which are both cheaper and easier to install; but 48V is still considered "low voltage" and does not require any special safety precautions.

48 is fine but then it need a dc dc step down power supply

Spheres can't be stacked together like terraced houses though.

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