# Why would the most energy efficient shape be a cube instead of a sphere?

The most energy efficient shape in our universe is a sphere. However, in another universe it is a cube. Why and how would this happen?

I came across this idea while thinking about Minecraft and why it’s all cubes. I thought perhaps this might explain it a bit but I’m more interested in why a cube would become the most energy efficient and how.

• efficient for what, spheres suck for packing efficiency.
– John
Commented Jun 12, 2020 at 12:49
• Atoms can self-organize into cubic crystals. Don't know if that fits your question, as energy efficient is not really well defined without more context. Commented Jun 23, 2020 at 16:08
• You could be living in a simulation. A cubical Matrix. The simulation running in a crystal. Commented Dec 8, 2020 at 22:43

This requires a fundamental rethink of the notion of distance in your universe, which dives into some pretty deep maths constructs. You have to introduce a preferred coordinate frame, and the fact that your preferred objects are cubes rather than any other shape indicates that 1) the axes are mutually orthogonal, and 2) the 'scale' of each dimension is the same. You would be able to tell if the scale was different in different axes by things like a spring having different force properties depending on which axis you aligned it in.

This universe can be imagined as 'pixelated' in the cartesian axes into a grid of voxels, but it can be continuous (or at least the size of a voxel can be minute, like the Plank length). And the most dramatic change is that distances are measured using Manhattan Distance.

The important thing to realise about this, however, is that to an observer inside the universe, minimal-energy objects still look like spheres! The surface of a sphere is the set of all points equidistant from a centre point, and in this distance metric the surface of a cube still meets this definition. Note that the cube is not aligned to the cartesian axes, they go through the vertices of the sphere. Only an observer from another universe who can see the space with eyes/instruments which measure in the Euclidian metric, can declare that the objects are 'cubic'. Objects which the Manhattan-folk declare to be cubic would appear to an Euclidian observer to be (smaller) spheres.

Now if you were to mix forces in one universe, such that some forces responded to distance in the Euclidian metric and some responded to distances in the Manhattan metric, you could start to get some seriously weird behaviour, like objects which were naturally spherical if 'uncharged' in the new mineforce but became naturally cubic (to a generally Euclidian observer) when charged. A planet which was naturally strongly charged with mineforce would tend to exhibit cubic tendencies. Unfortunately such a place would be extremely dangerous for an Euclidian observer to visit, because any accumulation of minecharge on their bodies would start to push things (like cells, atomic bonds and protein folds) towards the cubic form, probably fatally.

• Excellent answer (particularly noting that cubes under this metric look like spheres). Anyone wanting to prove that a cube is the minimal-energy shape for a self-gravitating object may want to modify the method used in physics.stackexchange.com/a/155955/56299, which explored the (more intuitive) case of a sphere under a Euclidean metric. Commented Jun 12, 2020 at 15:54
• Good answer, just a small nitpick: spheres in the Manhattan/taxicab metric are actually regular octahedra. To get cubes one should use the maximum norm. Commented Jun 12, 2020 at 16:32
• I was googling furiously to try and find something that would confirm or deny my deri-guessi-vation there :-) Commented Jun 12, 2020 at 18:00
• Manhattan is L1 you need Linf Commented Jun 12, 2020 at 22:23

The easiest idea would be to create the wanted universe in a minecraft-style. It could be a universe where every kind of movement (physical movement, energytransmission and so on) is just possible along lines parallel to some coordinate system. We can allow the steps along these parallels to be as short as we want (so we are still able to build near sperical objects), but as energy can enter or leave an objekt just along these lines, and the surface of our object is the same for a near-sphere and a cube, the cube would be the most efficient shape (defined as smalest loss of energy per unit of volume and timestep). I have to admit, physics in this universe could get realy weird.

I assume you mean the most structurally efficient shape.

Cubes are already better shapes than spheres for all sorts of stuff like stacking, packing matter or energy into a small spaces, etc. What spheres do better that cubes is the turn basically all outside forces into compression because thier shape naturally stacks material between the impact vector and resistance vector. Since most building materials are stronger in the compression state than the tension state, this explains why arches are better for doing things like holding up ceilings: because stone breaks much more easily once you start trying to bend or pull at it.

That said, you can also invert the arch like you see in suspension bridges when dealing with tension stronger materials. A suspension wire is stronger in tension; so, the upside down arch keeps it in tension for optimal strength.

To make cubes the way to go, you need to make your primary building material equally good in the compression and tension state. If your material does not have an advantage either way then curving it does not lend itself to being stronger. Instead, the best shape then becomes the one that uses the least amount of material: a straight line.

I would solve this by giving your other world a material as common as stone, that is just as good in tension as stone is in compression. Then you could have your other world use laminated lintels like the one shown below. When you apply pressure to a beam, the back of it pulls and the belly compresses. By alternating the right materials in the right places, you can put the stone into only compression and the "anti-stone" into only tension giving you the same effective advantage as the arch, but with a much simpler construction.