Real world usage of 4 dimensional geometry

Question

4d space and shapes, for example the hypercube, are quite strange, but is there any real world usage for 4d geometry? For example, do 4d graphs that use rules of 4d geometry exist?

Answer 1

Two cases come to mind.

1) First, 4-dimensional geometry can be used when working with 4-feature samples. Specifically, if you have a set of objects, each with 4 important measurements about them (e.g. length, width, density and age), and you wish to compare it to new objects with the same measurements, you would do so in 4-dimensional geometry. In this same way, N-dimensional geometry is used for any value N. Face recognition can work with over 10,000 dimensions, depending on the technique used.

2) Some 3D graphics stuff, such as that seen in video games, uses a fourth dimension to handle some wonky stuff. Translation can be expressed as a multiplication instead of an addition if you extend the system to work in four dimensions instead of three, which is helpful considering that rotation and scaling are both multiplications (and remain so when adding in the extra dimension), so the whole process can be expressed as one multiplication.

On the other hand, I don't think people construct four-dimensional graphs. Not to the best of my knowledge, anyway.

Answer 2

Another area where higher dimensional geometry appears in the real world is quasicrystals (https://en.wikipedia.org/wiki/Quasicrystal).

Certain lattice symmetries are classically forbidden but can emerge as projections of higher-dimensional lattices into our space. We get pentagonal, octagonal, decagonal and dodecagonal symmetries by projecting a four-dimensional lattice onto the plane, while icosahedral symmetry comes from projecting a six-dimensional lattice into three dimensions.

The icosahedral case has recently been found in nature, and various applications of quasicrystals are being explored.