Eli5 How come we know there’s only 3 dimensions in our world when math allows technically arbitrarily high numbers of them?

Question

We can’t physically see or understand how complex numbers exist or work in our world in a nice way, but we know they do exist. Because we’ve made massive advancements in science and technology off the assumption that they exist and work, and our understanding of many things in the world including stuff as basic as the solutions to quadratic equations would fall apart. By the same token, there are many problems for which vectors and problem spaces of nth degree are used, where n>3, and there’s that whole adage where time is considered a 4th dimension. In that way, we often solve many problems, even rudimentary linear algebra ones, using sets in R⁴, R⁵, etc, and there are many, many invisible forces at work in our world such as gravity. We know how easily our brain can trick us, we still are easily fooled by optical illusions even when we know they’re there and what they are/how they work, despite our visual cortex being the one of the most powerful and most used part of our brain. So the idea of forces and things which we don’t have the capacity to perceive existing in the world is not anything new or foreign. There are frequencies we can’t hear, colors we can’t see, etc which other animals can and do. So why is the concept of n dimensions in the world so widely rejected? There must be a simple reason, I have heard that it has to do with the volume of a gas in a container being proportionate to its dimensionality or something

In: 2497

Answer 1

Imagine you are holding a cube made of wires (along the edges, ie. 12 wires), outside in the sun. When you rotate the cube, its shadow (2d projection from 3d space) changes shape quite markedly.

Imagine now that there is a 2 dimensional creature with no perception into the third dimension looking at that shadow. They see an object which is changing shape and conclude that it is an object with varying geometry, not a fixed object simply being rotated.

Now search for “Tesseract animation”, you’ll see examples of 4d “Cubes” rotating in 4d space. To us, 3 dimensional creatures, it looks like these objects have varying geometry but they are fixed geometry objects being rotated, then projected into 3d space from 4d space.

The fact that we don’t observe objects changing shape like that suggests we’re not just 3d creatures living in a higher dimensional world only perceiving 3 dimensions, but are actually in a universe with only 3 ordinary spatial dimensions.

There may be higher dimensions, but they do not appear to ordinary ones like our known 3.