I'm hoping to construct a physical model of a 4D cube. However, I'm struggling to work out the proper size ratio between the inner and outer cubes. From the graphics I've seen, the inner cube seems about 1/3 the size of the outer, but I haven't been able to find any documentation for that. How would you go about working that out?
Thanks for your time.
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I doubt there is a correct answer.
It is equivalent to drawing a cube in perspective on a flat piece of paper. Just do what feels good.
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Suppose you are observing a point $P=(x,y,z,w)$ in $4D$-space projected to $3D$-subspace $\mathbb R\times\mathbb R\times\mathbb R\times\{k\}$ from the point of observation being $O=(0,0,0,0)$. Then the resulting point would simply be: $$ P'=\left(\tau x,\tau y,\tau z,k\right),\text{ where }\tau=\frac kw $$ So the distance of the observer in the fourth dimension to each point in your $4D$-object would determine the scaling of that particular point. Hence everything depends on your choice of configuring the size of the object, the point of the observer, and the distance $k$ from the observer to the projection space.
You can play around with this until you find a setup that suits you.
Go for what looks best. There is no mathematical reason for any particular size.
A 3D model of a 4D cube is a projection. It is like perspective in a photo or painting. Things that are further away are smaller, but depending on the distance of the observer, the relative sizes of two things in the painting at different distances can vary. Similarly, the "inner" cube of a classic hypercube projection is further away than the outer cube along the unavailable fourth dimension, and hence is depicted smaller. The size difference depends on how far away you as an observer are along that same dimension, so could be anything.