You probably already saw such a representation of the tesseract:

[![Tesseract][1]][1]

I did something similar on [my blog](https://laustep.github.io/stlahblog/posts/StereoTruncatedTesseract.html) for the truncated tesseract:

[<img src="https://laustep.github.io/stlahblog/posts/figures/truncatedTesseract_stereographic_asy.gif" alt="Truncated tesseract" title="Click to enlarge.">](https://laustep.github.io/stlahblog/posts/figures/truncatedTesseract_stereographic_asy.gif)

The vertices in 3D are the stereographic projections of the original 4D vertices. This point is clear. However I don't know how the varying radius of the bent tubular edges should be chosen. To get a bent 3D edge, I project the corresponding 4D edge on the 3-sphere and then I stereographically project the bent 4D edge. But how to choose the radii? The way I use on my blog consists in arbitrarily taking a radius "proportional" to the norm of the 3D point (e.g. I take $\log\bigl((1 + \lVert M \rVert)/4\bigr)/4$, because I empirically found this choice yields a pretty result). Is there a mathematical consideration that would justify to have a radius "proportional" to the norm of 3D point? What is the meaning of the value of the radius?

Schleimer and Segerman, the authors of the paper [Sculptures in $S^3$](https://arxiv.org/abs/1204.4952) (see Figure 3) have a different approach and I don't understand it:

> A better solution is to use tubular neighbourhoods in the intermediate $S^3$ geometry. For this we must parameterise the image of such a tube under stereographic projection. Here the circline property is very useful. The boundary of a tubular neighbourhood of a geodesic in $S^3$ can be made as a union of small circles in $\mathbb R^4$. (These circles lie in $S^3$, but are not great.) The small circles map to circlines in $\mathbb R^3$, which can be
directly parameterised. Computer visualisation of stereographic projections of 4-polytopes, in this style, are beautifully rendered by the program Jenn3d [8]. In Figure 3 we show four views of a 3D print of the 24–cell, ….

  [1]: https://i.sstatic.net/DxfzYb.png
  [2]: https://i.sstatic.net/YoHVj.png