Visualising the interior gluings of a 3D shape in 2D

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I have a small triangulation of a 3-ball that I'm trying to form a nice 'visualisation' of for a paper/talk.

The best I've got so far is a few rough sketches like the one below, where I've tried to colour the edges and faces of the 'new' tetrahedron added at each step. To me, this is not very clear.

Construction of a triangulation

I also made a simple 3D 'model' of it in Desmos.

I'm wondering if there's a better way to visualise/depict a 3D 'model' such as this one including it's interior components. In particular, I'm keen on software or program recommendations, as well as how to indicate stuff like depth in a 2D diagram