I did a Mathematica (MTM) analysis of several important papers here and here from Mehmet Koca, et. al. The resulting MTM output in PDF format is here and the .NB notebook is here.
![](https://theoryofeverything.org/theToE/wp-content/uploads/2022/08/DualSnub24Cell-2-1-917x1024.png)
* The (42) yellow have no overlaps.
* The (51) orange have 2 overlaps.
* The (18) tetrahedral hull surfaces are uniquely colored.
![](https://theoryofeverything.org/theToE/wp-content/uploads/2023/08/Dual-Snub-24-cell-1024x1024.png)
What is really interesting about this is the method to generate these 3D and 4D structures is based on Quaternions (and Octonions with judicious selection of the first triad={123}). This includes both the 600 Cell and the 120 Cell and its group theoretic orbits. The 144 vertex Dual Snub 24 Cell is a combination of those 120 Cell orbits, namely T'(24) & S’ (96), along with the D4 24 Cell T(24).
![](https://theoryofeverything.org/theToE/wp-content/uploads/2022/08/Snub24Prime.png)
![](https://theoryofeverything.org/theToE/wp-content/uploads/2022/08/Snub24CellPrime-hulls-566x1024.png)
![](https://theoryofeverything.org/theToE/wp-content/uploads/2022/08/DualSnub24Cell-2D-all-projections-1-450x1024.png)
![](https://theoryofeverything.org/theToE/wp-content/uploads/2022/08/orbitM-192-120Cell.png)
![](https://theoryofeverything.org/theToE/wp-content/uploads/2022/08/orbitN-288-120Cell.png)
the W(D4) C3 orbits T'(24)+S'(96)+M (192)
![](https://theoryofeverything.org/theToE/wp-content/uploads/2022/08/120Cell-600.png)
![](https://theoryofeverything.org/theToE/wp-content/uploads/2022/08/T-120-cell_with_600_vertices.png)
![](https://theoryofeverything.org/theToE/wp-content/uploads/2022/08/24CellLΦ2H4Anim81-1.gif)
![](https://theoryofeverything.org/theToE/wp-content/uploads/2022/08/H42JLΦAnim81-1-1.gif)