Thanks for the coordinates! Here's the very first render I did, from a viewpoint of <5,0,0,0>:
This looks awfully similar to the vertex-first projection of the tesseract, but actually, those hexahedral cells are not cubes, but parallelopipeds. Visibility clipping is on, so we can only see 4 cells, but actually, this shape has 20 cells. Here's a render with visibility culling turned off, and with edges on the far side of the shape rendered in blue so you can tell what's in front and what's behind:
This image begins to show the structure of the shape. It appears to be made of 3 layers: the top and bottom layers are tetrahedral arrangements of parallelopiped cells, 4 each, and these two layers are connected to each other by a middle layer of 12 parallelopipeds. This particular viewpoint shows the "tesseract-within-a-tesseract" projection of the 5D cube, which is kinda cool.
Next, let's take a look from the side (<0,5,0,0>):
I've left the edges previously colored blue still colored blue, in order to give a point of reference as to where the 4D viewpoint is. The black edges on top of this image trace the outlines of the 4 parallelopipeds in the previous images. I've turned on visibility clipping again so that the image is not too cluttered; so only 3 of the previous 4 cells are visible here. Furthermore, they look flattened because we're looking at them from a 90° angle. They are not quite exactly 90° with the eye-to-object direction, so they don't appear completely flat, but they look quite flattened.
Below these cells are another 4 cells, also parallelopipeds. The bottom one is actually one of the cells from the bottom layer (the other 3 are not visible from this viewpoint because they lie behind this cell).
Next, let's turn off visibility clipping so that we can see the rest of the edges:
As you can see, it looks rather messy (which is why I usually like to leave visibility clipping on). But by comparing with the previous image, we can somewhat see which edges are on the far side of the shape from the current 4D viewpoint. If you look carefully, you can see 3 blue hexahedra at the bottom of the image, which share the bottom faces with the bottom cell from the previous image. These 4 cells are the cells in the bottom layer. The remaining middle-layer cells are really hard to see, because there are a lot of edges here, so let's do a sort of reverse-visclip, i.e., hide the cells on the near
side of the shape from the current viewpoint, and show only those lying on the far side:
These are all the far-side cells. The bottom 3 cells on the far side are now much clearer. The middle layer cells are still a bit hard to see, but at least they are no longer overlapping each other's volumes, so if we look carefully, we can sorta tell where they are. If you look at the rhombus-shaped ridges at the top that has been foreshortened into narrow ridges, skirting around the top cell, you can see there are 6 of them in total, and below them are the squashed images of 6 parallelopiped cells. These 6 cells surround 3 cells at the center of the projection, which lie directly below the top cell. So that makes 9 cells in the middle layer, on the far side, which is 12 middle-layer cells in total.
Note that those 3 "inner" cells are actually the closest to the 4D viewpoint; the 6 squashed cells around the sides are being seen from a steep angle; they are near the limb of the projection image.
Finally, let's color the top and bottom layer cells so that it's easier to pick them out from the tangle of edges:
I've turned off visclip completely again, so that all cells are visible. Because there is no visclip, the cells overlap each other, but it shouldn't be too difficult to pick them out.
So in total, we have 4 top layer cells, 4 bottom layer cells, and 12 middle-layer cells. From the first and second images, you can see that these 12 middle-layer cells follow a rhombic-dodecahedral arrangement, so this projection of the 5D cube does show some kind of analogy to the rhombic dodecahedron. But it's not related to the 24-cell; instead, it has parallelopipeds for cells, with a combination of tetrahedral and rhombic dodecahedral arrangements. All in all, quite an interesting projection of the 5D cube, indeed!