I made this duoprism, a four-dimensional polytope, using Stella 4d. You can try this program for yourself, free, at http://www.software3d.com/Stella.php.
There are six regular, convex four-dimensional polytopes. Five of them correspond on a 1:1 basis with the Platonic solids (as the tesseract corresponds to the cube), leaving one four-dimensional polytope without a three-dimensional analogue among the Platonics. That polychoron is the icositetrachoron, also named the 24-cell and made of 24 octahedral cells. It also happens to be self-dual.
If, in hyperspace, the corners are cut off just right, new cells are created with 24 cubic cells created at the corners, with 24 truncated octahedral cells remaining from the original polychoron. This is the truncated icositetrachoron:
4-dimensional polytopes have 3-dimensional nets. These nets are shown below — first for the icositetrachoron, and then for the truncated icositetrachoron.
I used Stella 4d to create these images. You can try Stella for free at http://www.software3d.com/Stella.php.
There are objects in hyperspace known as duoprisms, which have prismatic cells. This one’s cells are 24 dodecagonal prisms. It was made using Stella 4d, available here.
The figure above, rotating in hyperspace, is an orthogonal projection of a four-dimensional polychoron known as the truncated tesseract. It is analogous to the truncated cube, one of the Archimedean solids. The image below is of the same figure, but is shown as a perspective projection.
Both images were created using Stella 4d, software you can buy (with a free trial download available, first) at http://www.software3d.com/Stella.php. It’s great software, and a friend of mine wrote it — but no, he doesn’t pay me to give his program free advertising, as some have wondered.
This is the hyperdodecahedron, or 120-cell, one of the six four-dimensional analogs of the Platonic solids. It’s been shown on this blog before, but this image has one major change: a much slower rotational speed. It is my hope that this will help people, including myself, with the difficult task of understanding four-dimensional objects.
This image was created using Stella 4d, a program you can try, as a free trial download, at this website.
One of the regular 4-dimensional polytopes, or polychora, is called the 600-cell. If you truncate it, you get the figure above, called “23-tex” for short. Its unit cells are 600 truncated tetrahedra, as well as 120 icosahedra. As shown above, you see only the cells, with space between them so you can see them, and vertices and edges rendered invisible. If the vertices and edges are shown, though, 23-tex looks like this:
Since these are four-dimensional figures, it’s not possible to see what these really look like all at once, in hyperspace, because you, the observer, are trapped in a universe where only three spatial dimensions are easy to perceive. That’s one reason I’m showing these figures with them rotating — as they spin, in hyperspace, the “slice” of the shape which appears as a three-dimensional projection, in our universe, changes. Over time, then, you get to see it all — but, in hyperspace, its appearance would be different. If you try to imagine a living creature living inside a horizontal Euclidean plane, and this creature trying to picture a three-dimensional shape, that can help with understanding the nature of the problem that arises when three-dimensional beings try to picture anything involving a fourth spatial dimension.
Three-dimensional polyhedra can be unfolded, and shown as flat “nets” made out of polygons. Well, what does the “net” of a four-dimensional polytope look like? The answer is one dimension “down” from four: the net is a bunch of three-dimensional polyhedra, stuck together at their faces, just as the nets of polyhedra are made of polygons joined at their edges. Here’s one possible net for 23-tex:
The three images above were produced using Stella 4d: Polyhedron Navigator, software you may try for yourself, here.
These tesseract views are all of the perspective projection-type, with the first one, above, being done in cell-first fashion.
The next one is projected edge-first.
The third one is projected vertex-first.
Although all of these are rotating in the same direction in hyperspace, the different projection-choices make the second and fourth images appear to be rotating in different directions. Why? I’m still trying to figure that out!
These animations were created with Stella 4d, software available at www.software3d.com/Stella.php.