Why though?
I think I'll write a blog post at some point explaining my process for visualizing 4D. Hopefully it should make it clearer what I mean.
I think most existing resources on the topic go about it in a way that makes it hard to build up a proper intuition. They start by assuming that humans can visualize 3D and then try to extend that one dimension higher. But humans can't actually visualize 3D, only 2D. We combine multiple different 2D perspectives together to "fake" an understanding of 3D. Our vision is also only stereoscopic 2D, not true 3D.
If you take a similar approach with 4D, trying to project directly from 4D to 2D instead of going through 3D as an intermediate step, it's harder to visualize at first but better in the long run for really understanding it.
While I don't dispute your method works for your purposes, would you say it allows you to visualize more than a single 4D shape side by side? What about interlocking shapes? Can you place this shape in an arbitrary 4D space among others and describe its relative position?
It's awfully hard to prove that real-world things are impossible, especially if there's no objective measurement of whether they've been achieved. (For example, if I tell you that I can visualize more than 3 dimensions, then how could you verify or disprove that?)
I don't really know. The first thing that came to my mind would be to ask to draw/model different cross-sections of a 4D object ("cross-volumes"?).
We can visualize 3D objects, and therefore can draw 2D cross-sections of 3D objects relatively well, and relatively easily. Like, sections of a human body, or a house. So, maybe someone who can visualize 4D objects in their head could also model 3D "cross-sections" of that object at arbitrary "cuts". And we could check if those 3D radiographies are accurate, because we can model those 4D objects on a computer, and draw their 3D cuts.
Just a simple idea. I'm sure there could be other ways of probing this.
Many people can draw cross-sections reasonably well, but I can't. Nonetheless, I believe that I can visualize 3D objects.
I'm not sure what that means, but my inability to draw is astounding.
I don't think that's true. For example, consider a regular octahedron: take a parallel pair of its faces and bisect the octahedron between those faces. What's the resulting figure? What happens to the figure as you tip the plane?
I mean, obviously the task I just set isn't impossible; and with a little reasoning anyone can give the answer in a few seconds; but it feels too me like the answer is not simply intuited merely by the virtue of our being 3D creatures.
Sure, part of the difficulty stems from that the octahedron (to most folks) is both less familiar and slightly more complicated than the cube. But the same applies to the hypercube!
If someone wants to have a look, feel free (but it is hard. You need time. If you do not have vr headsets but want to have a look, you can install browser add-ons thet let you simulate vr headset and controls):
https://www.brainpaingames.com/Hypershack.html
I have been planning to write a Show HN any day now for months, but maybe someday.
https://bigthink.com/starts-with-a-bang/time-yes-dimension-n...