I cannot resist replying to a post in Cosmic Variance: Why Can’t We Visualize More Than Three Dimensions? The comments section says much of this already, in addition to the wonderful:
Your better off just telling yourself your not god, even if you really are. Really you should think your selves lucky that you can’t imagine a world that has not even made up its mind whether you have been born or not. 3d living harsh enough, my advice is stay well away from the rest.
However I feel I do have some expertise here so I want to put my side.
Firstly do not make the mistake that as 4 is not much bigger than 3, 4D space is roughly the same. There is an awful lot more space.
Secondly we do not see 3d, we see 2d. Our brains have complicated tricks and tools to turn this into a 3d image, that go a long way beyond simple binocular vision. We know that these are tricks as they can be hacked in optical illusions. The book Mind Hacks
has some wonderful examples. For example how we trust shadows (and assume lights are fixed).
3D visualisation can be worked on and improved though (zometool is great for this). Architects and sculptors usually have a far stronger sense than mathematicians. Furthermore to answer questions in four dimensions it can be useful to add our visual processing to the analytical logical abstraction. However to do this we need to develop tricks. Some of these will work in one situation some in another. They all have to be treated carefully as sometimes as with the shadows they can fool us.
Common Tricks
1) Divide up the space. For my work on the Penrose tiling the main trick is to divide a 4D space into two 2D spaces and consider a pair of points. This view is beautifully illustrated in Rich Schwartz’s Lucy and Lily game.
2) Projection. Has the advantage of producing some stunning 3d objects:
120-cell
3) Slicing, possibly using time. This is the Flatland approach. Not as powerful as projection as it preserves less of the geometry and combinatorics of the object. In addition the choice and position of 3-space to cut introduces a new geometric choice.
Combining these three can help develop intuition and subconscious mind tools. However even the most dedicated 4d voyager will not get close to 3d perception as we use that every day to stay alive.
To finish here is another projection view of 4D, from my work “3d and 4d surfaces”. It is a 2d image, the projection of an approximation of a plane in 4D by cubes. However the shading and angles set off 3d processing. In my case at least it sometimes looks 3d and sometimes snaps flat as my brain cannot make sense of the geometry.
Ammann-Bennker Greys
The full piece appeared in this post.
There is even a prize, £50. Though I am lazy so the conditions of this are that the winner:
Posted by gelada 
