Today I Have Mostly Been Thinking About Icosa-10-Topes

I have! I really have. I'm trying to think of something useful I can write which will spit out information regarding higher-dimensions. The trouble is that apart from the lovely, rotating Tesseract up there (which is way beyond my ability - at least The Mandelbrot set stays still), there isn't much you can do to visualize such things. I could write something which takes in the n-dimensional coordinates of two points in n-space and returns the 1-dimensional distance between them. Distance is always 1-dimensional and despite this it still amazes me that distances between any n-dimensional space are always 1 dimensional. Standard Pythagoras is all you need.

My understanding has been greatly helped by working out how many faces and cubes a Tesseract has just by looking at a still version of the picture above. You can see all of the 8 cubes - the inner one - the outer one and the six which join these two together. However, it has not helped me see into the fourth dimension which I think needs some mental rigour in visualizing three dimensions. Actually thinking about this, we do it all the time. I can only see a part of the inside of my house at any one time and yet in my head I have a mental picture of the whole thing. Generally thoughts are abstractions of what they refer to to - the mind has ideas and concepts which only turn into words when they are voiced or written down. Similarly, the mind has images and maps of things in the real world - three-dimensional models as well as templates for recognition of those objects. So in a way we have the mental information to imagine four dimensions. It's just that we have no experience of four dimensions and just cannot imagine anything at right-angles to the existing three. Frustrating hey?

Looking again the rotation up there has made me wonder how it was produced. I suspect it is simply done in three dimensions, with the programmer allowing a plastic flow of the matter at the vertexes. It would be much better if the application had simply been programmed in 4 dimensions and then rendered the 4D information to the visible 3D. I am sure that computing power is quite capable of handling information in any number of dimensions. The program should be generic - starting a new project with "Enter number of Dimensions". Creating 4D objects and then examining them in 3D would go a long way to helping someone to actually "see" the 4D. Obviously the real problem is that to really see 4D we would need 3D retinas (I know our retinas are actually 3D already - they are really just 2D planes wrapped around a 3D sphere.) The problem is that light will not travel through solid matter. At this point I have to give up having to think about this; maybe there is a solution.