When you’ve finished with the tesseract you can have a go at the Klein bottle. This is like a Mobius strip with one extra dimension. You’ll remember a Mobius strip is where you take a strip of, say, paper then put a twist in it and glue the ends together. If you were an ant walking about on this you’d be aware that like any surface it has two dimensions (‘along’ the strip and ‘across’ it, if you like) but you’d also find that you can reach the entire surface of the strip - both the ‘back’ and the ‘front’ - without having to walk over the sharp edge - so really there is only one surface. It manages this by having the twist in the third dimension - the one the ant, which has to stay on the paper, can’t access. Well a Klein bottle is a surface which curves in three dimensions and has a ‘twist’ in the fourth. Now we are like the ant - we can’t access the dimension with the twist in - but we can have the experience of getting from the ‘inside’ of the bottle to the ‘outside’ without having to cross over any ‘lip’.
A Berkeley mathematician, Cliff Stoll, has set up an excellent, quirky, funny website about Klein bottles here http://www.kleinbottle.com/. He also makes them (in a sense) and sells them. I keep meaning to buy one. They’re not really expensive and might be a useful thing to pull out of your pocket if you ever want to create a bit of personal space at a party (watch everyone make their excuses and back away …).