I’ve read pieces on 2 dimensional worlds and how relating them to 3-d worlds can help to imagine what a 4-d world would look like? Are there any website anyone knows of that go into a bit more depth on the subject?
I could recommend you some good books on the subject. Classics are a 1884 novel by E.A. Abbott, Flatland. A Romance of many Dimensions (copyright expired, can be read online here) and C.H. Hinton’s An Episode of Flatland (1907), which are both set in a two-dimensional world.
Many non-fiction books address the topic; it’s done pretty well in J. Richard Gott’s Time Travel in Einstein’s Universe, which is full of interesting stuff but sometimes a bit hard to comprehend (at least to me).
This site has a good collection of links regarding hyperdimensional stuff.
Does a blind person ‘see’ in 4D? I only ask because they would be able to feel all the side of a cube at the same time. What would happen if they tried to draw it?
But surely a cube has three dimensions, not four?
Julie
Besides the books Schnitte has mentioned, you might also try Sphereland and (if you can find it) A.K. Dewdney’s The Planiverse, which takes Abbot’s “Flatland” idea and tries to imagine a real 2-D world with physics, chemistry, and viology, not just geometrical figures. (How do you make a 2D character with a digestive system without having it fall apart in two pieces, bisected by its esophagus?)
Google on “Hinton” and “cubes”. I first read about them in Martin Gardners “Mathematical Carnival”. They are a set of cubes developed by Charles Hinton that, supposedly, would help one visualize the 4th dimension.
Whoa!, hold on right there; we ALREADY live in a four dimensional universe… or, at least we can persive four dimensions of it, three spatial plus time.
But we don’t perceive time. We don’t have a ‘time sense’, we use our other senses to figure out how time passes.
this is getting a bit more interesting now. so what about using a 3d perspective of 2d to better understand a 4d perspective of 3d. is this physically possible? can such observations be interchanged so freely?
I don’t exactly understand what you are asking. However, if there are only three dimensions, you can’t perceive a fourth one because it doesn’t exist, of course.
It’s about imagining a twodeminsional world and the perspective of a person living in it (and the problems this person has when trying to imagine a threedimensional world), which could help us threedimensional beings imagining a fourdimensional world.
I’ve tried to visualize a fourth spatial dimension, and I can’t do it. You can, however, visualize the projection (or “shadow” ( of a 4-D object in 3-D space. (In fact some electronic orbitasls are apparently different projections of a 4-D shape into a 3-D universe). By analogy, a 2-D creature can’t visualize a cube, but it can view the shadow of a cube in 2-space. It’s sometimes a square, sometimes a rectangle, sometimes a hexagon, etc.
Similarly, you can “unfold” a 4-D cube (tesseract) in 3-D space, in various ways. Robert Heinlein described such a situation in his short story “…and he Built a Crooked House”. Salvador Dali did a painting of Christ crucified on such an unfolded tesseract. In an analogous way, a 2-D being can look at a cube “unfolded” in 2-space.
Finally, you can analyze 4-D shapes mathematically by noting progressions. A 0-D “cube” is a single point. A 1-D “cube” is a line, characterized by 2 points. A 2-D cube is a square, characterized by four points and four lines. A 3-D cube is characterized by eight points and 6 faces and 12 lines. So a 4-D cube (generated by translating a cube through the fourth dimension, perpendicular to the previous three the distance of one line) is characterized by 16 points and has 8 cubes for its “sides” (you can see them in Dali’s painting).
You can derives similar properties for a 4-D hypersphere or a 4-D Equilateral Triangle/Tetrahedron".
A good book on how comparisons between 2-d and 3-d can allow one to reason by analogy about what a 4-d world would be like is The Fourth Dimension by Rudy Rucker. Among other things, he has an illustration which shows (sort of) where the tennis balls went after they disappeared in the closet in the movie Poltergeist.
Rucker has his own website.
Rucker has observed that the person who came closest to truly picturing four dimensions was the mathematician Charles Hinton. Among other accomplishments, Hinton developed a memory system in which a person learns to imagine an object as being filled with small cubes which have been given names. With practice, one is supposed to be able to accurately imagine the object in any position. A man who used Hinton’s system said that it condemned him “to a living hell”.
Two stories which won’t really help one understand the possibility of a fourth dimension of space, but which are a lot of fun anyway, are There Was a Crooked Man by Robert Heinlein and The Plattner Experiment by H.G. Wells. I recall that Dorothy Sayers also discusses the idea briefly in one of the better stories about Lord Peter Whimsy.
Whoops. CalMeachumis correct: Heinlein’s story is called …and He Built a Crooked House.