Try visualizing a spinning sphere in the 4th D.. Way to easy with your talent, it looks like a sphere. My camera set to long exposer sees your 4th D with ease. Do you include perspective in your view of the 4th D. Should future time not be smaller even out of focus. Then again maybe you have a very unique ability to actually see the 3d projection of a 4D object. One could learn this skill with computer aided visualizations but you are still restricted by the projection matrix. How do you define a 4D lens? Mathematics and psychology don't mix well.
So, can you answer the riddle about how many different 4d shapes can be constructed with exactly 4 hypercubes? It's not a large number...
well this riddle is a bit open ended .. for instance can the shapes be redundant shapes but different in terms of direction? do the faces have to actually touch ? lets say we make sure faces have to touch that still leaves us a cube that is in essense 4 cubes .. each cube has 6 faces thats 24 possible faces each x 4 is almost 100 possible combinations (although this is also reduced if you take out redundant shapes due to direction) .. thats alot of shapes to visualize ... and overly tedious i gave up drawing them at about 20 something .. and im not a math wiz just a visualization person .. so yes i give up on the tedium .
well it all depends on where you are looking , how am i restricted by the projection matrix ? im very curious about this. well there are a couple ways to define a 4d lens .. for instance people are 4d lenses .. because they see objects 3dimensionally and store this information in memory .. so you can for instance see the progression of a 3dimensional object as you live.. a building being built and being demolished.. you can as a 4d lens go back and forth through your memory seeing individual slices of this 4d object. a 4d camera . would be maybe a stereoscopic camera , with a very long exposure time.. something that would allow someone to see 3 dimensions and 1 time. you could also have a 4d lens which does a a spheroidal laser scan of its surroundings and stores each as a 3dimensional frame in a hardrive or something.. and this way you could make a holographic movie. . i think a 5dimensional (2 time 3 dimensions) lens would be neat.. one that would allow you to see a multiverses floating around and each universe that you could see with the lens would have multiple histories connected to it .. because the objects are 3 dimensional and have 2 dimensions you would be able to see a sortof .. wave or topography of the multiple histories of a possible universe.. so universes would be split up into seperate things and the alternate histories of each universe would only interconnect between itself. and interestingly also you would be able to tell which universes were 5 dimensional and which were 4 simply by looking at how fat they are. well they do a little bit.. for instance creating computer graphics requires a little bit of math and a little bit of psychology
Sorry, I should have been more specific. Each shape is made of 4 hypercubes, attached squarely together - every hypercube must have at least one block (six faces) coincident with a block of another hypercube in the shape. I hadn't thought of only connecting faces - that would be unstable, like connecting cubes at the edges. If a shape can be rotated to look like another shape, then those two shapes are considered to be the same. If a shape can be reflected but not rotated to look like another shape, then those two shapes are considered to be different. There are less than 20.
You still seem hung up on time as the 4th dimension... are you aware that when people talk about the difficulty of visualizing 4D objects, they're talking about 4 spatial dimensions?
well assuming what you meant by six faces be coincident with eachother meaning that one of the possible cubes in a hypercube are in the same position as another hypercubes cube.. then i would say 7 because your sortof limiting the possibilities that the extra dimensions allow into essentially a 3 dimensional space.. i dont see why your so hung up on thinking of time and space as seperate things .. in my mind atleast in this context they are virtually the same thing .. also to answer an old question how to measure time in a 4d object with 1 time dimension .. it would be like any current method of measurment .. that is to say it will completely depend on how fine you want to cut time up into artificial pieces.. in essence there is no true way to measure time because it cannot be quantitized.. . its like pi you will never find a pattern to quantitize it .. time is infinitely smooth .. just like a perfect circle is infinitely circular.
Very good. Interestingly, the number of 4D shapes that can be made from exactly four hypercubes is one less than the number of 3D shapes that can be made from exactly four cubes. What about five hypercubes? The number of 4D shapes that can be made from exactly five hypercubes (same restrictions as above) is much more than the number of 3D shapes that can be made from exactly five cubes (27). Why? What's different in this situation? Is this situation still readily visualizable? Which leads to confusion, unless you're comfortable with Minkowski Space. For example: This is an example of confusion caused by trying to treat time as a spatial dimension. Unless you're comfortable with things blinking in and out of existence, changing shape and position as they do so, time isn't a good thing to use for 4D thinking. Particularly as 4D becomes more interesting when thinking about a 4D object in motion (particularly rotation), which can't be done if time is your 4th dimension.
okay, i know this may have come up in numerous conversations over the years, but WHAT IF the universe is a 4-D sphere, where the big bang is the 4-D center and the present is at a certain 4-D radius and the future is at a larger 4-D radius. Then travelling toward the 4-D center is moving back in time, and freezing the 4-D expansion of space actually halts time. Just a thought. Please Register or Log in to view the hidden image!
Is this how 4d objects stack? I'm reading this to mean that one of the 3d cubes of one hypercube overlap a 3d cube from another hypercube. That's an odd concept. I suppose you could say when you stack regular cubes that a 2d square from one cube overlaps a 2d square from another cube. But there is actually a space between them, even if extremely small. How is this represented in 4d? Do you have a link to a rough visualization of this?
According to Savov's theory of interaction 3D is the smallest number of dimension in which the all-building interaction remains always finite and so exists.
Here is a fun idea. Imagine you can move through time at any rate you want fowards or backwards. But restrict your motion through one of the spacial dimensions so that you can only move foward in it. Speeds are then measured by the amount of distance in this spacial dimension you have passed through. For example suppose I went backwards in time at a rate of 20 seconds per meter. I would be seeing different cross sections of the universe at different times. What's the point? Well I was thinking about the arrow of time and thought maybe I could understand it better if I switched which dimension we were passing through. I thought it kind of fit here since this is about visualization and all. I was going to make equations to transfrom moving objects like this as well. I'll get to work on that, then maybep people will understand what I am talking about.
From what I remember in my physics readings, there's a theory out there that mentions quanta of time. So time is not necessarily infinitely smooth. And I really wouldn't doubt that there exist quanta of time, if you think of the universe as a step function, stepping from one state to the next, it is easy to visualize quantum, or the shortest time possible.
well to make see that dimension is actually more than what you are perceiving .it was Lebesgue who gave dimension it's modern definition. in essence he defined dimension as a continuous function. remember the general concept of dimension does not have specific names for each dimension. so when you talk about 4-D forget about words like TIME and space and distance. as humans our eyes are very primitive, they only see 2 dimensions, the 3rd dimesion is a learned behavior, and to learn 4th dimension requires that you forget about the concept of right and left which is a very difficult for humans to do and after hard work it usually can be done in very short intervals. why is that the case, well you need to imagine your body floating in space and the moment you achieve that then you tend to try to hold on to something because you think you are going to fall so are back to 3-D i mmediately. now when you try to a 4-D object crossing your 3-D world is like adding of cross sections to get the full image unfortunately if your brain does not have that image already stored in your brain you see nothing. have you ever experience turning your head because you saw or you felt a shadow that is the crossing of a 4-D object that your brain was able to recognize. although we live in 3-D we really have movement in 2-D. so goto your dreams to experience how we really move in 3-D so do you feel the amount of fredom and movement you have now imagine if were able to move on to the 4-D. in 2-D to see your entire home you will have to walk through every room to see it all. in 3-D you can fly through every room to see it all , in 4-D you do not have to move at all to see it all. to really see 4-D objects it requires that your brain add up an infinite number of cross sections very fast and very quickly without any hesitation, a task almost impossible for humans to do. i hope this helps you.
and extra note. the 4-D can be interpredted in many particular types of problems 1) the Euclidean way (a hypercube, hypersphere) in other words your regular geometic extensions of the circle , the square, and so on. 2) as the coordinate system for imaginary equations such as (X+ i)2=0, so you let x=real, y=i, z=-real and w=-i 3) x=lenght, y=widht, z=depth and d=time 4)as defined by hamilton, he define the properties that would make a 4-D system be a field . in other words it has an identity, it has inverses except for 0., the associative property, the distributibe property and ..... property hold. and so on . but be careful here the definition of inverses is not like in the real numbers (it's more like a hyperbolic coordinate system) so you see that these are specific usage of a 4-D system, but trying to see 4-D objects in general has nothing to do with these specific definitions of the 4tth dimension in general. a good way see the fourth dimension is to draw a cube in 3-D but instead of using dotted lines to connect the sides that you do not see, use solid lines. and stare at it . What do you see? you should be able to see a box sometimes you see the bottom sometimes you see the inside, guess what that flipping of the box occurs in 4-D. not possible in 3-D unless you move the box up and down with your own hands, but if that movement happens just by staring at it . that can only happen if the flips in 4-D. easy example of seeing the power of 4-D.. ENJOY
For a truly unique visual experience of 4D Hypershift, check out my gallery/hyperspace website at www.okapistudio.netfirms.com (Specifically, the Hyperspace Gallery and the right-hand page links to hyperspace diagrams). There is also a free download of a neat little Spin Ship. patty (artist)
I have absolutely no problems imaging a 4D object in my mind, Its like a over exposed photograph. The only thing I dont get is why should time flow in forward or backward directions , can time travel in any other directions too ?
I once toyed with four dimensional tic-tac-toe. It was a normal tictactoe diagram, but each cell was split into three sub-cells. It was played with crayons. Three x's in a row would be like this: red x green x blue x all in the same cell (different subcells) or, red x green x blue x in three cells in a row - the color order matters (reverse okay) or, red x red x red x in three cells in a row (or three blue, or three green) Since the color order matters, it represents a well-ordering of the fourth dimension. Think "red is past, green is present, blue is future". In this sense, any quality like color or smell can be made to represent a dimension, assuming an order is placed on the different values. Just thought you might like another mathematician's perspective. Aaron
