Secret wrote:Progress 1
After tried many tesseract applets, i manage to interpret the 4D rotation properly (i.e. not seeing the tesseract turning inside out but instead rotating normally)
I Thought my interpretation is wrong at first but after watching this http://www.math.uconn.edu/~rogalski/4d/html/hypercube/interpreting1.htm I know that this is the correct interpretation
Secret wrote:My theory: Room shifting
(I might be wrong cause this extremely porous figure doesn't match with the one produced by spherating a 2-torus (which its shape is the same as the torus in the 2nd step of the gluing process))
PWrong wrote:Don't worry too much about ways to visualise things. Visual thinking can be useful, but it's not essential.
PWrong wrote:I'm not sure what you mean. 3-torus and spherated 2-torus should definitely be the same shape, but they might not look the same, depending on how you produced these images
PWrong wrote:I don't know what you mean by more porous. The shape on the bottom left is exactly right. You'll notice that if you follow one of the small black circles you'll get back to where you started. Same thing happens if you follow a large circle, or if you go all the way around. So it's the same as being in a cube with all opposite faces identified. The two pictures at the top don't look that dissimilar. I'm sure if you write down the equations for each they'll be equivalent.
PWrong wrote:I think I see the problem. You shouldn't have a solid 2-torus in the cross section, however you slice it. If the cross section is 3D and embedded in 3D, the original shape would be 4D and embedded in 4D. You want 3D embedded in 4D.
Are you spherating a hollow torus with a disk, or spherating a solid torus with a circle or something? You should be spherating a hollow torus with a circle.
PWrong wrote:That way the resulting shape is 3D and embedded in 4D
Umm... shouldn't the 3-torus a completely 4D shape?
PWrong wrote:Sorry secret, I tend to see a post and think "I'll reply to that later", especially when it's one that requires a lot of thought. Then I forget that it's there. The other posts tend to be easy questions or troll posts so it's much easier to write a quick reply. It's nothing personal.
PWrong wrote:We tend to distinguish between the dimension of a shape and the dimension it's embedded in. For example a sphere is a two dimensional manifold (longitude and latitude). That's why mathematicians call it a 2-sphere. But if you embed it in something, the embedding space is three dimensional. A space doesn't necessarily have to be embedded in Euclidean space at all.
The 3-torus has 3 dimensions, but exists in 4D space. If you fill it in, you get a solid 3-torus, which is 4D. The same is true for most 4D shapes, except the duocylinder which is 2D in 4D.
PWrong wrote:Sometimes it's not clear what frame we're talking about. For example if I say cube, you don't know if I'm talking about a solid object, a hollow cube, a wireframe, or 8 points.
PWrong wrote:I think we just called them pointframe, wireframe, sheetframe, swockframe, and generally n-frame. We also had minframe and maxframe. For example a minframe cylinder is a pair of circles. There's threads about this stuff in the geometry boards. Most of them are about homology groups (holes) of these things but I think there's a thread explaining frames and inventing a notation. I also invented an operator, which takes a k-frame shape and turns it into the (k-1)-frame shape. It basically just gives the set of non-smooth points in the space.
I like the idea of "rigid", but it could be more general. You could take a wireframe cube and make the wires thick. In general what you're doing is spherating by a solid interval.
PWrong wrote:So what would you call a duocylinder in your notation? Or a hypercube with just the squares filled in? In my notation the regular duocylinder is 2-frame or minframe. There are also 3-frame and 4-frame duocylinders.
I don't know of any software, I never use stuff like that. I like seeing the pictures but half the time I don't understand what they're supposed to be. I can get more out of a simple diagram or an equation.
PWrong wrote:Have you seen the 4D rubik's cube? I almost solved it once.
http://www.superliminal.com/cube/cube.htm
Secret wrote:From this interpretation, a 3D cube placed in 4D is like a necker's cube, where you'll see it's top constantly switching positions
Secret wrote:yup that's the difference between a topological dimension and a (forgot name) dimension of a object
wireframe, sheetframe, swockframe
Keiji wrote:* net space, the number of dimensions needed to define a point on the surface of the object (e.g. a sphere would have net space 2)
* bounding space, the number of dimensions needed to embed the object in Euclidean space (e.g. a sphere would have bounding space 3)
* nullframe, monoframe, diframe, triframe, tetraframe, etc., which define the net space (e.g. a monoframe cube has net space 1 because it's monoframe, but bounding space 3 because a cube is 3D)
Meloxicam wrote:The problem with visualizing 4D is the same as with visualizing 2D.People that try to understand higher dimensions are greatly overlooking the lower ones.It's easier to understand 2D so let's start with that.If 2D doesn't have any height,how do you visualize it?You don't,exactly the same with 4d,you don't.Drawing a 2D picture is still 3D in a sense cause the 2D picture is on a 3D surface.Perception is the key to it all,and it should explain that all dimensions are in this universe.
An earthworm that does not have any eyes,still operates in a 3D realm for us.The way he perceives it,is impossible for us to imagine,the same way it would be impossible to imagine what a 1 celled organism perceives.We can't even be sure how a 1 celled organism perceives this reality,it could even be that they perceive more dimensions than us.But it would be easier to imagine if you had eyes on every side of your head,capable of looking into all directions at the same time.Or eyes at the tips of your fingertips,functioning properly of course.Or even eyes inside yourself,and the outside,and if you could see future and past at the same time.If you could see time,if time even exists that is.
That ladies and gentlemen,is what dimensions are,perception.I have often thought,what it would be like if we were able to see radio signals or sound waves.When coming to the conclusion that most particles can behave not only physically,but also as a wave.Imagine if we saw sound and heard matter,can you imagine it?...i don't think so.We can understand so little about this universe,but it's quite certain that reality is a lot more than it seems.Technology is what allows us to comprehend these higher "dimensions".
You can't see the infrared spectrum,UV or hear low frequency/high frequency above our capacity.But we have complex interdimensional stargates(devices) that interpret this information into a form that we understand.Like a TV screen that can turn signals captured with an antenna,into pictures that we see visually.
To put it more simply,energy,information,matter however you like to call it,is constantly changing it's shape and form and can be experienced or perceived in an infinite number of ways.That is in my opinion what defines dimensions.Not to mention i just experienced Deja Vu,which doesn't physically even exist.The more and more science discovers,the more and more questions it will find.Simply because our perception of reality/information around us is constantly changing due to new discoveries and findings.Order is made out of chaos and vice versa,somehow our universe maintains a balance between it all.
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