Attempts to visualizing 4 depth

[Secret]

I always have a interest in higher dimensions and want to one day travel within them when i was 7
But browse through 4D related materials just recently as I cannot understand the meaning of dimensions at a younger age.

After nearly a year of imersing into 4D games, projections ,info etc.
I have the following (supposed) progress

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

(Progress 2 ?) Theory
one day when I was messing with tesseract drawings I have the following idea
http://img143.imageshack.us/img143/1149/tesp.jpg
Large image replaced with link ~Keiji

Fig. 1 A Hand drawn tesseract projected into 3D then to 2D using mspaint
White: the 3D axis
Pink: the w axis
Yellow cube: 3D "card" placed at the center of the tesseract

My theory: Room shifting theory

Moving in 3D in the tesseract is indicated by the white axis
Moving ana kata only is like a 3D person (in this case a point in the yellow cube) get temporary locked in place in the x y z directions within the yellow cube, and then the person+cube complex shift along the ana kata directions (Using the figure, it is like as if the extra 8 lines that produces when you extrude a cube into a tesseract act as rails where the middle yellow cube is confined in and can only move "forward" or "backward" in the figure) (Of course the yellow cube never moved!, it's just an interpretation)

So moving in all 4 axis is like the person is free to roam around within the yellow cube and with the yellow cube+person moving to and fro within the tesseract framework
Then I tried "dry running" the process
And the results indicates that I can access to any points in the tessseract (projection)

I tried putting the theory further into a penteract and hexaract and it works well

BUT my question is, Is this really the correct interpretation of the 4 volume and 4 depth?
As I haven't learned higher dimensional geometry and matrixes yet, I can't check it myself

P.S. Progress yet to be achieved
a. Would not get confused when rotating a tesseract (projection) simutaneously in 3D and 4D
b. Able to draw a meaningful projection of a 3 sphere and higher
c. Understand complex 4D shapes or higher such as cublinder. sprinder. duocylinder etc. when they tumble in their space (and projected back to 3D)

P.S.2
Also don't say it is impossible to perceive 4D unless you have a solid mathematical proof
i believe it's impossible to travel into 4D if it doesn't exist, but that's another issue
4D = tetraspace = spatial 4D
=/= Time

[Keiji]

Finally, someone new who understands it properly we need more people like you. Welcome to the forum!

The cube-in-tesseract thing reminded me of a print head. The head moves in one dimension along the paper's width, and the paper moves in another dimension to let the head access its entire length, allowing two dimensions to be accessed. Here, you could consider the tesseract the paper, and have the head move in three dimensions instead of just one, so four dimensions can be accessed.

I wouldn't really call it a theory though, more of an interpretation.

[Hugh]

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

Wow! Thank you for this!

After clicking on the links and seeing how the yellow dots move around on the 2D plane with the 3D cube, I can actually see the 3D cube rotating around in the tetracube as it should rather than it turning "inside out" on the 2D plane.

My theory: Room shifting

This idea sounds similar to what I experience with Visual Reorientation Illusions (VRIs), which I've been trying to explain using higher spatial dimensions for a long time.

What I've been lacking is a better way to understand the rotation of the 3D limited cube viewpoints and how they fit together.

This gives me food for thought.

Thanks Secret for posting your thoughts, and welcome to the forum.

[Secret]

Attempt 1: Visualize the 3 torus

(Made entirely in powerpoint and by hand (few hours making time, a week thinking and checking time))
http://img831.imageshack.us/img831/567/ftuf.png (Side view (from ???))
http://img709.imageshack.us/img709/435/14432638.png (Ana view???)
http://img843.imageshack.us/img843/7679/ftvn.png (??? view)
Descriptions:
1. White translucent 2-torii are 4D empty space
2. Solid white 2-torus is 3D empty space
3. Pale purple 2-torus in the middle is a punctured 2-torus (2nd step of the 3-torus gluing)
4. Purple 2-torii are solid 4-volume of the 3-torus

Gluing attempt:
http://img841.imageshack.us/img841/4674/50859784.jpg
(pale semi translucent purple parts are the sticky ends)

if our universe is a 3 torus, this is what it'll look like

(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]

Your last picture is really cool, and exactly right, as far as I can tell. It's very difficult to visualise a 3-torus from the outside, but actually quite easy to imagine it from the inside. It's a bit like being in a room full of mirrors, except your image in each mirror gets further away as you walk towards it.

Don't worry too much about ways to visualise things. Visual thinking can be useful, but it's not essential.

(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))

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.

[Secret]

Don't worry too much about ways to visualise things. Visual thinking can be useful, but it's not essential.

Although math formulae are useful. in order to let the general public to understand nerd ideas more easily as well let them to grasp them, visuals are an easier way than math formulae. Besides, to me, what will be facinating about higher dimensions is you can marvel the structure of these alien objects directly, and maniputcating them when you finally have a chance to imerse into their world.

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

http://img443.imageshack.us/img443/2088/16575422.jpg

3 torus from spheration: only one 2 torus 4-hole (4D empty space)
3 torus from gluing: 1 2 torus 3-hole and 2 2-torus 4-hole

So the 3-torus from gluing is much porous than the one from spheration

[PWrong]

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.

[Secret]

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.

After another few hours of paperwork, it seems that the difference between the glued and the spherated 3-torus are their uniform cross sections
http://img837.imageshack.us/img837/7135/31944622.jpg

here's some of my paperwork when i tried to "fold" the 3 torus
http://img340.imageshack.us/img340/2923/26874247.png

Will work on the equations later cause I need the knowledge of quanternions to do so

[PWrong]

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. That way the resulting shape is 3D and embedded in 4D.

[Secret]

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.

Details in this image
http://img186.imageshack.us/img186/1015/31439107.jpg
Yup I've no idea how to name these 2-torus varieties

That way the resulting shape is 3D and embedded in 4D

Umm... shouldn't the 3-torus a completely 4D shape?

Also to prevent the topic getting bland, I'll start to post my insights/ideas of 4D slicing seen from the outside

My paperwork on 4D slicing
http://img814.imageshack.us/img814/1015/88012633.png
(better and more convincing graphics will arrive once a suitable software is found for drawing them)

Btw, powerpoint no longer can satisfy the requirements in making convincing 3D/4D images. any freeware or shareware can suggest to me that can draw translucent 3D shapes?

[PWrong]

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.

Umm... shouldn't the 3-torus a completely 4D shape?

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.

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.

[Secret]

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.

Understood
Well I guess I've to pm some reminder after 1-2 weeks in case you forgot about them
Then you can check your pms to see what you've missed (might post pms of some potentially worth investigating topics of some other users)

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.

yup that's the difference between a topological dimension and a (forgot name) dimension of a object
For the duocylinder, rough paperwork shows that it is indeed a 2-torus (with it's center plugged) which got servely flattened in the 4th dimension. Further paperwork or understanding is needed to fully understand it's structure

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.

Yup the frames are quite confusing
So I've a suggested glossary

1. Verticed: Only points are shown, nothing else (seldom use this at least for me)
2. Wireframe: only edges are drawn, no surfaces or higher dimensional elements
3. Hollow: For a convex n dimensional object, it is composed of (n-1) elements of negligible nth dimensional thickness enclosing a nD space
4. Rigid: Same as 3. except all (n-1) elements have non negligible thickness
5. Solid: Same as 3. except the entire nD space within is filled up

e.g.
Verticed cube: 8 points arranged in a cubic shape
Wireframe cube: A series of lines forming a cubic shape
Hollow cube: 6 squares folded into a cube, which the inside is empty
Rigid cube: Like a crate, which it's sides have some thickness
Solid cube: Like a bean curd, no empty space within

Btw any good 3D drawing software (best if it is shareware or freeware) that can draw transparent 3D shapes? Cause powerpoint cannot support drawing complicated shapes without losing the 3D illusion

[PWrong]

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.

[Secret]

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.

Very sorry, I've checked the threads, but are utterly confused of the maths notations (as I haven't learnt topology before), thus I can't use the suggested notation...

therefore i can only continue on my notations....

Rigid: A solid nD shape with it's insides scooped empty into an empty space with the same shape as the original object but smaller
Hollow: A rigid nD shape which it's shape of the empty space is congrunt (exactly the same) to the original shape
Thick wireframe: A nD wireframe shape with it's edges (lines) thickened into n-prisms

[PWrong]

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.

[Secret]

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.

Damn
My notation cannot handle the only squares tesseracts
Also I still yet to understand the duocylinder's structure completely (This thing is more complicated than the 3-torus to me)

hmm....
It seems I've to find a way to learn the higher dimension geometry in order to continue on the 4D investigations....
Actually , I 'm trying to perceive 4D and higher dimensions with little help from the maths
Since most people are afraid of maths, I"m trying to find a simple way to let people understand 4D

Maybe i'm wrong all the time , but idk
But this journey in reading 4D stuffs from the internet (including the alkaline site) does gives me a richer understanding of 4D interpretation
I can 95% understand how 8 cubes fold into a tesseract and how 5 tetrahedron folded into a pentachoron
Or rotations by a plane in general

My interpretation on the folding:
Imagine these 7 square faces of this figure snipped from this website
http://www.learner.org/courses/mathillu ... dimension/

when folding, are going outward of your computer screen.
You'll see how actually the 8 cubes sticked together in a hyper perspective
(The final fold of the top cube is a little incorrect as it should never break into pieces)
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
(If the interpretation is wrong, please tell me, causes i still lack the maths to check my interpretations)

[PWrong]

Have you seen the 4D rubik's cube? I almost solved it once.
http://www.superliminal.com/cube/cube.htm

[Secret]

Have you seen the 4D rubik's cube? I almost solved it once.
http://www.superliminal.com/cube/cube.htm

Although I know how it works, I suck at rubik's cubes
I can't even solve the standard 3D one

[Hugh]

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

I agree, and this is what I've always thought was the 4D explanation for our experience of Visual Reorientation Illusions (VRIs), where everything in our 3D spatial orientation gets flipped around in an instant (you may have heard of people getting "turned around" in their bearing sense), because we may actually be 4D but with only a limited 3D cube spatial awareness.

There is a 3D necker cube type of flip that would happen in actual 4D space, because of the shared common 2D planes of our 3D cube orientational viewpoint.

[Keiji]

yup that's the difference between a topological dimension and a (forgot name) dimension of a object

wireframe, sheetframe, swockframe

Sorry to intrude on your discussion (which is very interesting by the way ), but the terms used on the wiki are:
* 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)

[Secret]

* 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)

thanks for the notations, i'll make use of them in the future

Also here's the promised half folded tesseract
http://img824.imageshack.us/img824/7573 ... seract.jpg
The net of the tesseract
http://img178.imageshack.us/img178/9558/half1t.jpg
(4D?)Bird's eye view of a folding tesseract
http://img693.imageshack.us/img693/4803 ... ctfold.jpg
The whole process, note that the net you see earlier is actually 2D (i.e. flattened)

(Above are results of some paperwork modeled using some 3D software)
Above simulated 4D (3D->2D projections) are based on the necker cube hypothesis (made that up)

Necker cube hypothesis:
Any 3D bounded and netted shapes are 'flattened' when placed into 4D space, and with their 3D interior and exterior become ambigurous

[PWrong]

The half-folded net is really cool, I haven't seen something like that before.

The usual name for the bounding space is the embedding space, but that's a bit more general.
http://en.wikipedia.org/wiki/Embedding

monoframe, diframe etc does make more sense. I liked wireframe simply because everyone knows what that means, but sheetframe and swockframe obviously don't have that advantage. Apparently they're sometimes called n-skeletons, but that terminology is only used for simplicial complexes.

For a general shape, you can decrease the frame or skeleton number by 1 by taking all the points at which the shape isn't smooth (differentiable infinitely many times).

[Secret]

More 4 dimension insights:

By Necker Cube Hypothesis, you can 'fold' any 4D shapes you want

http://img521.imageshack.us/img521/6803/4dfolding.jpg
Swock and paper folding

http://img529.imageshack.us/img529/4803 ... ctfold.jpg
Enhanced tesseract folding as seen in xyz view
http://img835.imageshack.us/img835/4803 ... ctfold.jpg
The same process in full 4D (faked) view
http://img151.imageshack.us/img151/9297 ... oronfo.jpg
Xyz view of pentachoron folding
http://img139.imageshack.us/img139/9297 ... oronfo.jpg
Full 4D view of pentachron folding

Made using google sketchup

P.S. Just checked the paperwork, actually the 3-torus looks sort of like a 3-frame 2-torus (The punctured solid 2-torus) with its cross sections 2-frame 2-torii
P.S. 2 Still working on the xzw view of the duocylinder
P.S. 3 Cross section seen from outside cooming soon

[Hugh]

Here's a great page with similar animations in movie form which you might enjoy.

http://alem3d.obidos.org/en/cuberot/movfold

Quoted from that page:

This movie shows an unfolded cube (red) in space, together with its shadow (pink) on the plane below. The white dot at the top of the image represents the light source. As the red cube folds up, we can follow the results in the shadow below. As the sides of the cube begin to fold up, the shadows of these squares become distorted (one edge is closer to the light than the opposite edge, so one edge has a larger shadow than the other). As the edges of the squares come together in space, so their images come together below. Then the top folds over to complete the cube. In doing so, we see the image get larger (as it moves close to the light), and turn inside out as we go from seeing one side to seeing the other side of the top square. As the top closes up, its shadow forms the well-known "square within a square" view of the cube in perspective.

At this point, we rotate the whole arrangement so that we see only the shadow of the cube and must imagine the three-dimensional cube unfolding that causes these shadows. This is good practice for visualizing the hypercube folding up, as seen in the movie below.



This movie shows the analogous sequence of three-dimensional shadows of a hypercube folding up in four-space. Just as we can visualize the cube folding in space using just its shadow (as is done at the end of the previous movie), we must use these thre-dimensional shadows to try to imagine the hypercube folding up in four dimension.

We begin with eight cubes forming a cross-like shape. Some faces are partially removed to make the interior structure easier to see. The central (yellow) cube will be the bottom of the hypercube, and the purple one will be the top. The remaining six cubes form the faces of the hypercube that join the bottom to the top. As these begin to fold up in the fourth dimension, we see their shadows become distorted in three dimensions (as one face of the cubes moves closer to the light source, its shadow get larger). Eventually, the faces of the cubes come together and are joined, just as the edges of the squares that form a cube are glued when they are folded together. This leaves just the top remaining to fold into place.

When the top folds over, it gets larger (closer to the light source), and eventually seems to turn inside out (we go from seeing one side of it to seeing the other). As it closes in to join the six other faces, we are left with the well-known "cube within a cube" view of the hypercube in perspective. The small yellow cube is farthest from the light source, while the large purple one is closest. The remaining six faces appear as truncated pyramids joining these two; these are views of cubes in (four-dimensional) perspective.

[Hugh]

Here's another picture of a folding hypercube I saw a while back...

[Secret]

Actually i've seen those before
Also in order to produce a more convincing 4D illusion, the final cube side of the tesseract should NEVER go pass through the other cubes while folding. (else it's just the shadow)
In 4D, 3D objects are essentially 'flat', hence the Necker cube hypothesis

Btw, here's the 3D slides viewed from 'outside'



Consider a 3D blade slicing across the tesseract:
Right pic: cross section
Left pic : cross section from the corner

To work out the cross section:
Notice that you can consider the entire cuboid blade (3-plane) as 3 orthogonal pairs of 2D blades slicing the different cubic sides of the tesseract. By consider the 2D blades and each cubic sides one by one, you can work out the partial 2D cross section formed on the cubes. By joining all partial cross sections together, you can obtain the 3D cross section
In general, (i think) you can do this treatment to different shapes of the slicing blade and the object being sliced

To work out the orientation of the slicing 3-plane/3D blade from the 3D cross sections
Work on the 2D sides of the 3D cross section one by one, extend the sides to produce a 2D plane coplanar with the side. Repeat on other sides of the 3D cross section. the 3D shape formed by the intersections of the constructed planes shows the orientation of the 3-plane in 4 space

I never tested that on all other shapes, and the tetrahedron cross section is a little inaccurate. Correct me if my theory turns out to be wrong, thanks!

[Secret]

A link of a VR project of a university found by googling
Some sections are about visualizing 4D
Check out the 3-torus wireframe there
http://www710.univ-lyon1.fr/~sbrandel/en/research/VR/

[Meloxicam]

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.

[Secret]

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.

Spacial dimension is also perception
That's why it is theoretically possible to 'see' higher spacial dimensions from lower dimensional drawings/projection
All we need is just the 'device', the correct method to interpret nD depth from (n-m)D drawings/projections
Then we can see them
Of course science/nature always surprises us with unexpected things
Until that day we discover macroscopic higher dimensions, we'll treat the mathematical ones as 'real'
and try to perceive them.

In other news:::

A 'tangent' on a line is a point (???)
A tangent on a plane/curve is a line
A tangent on a solid or 3D curve is a line or a plane
Question:
What does a 'tangent volume' look like on a 4D object, e.g. 3 sphere
Also how will it look like in some massively enhanced (depth curing, hidden surface removal etc.) perspective/parallel 3D->2D projection?

[Secret]

http://www.hipercubo.mobi/

while messing with the tesseract in this game, I noticed if I consider the trapezoid squashed cell at the very far side of the projection instead of the center cell as the furthest cell in 4D, then for a brief moment i see the tesseract becomes hollow

wonder if this is an illusion or is real?