Why Humans Will Never Understand 4D Space

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Re: Why Humans Will Never Understand 4D Space

Postby Hugh » Wed Mar 08, 2017 7:51 am

Klitzing wrote:I'd say that slicing the cube is even worse for understanding the cube by mere square sections: the first and the last square actually are faces of the cube, but all the other sections would lay out the inner part of the cube. It is only their sides which lay out sections in turn of the lacing squares of the cube. And this disconnected lower-dimensional sectioning of the lacing elements is what makes the sectional representation so hard to grasp.
--- rk


That's why I think the key in understanding is to sweep the axes to the new perpendicular direction and include all the "new space" in between into the totality of it. :)
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Re: Why Humans Will Never Understand 4D Space

Postby Klitzing » Wed Mar 08, 2017 8:01 am

SteveKlinko wrote:Id like to thank everyone for their comments over the weekend. I want to say a few more things about the Tesseract. It is commonly said that the sides of a Tesseract are made out of cubes. In fact Tesseracts are not made out of 3D cubes like the ones that exist in our 3D Space. The Tesseract cubes have an extra dimension. They are in fact something different than our 3D cubes. They are Flat 3D Objects. If you look at them on edge, in a 4D Space, they will look like 3D cubes. But these cubes have a Flat characteristic that is the main characteristic of a cube in 4D Space. These Flat 3D Objects must be oriented so that their Flat sides form the sides of the Tesseract. They connect to each other at 2D Objects that have a Line like characteristic. Yes 2D Objects in 4D Space have a line like characteristic. Two Tesseract sides connected on these 2D Lines can pivot about each other like they were hinged together. This is why a 2D object can rotate in place like an axis. My Animations show this in some detail. Bottom line of all this is that a Tesseract is a simple box with thin sides in 4D Space. It seems like projections, slices, etc. make them into something more complicated than is needed. I'll have to speak for myself and say that I can not make a Tesseract out of bunch of cubes in 3D Space. Just like the 2Der will never understand a cube by looking at a bunch of squares in his 2D Space. He would have to see the Flatness of a square but he is always looking at the squares on edge. Just in case here is the link to my Animations:

http://www.theintermind.com/ExploringTh ... ations.asp

I'd like to point out some Problem in your terminology. While going up the dimensions you often encounter problems of refering to k-dimensional elements either as 0+k or as d-(d-k), where d is the embedding dimension. In 3D we are well used to mentally distinguish these different usages without any further mentioning. But when transfering according terms into higher dimensions, this natural distinction gets lost completely. Therefore it is much better to use different terms for the respective meanings.

I'd would like to direct you to Wendy's PolyGloss where a very high emphasis on these topics is taken.
Esp. such co-dimensional terms like margin > ridge > peak > spire would be of high value here, I'd think. - Wouldn't you think so too?

--- rk
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Re: Why Humans Will Never Understand 4D Space

Postby Klitzing » Wed Mar 08, 2017 8:28 am

Hugh wrote:In order for the 2Der to understand the fullness of 3D, he has to take his 2D slice of 3D and turn it "all the way around" the 1D line that he understands into a "new direction". He understands forward/back and right/left, but just has to understand that there is a new perpendicular direction to each of these that is available. Once he realizes that, he can appreciate that there is an up/down that any of his axes can be re-oriented in, and understand the "new space" that is made available because of it.

I would not fully support that view. If we 3D-humans look into 3D honeycombs (like the cubical honeycomb x4o3o4o etc.) we would just recognize that it fills all of our perception space. At first glance there is nothing "above" or "below". The same would be for 2D tilings (like the square tiling x4o4o). As such they just fill the flat 2D space. Ant for ant like beings of flat land there would be nothing "above" or "below". But when taking such a tiling in use as boundary element of some 3D tesselation, e.g. when building the according prism, i.e. a structure with just 2 vertex layers, we 3D beings well percept some above and below. And it gets even worse when entering 3D hyperbolic geometries. There not only finite polyhedrons can be used as cells, but also euclidean flat tilings would be allowed as limit tangential cells. And even any hyperbolic tiling with lesser or equal curvature than all space curvature could be used too. That is, in these contexts flat euclidean tilings well behave just like polyhedra, having an outside and an inside. And in fact, we just asign an "above" as outside and a "below" as inside to them, and consider them to have just a limit vanishing curvature towards the below direction.

The same holds true for euclidean 3D honeycombs. We could asign some "above" in ana direction and some "below" in katha direction, and consider them as being limit vanishing curved into katha direction. And right this perception then is what one could think of 4D polychora as well. One simply blends out the outer and inner space, that is the radial direction and reduces the whole structure - like a crawling ant - to its mere 3D surface topology. Then you'll get there some cell complex, which locally is not too different from what one is percepting in 3D honeycombs. Just that the cells are a bit distorted, due to the corresponding curvature of space to be used.

So, you are right, cells, as polychoral building elements, are clearly flat objects within the 4D embeding space. But with respect to mere surface geometry they are full bodied 3D objects. And 3D humans are well suited to understand 3D surface geometry. We do not need to know something about the outside and inside - at least for that purpose of the cells connectivity structure of such 4D polychora.

--- rk
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Re: Why Humans Will Never Understand 4D Space

Postby ICN5D » Wed Mar 08, 2017 10:50 pm

SteveKlinko wrote:In fact Tesseracts are not made out of 3D cubes like the ones that exist in our 3D Space. The Tesseract cubes have an extra dimension. They are in fact something different than our 3D cubes. They are Flat 3D Objects. If you look at them on edge, in a 4D Space, they will look like 3D cubes. But these cubes have a Flat characteristic that is the main characteristic of a cube in 4D Space


Actually, the 8 cube faces on a tesseract are just good ole' regular 3D cubes. There's nothing different about them. Take this image, for example :

Image

This is the well known shadow, or projection of a 4D cube. We can see 8 cubes in this image. There are two, 'normal' looking cubes as the big one on the outside, and little one on the inside. Then, connecting them, are 6 more cubes, that are squished and skewed.

The 2 normal looking cubes are parallel to the 3D wall that the shadow is projected on. The 6 skewed cubes are the ones sticking away, and into the 4th dimension, which join the 'top' and 'bottom' cubes. These 6 cubes are seen edge-on, from a higher dimensional direction, which makes them appear flatter than a normal cube.

This flatness of the cubes you are seeing is not a property of the cubes themselves, but from seeing them edge-on, from a higher dimensional perspective. It's a matter of viewing angle.
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Re: Why Humans Will Never Understand 4D Space

Postby Klitzing » Thu Mar 09, 2017 6:23 am

ICN5D wrote:Image

This is the well known shadow, or projection of a 4D cube. We can see 8 cubes in this image. There are two, 'normal' looking cubes as the big one on the outside, and little one on the inside. Then, connecting them, are 6 more cubes, that are squished and skewed.

In fact, this is nothing but the Schlegel diagram of the tesseract, very similar to the one of the cube, which I showed some posts before. The viewpoint again has be taken close above one of the cubes. Therefore all the further structure becomes projected into that very cell.

Alternatively these Schlegel diagrams allow for a different interpretation: consider the outer cube would be inverted, i.e. being all the outside of the provided diagram/picture. Then you'll get rid of the double cover of the visible "halves". - But that's a matter of taste, of Interpretation of the result. Not of applying the projection.

Sure, being a projection from 4D onto 3D there is no fourth dimension left visible. But, just as ICN5D already pointed out, you can reconstruct that further "depth" from the applied scaling factor within the result: the smaller parts were farer away, the larger parts were nearer to the observer. - Sure this highly depends on all unit edged polychora. (Else this feature would get burried under the different edge length already existing before projection.) - Note that in this picture even the edges themselves (as well as the vertex balls) show up this depth: the nearer parts are depicted fatter, the farer parts are depicted much thinner.

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Re: Why Humans Will Never Understand 4D Space

Postby SteveKlinko » Fri Mar 10, 2017 4:58 pm

Let's look at a 2Der trying to understand the Flatness of a solid Square. To him a Square is only viewed on edge. He sees the Width and Depth of it but, the inside points are always unseeable as they are blocked by all inside points in front of them. A Square is a nice solid object. He knows theoretically that from a 3D point of view he would be able to see all the inside points of his Square at the same time. He also knows theoretically that his Square will look Flat. But he does not have the 3D view advantage that we do. He is unable to travel around into the 3rd dimension and see the Square from all those viewpoints you would get by doing that. Instead he must only theoretically rise out of 2D Space into 3D Space and then theoretically look down on the Square. He can then move accross the Square and make a collection of views. He can then lay them out around in his 2D Space and try to make sense out of it all. If he goes around the Square in a symetric way he will have a collection of Lines. All he can do is see one Line at a time because that's how his Visual system works. The act of theoretically obtaining these views tells him nothing about where that extra direction into 3D Space is. All he has is the collection of Lines but still has no clue how to put them together so that he can see all the points at the same time. It will still boggle his mind as to how the Square could ever be viewed as Flat. To the 2Der something Flat is a Line like object. To him something Flat would for example have measurable Width but no Depth. In his mind by saying the Square is Flat would make him try to put all the Line views of his Square onto a single Line which is a physical impossibility in his 2D Space. He will never really be able to view 2D Flatness as it is.

The situation is analogous for us 3Ders trying to see how a solid Cube could ever be viewed as Flat. Theoretically we know we can go into 4D Space and look down on the Flat Cube that's back in our 3D Space. By analogy we will be looking at different views as we look down and move accross the Flat surface of our cube. Each view will be a slice of the Cube. When we are done we will have a collection of Squares that are cross sections of the Cube. By doing this we have no insight as to where we went to get the collection of Squares. We have a collection of Squares and still have no knowledge of how to connect them together in a Flat way. Take 2 adjacent Squares. The mental task is to arrange them so that they are in the same plane but yet corresponding points in the two Squares are able to touch. This is impossible for us to imagine, but this is what is possible when you have a 4th dimension.

I must insist that the problem with understanding 4D is a limitation of our Brains. We do what we can but can't really get there. At least not the "there" that I want to be at. Again, a Tesseract is just a simple empty box. All the discussion, argumentation, and mind bending Projections and Slice analysis shows us that someting is basically missing in our ability to comprehend it. A Tesseract would be a comletely simple, intuitive, and obvious thing for even the dumbest 4Der to comprehend. I can't help it I have to say it again: It's just a Box! A 4Der would see it as being made out of Flat 3D Objects which in our Space are Cubes. If you were to ask a 4Der to describe how the sides of a Tesseract look, he would say that they are Flat Objects that have 3 dimensions. He would have to strain to see the Cubeness of the side. He would recognize that two sides of the Tesseract would connect at Line like objects (to him), but these Line like objects would be two dimensional. The Line like objects correspont to the faces of the Cubes that we would see by Projection or Slicing. He doesn't see faces of Cubes he sees Line like Objects that are two dimensional. The sides of the Tesseract can pivot on these Line like Objects as if they were hinges. This means that rotations in 4D take place on 2D Objects not 1D Objects like in 3D Space. So if the sides of a Tesseract can pivot on these 2D Objects then this brings a little more insight into the possibility of unfolding a Tesseract into 3D Space into that cross looking Object. But to show the cross Object made out of Cubes connected at their faces is still completely non intuitive until you understand Flat 3D Objects and rotations on 2D Objects.

Now for the self serving promotion:
To see my Animations that show Flat 3D Objects and rotations on 2D Objects please go to: http://bit.ly/2mQxRw1



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Re: Why Humans Will Never Understand 4D Space

Postby SteveKlinko » Fri Mar 10, 2017 5:10 pm

[quote="Klitzing[/quote] Yes I have been wondering about terminology. Thank you for that reference, but I'm still not sure what to call the Flat 3D Object that I talk about. A Cube is a Cube in 3D but when you add the Flat 4th dimension what do you call it? Also what do you call a Square that has now picked up a 4th dimension and has Line like qualities in 4D? Thank You
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Re: Why Humans Will Never Understand 4D Space

Postby Klitzing » Fri Mar 10, 2017 7:06 pm

SteveKlinko wrote:
Klitzing wrote:
Yes I have been wondering about terminology. Thank you for that reference, but I'm still not sure what to call the Flat 3D Object that I talk about. A Cube is a Cube in 3D but when you add the Flat 4th dimension what do you call it? Also what do you call a Square that has now picked up a 4th dimension and has Line like qualities in 4D? Thank You

As far as I 've learned,
  • a solid bodied polytope in D dimensions is bounded by
  • D-1 dimensional face or facet polytopes. These in turn are bounded by
  • D-2 dimensional margin or ridge polytopes. And these in turn are bounded by
  • D-3 dimensional peak polytopes. And these then by
  • D-4 dimensional spire polytopes. Etc.
That is, for a 4D tesseract the cubes would be its facets, the squares would be its margins. The edges are the peaks, and the vertices are the spires.
Whereas for a 3D cube the squares would be its faces, the edges its margins, and the vertices are peaks.

"Dihedral" angle then would be better called margin angles, i.e. the angle between 2 adjoined facets, measuread across the margin.

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Re: Why Humans Will Never Understand 4D Space

Postby ICN5D » Fri Mar 10, 2017 11:58 pm

There's nothing different about a cube in 4D space. It's still just a cube. The flatness is a matter of viewing angle.

Maybe the term you're looking for is a cube embedded in 4D space? A cube embedded in 3D space is what you consider a regular 3D cube. A cube with this '4D Flatness' as you put it, is a cube embedded in a 4D space.

The cube doesn't change. It's the number of dimensions that you can move it around in, that changes.

You can go even further, and embed a 3D cube in a 5D space, which would give the cube a line-like quality. You can view the full, complete cube from an infinite number of angles, like walking around a flagpole.

To a 5Der, a 3D cube is a line segment object, as thin as a strand of hair (but actually infinitely thin).
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Re: Why Humans Will Never Understand 4D Space

Postby wendy » Sat Mar 11, 2017 1:04 pm

You really don't need a brain capable of holding 4d images directly. All you need to do is something like feel the space.

Extent

The dimension analysis of extent, is that one can replace solid space of 3d with Nd, and those dimensions of extent count down from N, rather than three, dimensions. This means that you need to be aware that words are not just intense (counted from 0), but equally extense.

Mathematically, a bounding surface equates to an equal sign, eg 'x = 1'. In any given space, this has a dimension one less. The extensive spaces then equate to how many equal signs are needed, thus a part of the plane x=1 is a boundary x=1, y=1. And so forth. A point in this reason, is a space of N extensives, eg w=1,x=1,y=1,z=1.

Each word used, can be used as an intensive or extensive. The distinction is usually not made, because plane geometry is lines in the ground, and solid geometry is solids on the ground. For people with 2d or 3d needs, it suffices. People make a hand-wave stab at four dimensions, supposing that surface suffices as a 2d thing in 4d, where it doesn't.

We are essentially 3d things, but gravity dominates our world. So we stand on the ground or plane, and because our abilities of flying are not crash-hot, we generally mill around on the ground. So our three dimensional world looks like ++= (ie h=0 of the ground). This is the 'plane' or 'plain' we walk on. On this plane, we draw 'lines' that are supposed to divide the land up, ie +==. Lines in the sand, lines not to cross (less ye be 'dead'), or to delineate, are lines of == (ie two levels of division from space). The front line is here too. A bee-line, is as the crow flies, a single +, that is, a path from A to B.

Edge is used as a kind of line, but the 'edge of the universe' is shown as 2d, which means the root meaning is extensive (ie =, rather than +), and the edges are actually divisions of the surface (ie += ), rather than lines. A knife edge in 4d, is intended to split an object with its sweep, which means that solid ÷ division ÷ sweep = edge, so the binding meaning of a knife edge is extensive (== ), and therefore must be read as ++== , a 2d thing.

The polygloss looks carefully at the common meanings of these words, and decides if they are intensive or extensive, and gives meaning accordingly. A face is a division, the face of the moon is that part of the moon that stops us from seeing in. Facet is to make faces, but as a noun, is notionaly a cut, and thence a view through that cut, to see a different aspect of what's inside. But an image is an extent of partion, and therefore is a single equal sign.

One must always be mindful of this, and in the PG, the bulk of the divisions are made by making new words for the intensive forms, eg hedrix for 2d-cloth.
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Re: Why Humans Will Never Understand 4D Space

Postby SteveKlinko » Sat Mar 11, 2017 4:23 pm

It's always good to start with 2D Space. In 2D Space there is the Square Object. The 2Der looks at it and says look at the beautiful Width and Depth of the thing. He has no knowledge of the Height of it. (Could have used other orientations) In fact in a 2D Space there is no Height Dimension. We always look at 2D Space like it is embedded in our 3D Space. But this is not true. A 2D Space is an entity that exists on its own without being embedded in a higher Dimensional Space. The Square picks up a Height Dimension (although empty) if we all of a sudden proclaim "let there be 3D". But the Square does not have a Height Dimension (not even an empty Height Dimension) when it exists in 2D Space. The Height Dimension literally does not exist for a Square in 2D Space. Similarly you cant say that the 3rd Space Dimension itself exists even as a potential Dimension if the Universe is 2D. If you think a Square in 2D Space actually has a Height Dimension then it must have all the Dimensions or an infinite amount of Dimensions. This can't be true.

If we suppose a point Particle is bouncing around in a Square in 2D what keeps it inside the Square? The 4 walls do it for the 2D Space itself. But if the 2D Space is embedded in 3D Space then of course with even the slightest perterbation in the 3D direction the Particle will eventually bounce out of the Square into 3D Space. If the Space is pure 2D then the Particle stays inside the Square because there is literally no other place for it to go.

I think it would be interesting to try to experience 2D Space like a 2Der. But I think our 3D Brains and Visual systems again prevent us from properly being able to do this. Obviously we cant look down on the Space from our 3D vantage point. The best we can do is look at the 2D Space on an edge view of it. It will always look like a line embedded in our 2D Visual image. We will never appreciate the absolute non existance of our 3rd Dimension to the 2Der. But the 3rd Dimension not only does not exist to the 2Der it really does not exist in the Universe. These things are hard to imagine.

I look at Objects as Physical Objects and Mathematicians look at Objects as pure Mathematical objects. The Mathematician would say a pure Mathematical Square has the same properties in 2D or 3D. A Mathematical Square can exist in 2D or 3D. But if you are thinking about Physical Properties a Physical Square is a solid real thing in 2D but is not even a real Physical Object in 3D. The Mathematical properties stay the same, but the Physical Properties, in going from being a real Object to not being a real object, certainly are hugely different. So I think there is a limitation in jumping to Higher Dimensions with Physical Objects that does not exist with pure Mathematical Objects.
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Re: Why Humans Will Never Understand 4D Space

Postby SteveKlinko » Sat Mar 11, 2017 7:02 pm

I'm beginning to think that even Mathematical Squares are different things in 2D Space and 3D Space. In 2D Space a Square is a 4 sided object. If a Square is considered to be an infinitely thin Cube in 3D then a Square has 6 sides in 3D. A Square is a degenerate Cube in 3D. There is no way you can say a Square has 6 sides in 2D. So a Square can be different things in 2D and 3D. The 4D Animations show these kinds of things:

http://bit.ly/2mQxRw1
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Re: Why Humans Will Never Understand 4D Space

Postby SteveKlinko » Sat Mar 11, 2017 7:37 pm

Taking the Cube as an example might be more intuitive. A Cube has 6 sides in 3D Space. There are no other dimensions for the Cube when you have a 3D Space. Any other Dimensions or properties literally do not exist. Embed the 3D Space into 4D Space and then you can say the Cube has 8 sides. A Cube is a degenerate Tesseract in 4D, in the same way a that a Square is a degenerate Cube in 3D. A Cube is Flat in 4D. There's no way a Cube can be Flat in 3D. It is the solid Object that we all intuitively know.

You actually cannot just put a Cube in 4D Space you must do something to create the 4D Space around the Cube. This is because Space is a thing not just a Mathematical concept. There can be 4D Space, 3D Space, 2D Space etc. There can be no Space. Try to imagine that. When you add the 4th dimension you have to modify the Cube so that it now has 8 sides. Something changes when the Cube is in 4D Space. The Cube cannot be in 4D Space and only have 6 sides. While in our 3D Space the Cube can not be thought of as having 8 sides. There are no unseen sides lurking within a Cube in 3D. If you say that then the Cube should also have secret sides for the 5th dimension and higher making the Cube an infinite dimensional object with all other dimensions besides 3 being degenerate. The same is true for the Square in 2D Space. There are no secret extra sides that exist for it in 2D it has 4 sides. That view you have always had of looking down at the Square from a 3D advantage is a false view of the situation in an actual 2D Space Universe where there is no third dimension that exists. The Animations show these things in a Visual way:

http://bit.ly/2mQxRw1
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Re: Why Humans Will Never Understand 4D Space

Postby SteveKlinko » Sat Apr 29, 2017 2:55 pm

I would like to mention an overlooked point about why viewing 2D Space from a 3D perspective is physically impossible. A 2D Space is so easily drawn as a diagram on paper that we just assume this can be done in reality. I think the problem is with ideal Mathematical thinking versus practical Physical World realities. If we are talking about a 2D Physical Space embedded in a 3D Physical Space then we have to deal with the fact that the 2D Physical Space that we are viewing has zero extension into the 3D Space. The whole 2D universe would break apart at the least perturbation from 3D Space. You must assume some kind of Star Trek like containment field holding the 2D Space together. It's just not something that you can do with Physical real world Spaces. The same thing holds for the 4th Dimension. You can not have a 3D Universe existing in the context of 4D Space. You ether have 3D or you have 4D. A 3D Space could not Physically remain intact in a true 4D universe.
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