ICN5D wrote:I think you're giving up too soon, Steve.
Slices, intersections and reflections don't do it for me anymore. For example I want to see all the points of a 3D cube simultaneously with all the connections between points to other points without slicing it up. It must also seem flat in this visualization that I am looking for. Can you really do that?quickfur wrote:While our physical senses will never see 4D directly, nothing theoretically prevents us from mentally grasping it.
Can you really see all the points of a solid 3D cube at the same time while seeing all the connections and without slicing the thing up? It must also give the sensation of a flat object. I don't think anybody can do that. But this is the requirement I put on my study. I was not able to do it.wendy wrote:What you see is only part of the picture.
But can you really see a single 3D Hyper Plane like a 4D Being would? I'm interested in knowing how you do that.Klitzing wrote:In fact, the best way to get from grasping one dimension towards the next, is by considering prisms or pyramids of things of the lesser dimension.
By rotating the extra dimension into our 3D Space you are doing what I did in my animations. It works for a period of time but eventually I needed more. You need to see all the point of a solid 3D cube simultaneously while seeing all the connections of the points to each other and not slicing things up and it must have the sense of being flat.gonegahgah wrote:I love the thought of 4D space. I just wish I had more time to spend there...
For me it was a case of understanding the relevance and nature of extra side-ways and the relation of those to our common actions.
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SteveKlinko wrote: wendy wrote:
What you see is only part of the picture.
Can you really see all the points of a solid 3D cube at the same time while seeing all the connections and without slicing the thing up? It must also give the sensation of a flat object. I don't think anybody can do that. But this is the requirement I put on my study. I was not able to do it.
I Googled the name of your second figure to learn more about it. I can see that you have done a lot of good work on these topics. I do appreciate the approach that you take, because it does provide a certain way of visualizing these things. But I am simple minded and I have to understand the first more simple principles of methods and theories before I can move on to more complicated aspects. I admit I am stuck on the 3D Hyper Plane and feel that I cant move forward without understanding it better. I understand it conceptually but I want to understand it like a 4Der would, which to me is the next logical step. I am looking for a representation for the cube that will give me the insight that a 4Der would have. First the cube would be flat. Your diagram shows that, but things are overlapped. The 4Der would not see any overlap as long as he is looking at the cube from outside our 3D Space. Next thing is that the six faces of the cube would all look the same size and shape and would surround the inside points in some way. All points of the cube, even a solid cube, would be visible and unobstructed by any other points of the cube. The 4Der would clearly see all the inside points at the same time with all the connections of the points to each other spread out over a flat 3D area. Some how the points that make up the six faces would clearly be seen as the surface of the cube. If we were talking about a sphere the situation is even worse. I want a representation that would clearly show all the surface points drawn on a 3D flat surface with all the connections between the points clearly visible with no overlaps. The surface points of the Sphere would have to clearly be shown to contain the inside points with no overlapping. All points on the surface and inside the Sphere would be spread out over a Flat 3D Hyper Plane from the point of View of the 4Der. He sees inside the Sphere. I still maintain that our Brains will not let us see these things in this way. I hyped it up by observing that if we did have a 4D Brain that we would have 32000 times more neurons and so be 32000 times more intelligent. The added intelligence is good but probably not necessary for understanding 4D Space. However a 3D Retina and 3D Visual Cortex is absolutely necessary. This means that a 4D Cat would understand things about 4D Space that we can not. The understanding would be Visual not theoretical. In this sense a 4D Cat might be smarter than we are. I think we can understand 4D Space the best that 3Ders can do, but we can never understand 4D Space like 4Ders would. I think I should have emphasized the stipulation "Like a 4Der" in my original post.Klitzing wrote:I agree, simply slicing 3D bodies and viewing disconnected slices as some series does not provide a good understanding of 3D by means of simple 2D objects.
By analogy if you circularly tile six squares in 2D Space you can never get a cube. In fact the 2D observer will still have no idea what a cube is by doing this. If you take all the techniques that we use to try to understand 4D Space and apply them to a 2Der trying to understand 3D Space you will see how the 2Der can never really see a cube for what it is. Even more importantly the 2Der will never even understand the flatness of a square in 3D Space.wendy wrote:You don't necessarily have to view it as a 4d person ought. It is understanding, rather than physical form that matters.
There has been a fierce debate here on whether the spherical tiling of eight cubes, constitutes a tesseract or not. It has no terrage, and thus is not a 4d figure, but a 3d tiling.
Of course, things like lace cities and various other projections help. With lace cities we have been able to reach eight dimensions. With laminate tilings, twentyfour.
But i digress.
I think these diagrams show connectivities but completely obscure the real appearance of objects. These diagrams seem to eliminate the insides of the object. I think you need the 4D view even with these diagrams. Take the Tesseract for example. from my 3D perspective the central small cube seems to be completely surrounded by the other warped cubes and completely inside the large outside cube. There is no inside with these diagrams. Also this kind of representation of a Tesseract continues to show the Tesseract as being made out of cubes. While technically true in 3D Space it is not what a 4Der sees. The 4Der sees 8 Flat 3D objects connected together to form the Tesseract. If you put a test particle inside the Tesseract and let it randomly move around it will eventually bounce off of all the walls. At the point of impact on any wall and at any position on the wall the particle will be only one point (the thickness of the wall) away from the outside of the Tesseract. By making the sides look like cubes you would get the impression that the test particle would be hitting the sides of cubes and never hitting any points inside any cube. By making the sides look like cubes the diagram is completely obscuring the reality of a Tesseract. I think the key is in visualizing how things like cubes and spheres are actually Flat in 4D Space and then putting them together.Klitzing wrote:Do you really need to have a real 4D perception in order to grasp the local connectivities of 4D polychora?
To see if I understand what you are saying let's apply this to a 2Der trying to understand a cube. Let's slice up the cube into a bunch of squares and put these squares into the 2Ders 2D Space. He can only see them from their edge views. This deconstruction of the cube gives him no idea how they connect to each other. He sees them as nice solid squares with beautiful width and depth. In order for him to see how they connect he must see them as Flat objects but in his world a square is not a Flat object. it is a solid impenetrable object that he can walk all the way around. He is unable to see the Flatness of these squares. He can know in theory how to do it but he will never really know how to stack them up. The situation is the same for a 3Der looking at a bunch of cube slices of a Tesseract Unless the 3Der recognizes the Flatness of a cube in 4D Space I don't see how looking at 3D cubes gives him a good way to understand 4D Space. I agree it's a start but the missing element is in recognizing the Flatness of 3D objects in 4D Space. When you can see how they are actually Flat they will stack up in a more understandable way. The problem is in seeing the Flatness and I think this is a limitation of our 3D Brains.Hugh wrote:Seeing a particular viewpoint from a different perpendicular direction indicates the existence of a higher dimension.
SteveKlinko wrote:To see if I understand what you are saying let's apply this to a 2Der trying to understand a cube. Let's slice up the cube into a bunch of squares and put these squares into the 2Ders 2D Space. He can only see them from their edge views. This deconstruction of the cube gives him no idea how they connect to each other. He sees them as nice solid squares with beautiful width and depth. In order for him to see how they connect he must see them as Flat objects but in his world a square is not a Flat object. it is a solid impenetrable object that he can walk all the way around. He is unable to see the Flatness of these squares. He can know in theory how to do it but he will never really know how to stack them up.Hugh wrote:Seeing a particular viewpoint from a different perpendicular direction indicates the existence of a higher dimension.
SteveKlinko wrote:The situation is the same for a 3Der looking at a bunch of cube slices of a Tesseract Unless the 3Der recognizes the Flatness of a cube in 4D Space I don't see how looking at 3D cubes gives him a good way to understand 4D Space. I agree it's a start but the missing element is in recognizing the Flatness of 3D objects in 4D Space. When you can see how they are actually Flat they will stack up in a more understandable way. The problem is in seeing the Flatness and I think this is a limitation of our 3D Brains.
SteveKlinko wrote:To see if I understand what you are saying let's apply this to a 2Der trying to understand a cube. Let's slice up the cube into a bunch of squares and put these squares into the 2Ders 2D Space. He can only see them from their edge views. This deconstruction of the cube gives him no idea how they connect to each other. He sees them as nice solid squares with beautiful width and depth. In order for him to see how they connect he must see them as Flat objects but in his world a square is not a Flat object. it is a solid impenetrable object that he can walk all the way around. He is unable to see the Flatness of these squares. He can know in theory how to do it but he will never really know how to stack them up. The situation is the same for a 3Der looking at a bunch of cube slices of a Tesseract Unless the 3Der recognizes the Flatness of a cube in 4D Space I don't see how looking at 3D cubes gives him a good way to understand 4D Space. I agree it's a start but the missing element is in recognizing the Flatness of 3D objects in 4D Space. When you can see how they are actually Flat they will stack up in a more understandable way. The problem is in seeing the Flatness and I think this is a limitation of our 3D Brains.Hugh wrote:Seeing a particular viewpoint from a different perpendicular direction indicates the existence of a higher dimension.
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