http://www.deviantart.com/deviation/11263772/ click on image for the fullsize image.. *note for simplicity most cubes were drawn in an orthogonal prespective* ok well i figured i should put my money where my mouth is and actually draw out some visualizations of hypershapes .. now i could also draw what you were saying of a 4d cube where its a 3d cube that you can see both sides of .. this is very easy you simply draw two cubes, shade them oppositly , then cross your eyes and your steroscopic vision will allow you to see both sides of a shaded cube.. this is mostly to show you my ideas on hypercube visualization i dont think its anything that new .. i made a "shorthand" method of drawing general hypershapes which i describe over at the page.. and if you will look to the far left underneath the blue hypercube which im not quite sure as to how many dimensions it is .. is it 6 ? i get confused with numbers not images Please Register or Log in to view the hidden image! anyway there is a 3d cross of 6 cubes.. this is what i believe to be the answer to unpacking a 4d hyper cube i further unfolded that 3d cross *its to the right of it* which was very easy .. i imaging a 5d hyper cube would be much the same way except you might have to have certain parts not connected .. basically what your talking about is in 3d world called uv mapping.. where you take each face of a 3d object and flatten it in a sensible way so that you can put textures on the 3d shape.. i was thinking prolly a more proper way to do this would be to use the vectors instead of faces for this problem since when you do it by face you replicate vectors.. but both ways can work i actually did the face version and not the vector version to show you its possible. why does a 4d cube convert to a 3d cross ? well because a 3d cube converts to a 2d cross thats why .. each time you flatten all your doing is figuring out a way to represent a higher dimensional shape in a lower dimension and this problem only needs you to lower by 1 so yea.. i hope this agrees with your ideas of visualization as well.. i thought this was a pretty fun problem actually ..
Planaria: Your version of an unfolded 4D hypercube is a 3D cross consisting of only six 3D cubes. A 4D hypercube unfolds into a 3D figure consisting of eight 3D cubes. This discrepancy suggests that your ability to visualize a 4D hypercube is a bit suspect. The eight 3D cubes representing an unfolded 4D hypercube can form various 3D objects. You can start constructing some of these 3D objects by stacking four of the cubes to form what can be viewed as a four story building (actually a 1X1X4 parallelopiped). The four additional cubes can be attached to the four exposed faces of the cube forming any one of the four stories. All of these objects are symmetric cross-like structures. Starting with the four story building object, various other representations can be constructed by attaching the four additional 3D cubes in various unsymmetric positions. Not all such objects represent an unfolded 4D hypercube. I think that each of the additional four 3D cubes must be attached to a different side of the four story building. The situation is analogous to unfolding a 3D cube into six squares. The cross-like figure is not the only possible configuration for the six squares. You can start with four squares forming a 1X4 rectangle and attach the remaining two squares, one on each long side of the rectangle at any one of the four possible positions. If you attach the 5th & 6th squares to the same long side of the rectangle, the resulting figure cannot be folded to form a cube.
actually you can make a 4d cube with this many cubes.. if you look at this instead as 2 2d crosses and you use each 2d cross to construct a cube the edges that connected and made the 3d cross create the necessary hyperness that connects the two cubes .. meaning those edges are what create the cubes that connect to the cubes. if you really really need me to i will draw it out step by step.
Planaria: You are a piece of work. You claim to be able to visualize 4D, 5D, & 6D objects. You also claim you could visualize 7D objects if you had enough patience. After making such claims you direct us to your web site where there is an erroneous perspective view of an unfolded 4D hypercube. As mentioned in my previous post, a 4D hypercube unfolds into a 3D object consisting of eight 3D cubes, while the object shown at your site consists of only six 3D cubes. Your last post seems like some attempt at obfuscation. The above does not make sense. Perhaps English is not your native language.
If it is so easy to "visualize" then why don’t you draw it? That’s what they mean by hard to visualize.
cato .. http://www.deviantart.com/deviation/11263772/ this was posted earlier btw .. and dinosaur i will post a detailed drawing explaining how my method works. what i realize is that your method deals with the problem in terms of cubes and i deal with the problem in terms of vertice/edges/faces a subtle difference but important nonetheless. actually both methods work your method constructs my method folds and joins. dont worry if this explanation is confusing the drawing will clarify.
Its not really 4d, its just 3d that you give some qualifications to. I understand what you mean, and how to think of those shapes as 4d. However, what I meant was that you can’t do something like say... draw x, y, z, and t axes all orthogonal to each other.
thats true but you also cant draw a 3d cube with all orthogonal lines.. you can only do that in 3d space. you can draw a 2d square with all orthogonal lines however.. because that is the realm 2d lives in .. this bring up a thought however, you can represent everything orthogonally if you note the vertex coords next to each vertex, for instance if you draw a 2d square but then on the 4 visible vertice you simply write both xyz coords for the vertice that is theoretically behind that vertex you can represent.. show a 3d cube on a 2d surface with all orthogonal lines correctly .. now this should work with a hypercube as well correct ? the number would just jump up to 4 coords for each visible vertex.. ? this should also work if your thinking about a hyper cube in terms of faces or cubes.. although this starts to shift from visualization to a combination of visualization and notation..
Planaria: The following looks like obfuscation or an attempt to baffle with bull**it rather than dazzle with brilliance. While awaiting the drawing, I am wondering about the difference between dealing with cubes instead of vertices, edges, & faces. I still claim that your erroneous 3D perspective view of an unfolded 4D cube indicates a lack of the ability to visualize a simple 4D object such as a tesseract (4D cube). If somebody showed me a 2D object consisting of 4 squares and claimed that it represented an unfolded cube, I would doubt that he understood what a cube looked like.
Infinity, eternity, existence You can plot a hologram in space easy enough, but what if you plotted it in time as well. Wouldn't you get a ghost of the positional features in the time references, if somehow you could color-code the different plots. The 4th dimension may not exist as we might expect from our current mathematics because we have modeled the world in three dimensions, except for the space-time thing, which is simply an extension of the current logic as much as the xyz plane is an extension of the positional features of space. Space-time simply goes it one better by including time as a 4th dimension, when it is really an extension of the two dimensional coordinate system into the third plane, not into a 4th. Rather, it is a combination of the two extensions that may bring us a 4th dimensional plane, the wxyz plane, and in that is existence as well as space and time. We are at a place in time, but we are also an apple. That's my view to date. I'm still working on it. What do you think?
hi there ,, i have been unable to get to a scanner .. here is the picture.. http://www.deviantart.com/view/11465146/
I like the cubes marked 4D especially because of the shadows that help illustrate the different sides, and the ability to see through them. It would be helpful to color-code the planes (of existence I might add, as well, those being not only the planes of the inner cube, but also of the outer cube). I begin to see how well various particles and forces of atomic structure could be illustrated, animated, and maybe even manufactured.
It's the next day. I don't think this goes far enough. The combination of the two extensions would still lack the breadth of a borderless plane of existence, and I'm not sure it could be described simply by adding a single reference (w). Could be that it would require more than one. When I think of xy I think of x as time and y as space. z would be existence, and the combination of the three would enable a 'map' of the the three dimensional universe in part or in whole. In other words, the universe is time, space, and existence, and each of those is a domain of its own. Hence, the nature of three persons in God, as declared in some Christian religions. DenniSys.com
Planaria: Once again you are showing a perspective view of a 3D object consisting of six 3D cubes forming a cross, and claiming (at least implying) that it represents an unfolded 4D cube. This is an erroneous representation, clearly indicating that you cannot really visualize a 4D cube. Again, I point out that a 4D cube unfolds into a 3D object consisting of eight 3D cubes. Note one of the last drawings in your montage: It shows a 3D cube inside of a larger 3D cube with the corners of the two cubes connected. This is a well known representation of a tesseract (4D cube). A proper interpretation of that representation clearly indicates the eight 3D cubes comprising a 4D cube, namely the inner 3D cube, the outer 3D cube, and the six cubes connecting the faces of those two cubes. I do not claim to be able to visualize a 4D cube, but my concept of it is better than yours. BTW: It boggles my mind when I try to visualize how four straight lines can be mutually perpendicular, which is what occurs at each vertex of a 4D cube. It boggles my mind even more when I realize that five planes meet at each vertex, and that each plane is perpendicular to the other four.
i just told you that i drew My way of unfolding a hypercube.. my way deals with hypercubes not in a method of connecting Cubes together but a method that connect Vertice and Edge together. i drew that common method of showing a hypercube to show you what i thought you were talking about, i could see that from this angle of a hypercube that you could easily create the solution of 8 cubes necessary for creating the hypercube.. however from the view i was looking at i seem to have created a different solution. my method creates cubes as they are being constructed into a hyper cube so the vertices and edges that are assembled Create the extra cubes needed. they are a consequence of the folding action.. i could even make a more slimmed down version of this folding excersize if you said that you could also not reproduce any vertice or line. if youll note the drawing after the last folding is a hypercube with 8 cubes.. and then there are rotated drawings of that same cube to show it is so . can you see that from this drawing? look even you yourself said there are many solutions to a problem such as unfolding geometrical objects (such as 3d cubes as was the example) can you not fathom that i have found a method that requires 6 cubes instead of 8 ? it is you who cant visualize my ideas not the other way around. ****Again, I point out that a 4D cube unfolds into a 3D object consisting of eight 3D cubes.***** again i point out that in a previous post i AGREED with you that your method works , but that also my method works i do not understand how you cannot see this after i went through the trouble of drawing it out for you .. can you tell me what confuses you atleast? you only seek to dissprove.. ****BTW: It boggles my mind when I try to visualize how four straight lines can be mutually perpendicular, which is what occurs at each vertex of a 4D cube. It boggles my mind even more when I realize that five planes meet at each vertex, and that each plane is perpendicular to the other four.**** try to visualize a 3d cube with all perpendicular lines, you cant and yet you can visualize a 3d cube just fine..
Planaria: I give up. I will not waste more of my time on this nonsense. If you still think a 4D cube unfolds into a 3D object with six 3D cubes, there is no convinincing you that you are dead wrong. Mathematics is not a matter of opinion: You can be wrong. The following is nonsense. Yes, I understand that you found a way to unfold a 4D cube in six 3D cubes. You have claimed this several times and have now shown me the same perspective view of the erroneous object at least twice. What I also understand is that you are mistaken when you claim that the unfolded object represents a 4D cube.
You can Google 'tesseract' and find Java applets that fairly illustrate various concepts surrounding the subject. Maybe it is more important to describe the use of such information. We usually approach a problem and then find a solution. The establishment of mathematics affords an organized method of problem solving. Mathematics itself, as with most science, is useless without application. Once the problem is described the utilization of the methods define the meaning of the organizational elements. Basically, it helps to have a problem in mind before trying to visualize a solution, and with that the solution may be reduced to its necessary level of complexity, wherein elements are given meaning. You can define the xy coordinate system as anything you want and visualize it any way you choose within the limits of its usefulness, so that the assignments that you make become important only with reference to the methods used and the outcomes they produce. What use is it to be able to define a four, five, or more dimensional system if there is no application to apply it. The answer is that there are applications that are suited to the methods, and each application may take a discrete or unique viewpoint on the meaning of the organizational elements. By understanding the application the visualization then becomes a done deal. So the importance exists within the concept or concrete object that you are attempting to model. If I am trying to plot a belief rather than an apple, then I see a belief system made of ideas which have behavioral significance with an anatomic source. This belief object determines the use I make of the xy coordinate system, and having assigned meaning to the coordinates I can visualize some sense of the object. The study of mathematics is concerned with assigning the coordinate system the values of (x) as space and (y) as time, so that those planes of existence can be measured. I am concerned that the term 'planes of existence'' contains the phrase 'of existence' as giving existence a meaning and identity of its own where time and space are subordinate components. I want to give existence an equal share and build it into the coordinate system, because the three elements, and no others, comprise the universe as we know it. That is, you would be hard pressed to build an index of importance for the three objects.
what you dont realize is my erroneous object is your tesseract just in a different view .. your tesseract and my drawing are the same thing just in different states of visualization , i know this because for one thing there are several java applets that will cycle between them in a smooth tranformation .. just admit that were both right Please Register or Log in to view the hidden image!
