Prashantkrishnan wrote:I have tried these methods. These do help me to visualise projections of some rotachora like tesseract, cubinder and spherinder. As for the duocylinder and glome, we have already discussed that in another forum. But don't these methods get more complicated with an increase in the number of dimensions? Suppose we are visualising a 6-space in a 7-space with all geometric properties. We have to first visualise a cube, then visualise various frusta to completely visualise a tesseract, then some hyperfrusta(?), then a penteract and so on until we get to visualise a 6-space. Moreover, we visualise only a 3D projection of a 4D projection of a 5D projection of a 6-space. Still, this can be helpful in many ways.
Yesternight, I vaguely visualised two spheres intersecting at their circle shaped cross sectional planes in 4-space. When I tried to visualise more clearly, both the spheres came to the same realmspace. I understood the concept of chains of spheres in 5D mentioned by Mrrl in the page on knots.
wendy wrote:Using the language of geometers to describe 3-hyperplanes etc will only get you into more trouble. They do not make the crucial distinctions necessary to do this.
Plane geometry is something you do on the ground, and Solid geometry is something you do with solids. So to make the occasional foray into four dimensions, hyper-this and hyper-that suffice. If you are really going to look at looking at four dimensions as 'solid space', you need to put a new view on things.
Things that divide are like equal signs. So eg, a line divides a hedrix, and a hedrix divides a chorix, can all be represented by a single equal sign: z=0. We feel gravity, which puts us in the space of air, where no more fall is let, gives us z=0. A fence stops things wandering across the plane, so it's of limited height, say z=0 to 1, and has a second equal sign, y=5. It's the intersection of a plane z=0, and a wall at y=5. z=0,y=5 gives a point in 2D, a line in 3D, a hedrix in 4D.
Geometers don't think like this. Instead, things have 'right angles', that is, are of fixed dimensions. So a wall as we see it is a hedrid thing, So it must be a hedrid thing for everyone. So the chorid thing needs a new name. Railway lines unite, and stay 1d in all space,
Drop the hyper-nonsense, sit down and work out whether things unite or divide, because common language is more a case of division, rather than union, and call things by the meaning of unites or divides. So a wall is not 2d, but N-1 d, and a knife edge (whose sweep divides something), is not 1D, but N-2 D. Once you get this idea, it's not too hard to get N-1 to be a plane, and grasp 3d space as a photo or map in 4D.
quickfur wrote: And how do we represent intersections between two or more hyperplanes? We simply draw them as intersections between cubes, which in general would be squares,
ac2000 wrote:Sorry, when I'm asking a question about a four year old post. But as there where some recent posts on this thread this year, it should be OK, I guess.quickfur wrote: And how do we represent intersections between two or more hyperplanes? We simply draw them as intersections between cubes, which in general would be squares,
Quickfur: I did not understand that bit from your first post.
Why would the intersection between cubes be squares?
When I imagine two intersecting cuboids the intersection would look to me like a parallelepiped (or if the cubes intersect each other in a rectangular fashion more like a rectangular cuboid).
quickfur wrote: To illustrate what I mean, look at this diagram of two intersecting 3D planes:
Note that the intersection here is a line (segment), but only when we interpret the two rectangles in a 3D way! If you interpret them in a 2D manner, then they would be parallelograms, and their intersection is certainly bigger than the line segment depicted above.
quickfur wrote:
Similarly, when we draw intersecting cubes in 4D (which are actually parallelopipeds, not regular cubes), we draw them with a rectangular intersection because they are to be interpreted in a 4D manner, not as intersecting 3D volumes.
ICN5D wrote:Since we are talking about linear operations, I am inclined to use a notation system that I have developed. There is another one on this forum that has been in place for a long time, and I respect that. However, when comparing shapes and linear processes, I find my notation just slightly more intuitive, with my full respect for the original notation. In paying homage, I will include both.
ICN5D wrote:Okay, well, I just had to go out and celebrate my 31st birthday, which is today, and enjoy myself while I'm still young enough to be able to enjoy myself,
ac2000 wrote:quickfur wrote: To illustrate what I mean, look at this diagram of two intersecting 3D planes:
Note that the intersection here is a line (segment), but only when we interpret the two rectangles in a 3D way! If you interpret them in a 2D manner, then they would be parallelograms, and their intersection is certainly bigger than the line segment depicted above.
I think what you mean here by "bigger intersections" are the two triangles made of the mixed colour, right?
But then the 2D parellograms would be different objects right from the start (not the assumed rectangles in 3D representing planes), and above all, they would only have these triangular intersections if they were rotated within the 2d plane and not if they were rotated into 3d space to intersect.
Maybe I misunderstood what you meant with "interpret them in a 2D manner": to me they only *look like* they have these big triangular intersections, but when we would really "interpret them in a 2D manner" the planes intersecting in 3D would have no intersection in 2D whatsoever, or would they?
quickfur wrote:
Similarly, when we draw intersecting cubes in 4D (which are actually parallelopipeds, not regular cubes), we draw them with a rectangular intersection because they are to be interpreted in a 4D manner, not as intersecting 3D volumes.
So, if we had two rather flat parallelopipeds sharing the same 3D space and than rotating one of them about 90° into the direction of the w-axis (so that it sticks out in the ana/kata space), then the resulting intersection would have only a "next to nothing" extension into w, and would therefore resemble a line in 3d?
But the rest of the cube sticking out into ana/kata should then be drawn as rectangle too, no? Because it lacks the same dimension as the stuff inside the intersection.
ac2000 wrote:[...] I suppose each of the 3D => 4D spinning objects then rotates around the plane which is a slice in its center or something? But I can't really imagine this 4D rotation behaviour, where things rotate around a plane (instead of an axis), it's just too complicated for me.
[...]
thank you for the detailed introduction and explanation of your notation. I've understood more of it than I had expected. Before, I was always at a total loss when I had seen all those strange notation signs somewhere on the forum.
ICN5D wrote:Klitzing: thank you for the kind words!
I like to describe shapes by their hollow form, and associate the N-1 elements as being surface panels paired together, on a respective axis.
* Generalization: For an N-D shape, there are (N-1)D flat surface panels joined together, encasing a central void of N-D space.
This means a 3D shape has 2D surface panels encasing a central void of 3D space. A 4D shape has 3D "flat" surface panels encasing a void of 4D space. A 5D shape has "flat" 4D surface panels, encasing a central void of 5D space, etc.
ICN5D wrote:I posted a decent definition of the spin/lathe for Wendy some time ago, here it is. However, words can only do so much, I need to illustrate it! :
In reality I suppose there are two ways to lathe non-toric shapes:
- Hold a shape in place and rotate into N+1 around an axis that divides the shape in half : "lathe"
- Hold a surface element in place ( most likely one on the previous constructed axis ) and rotate into N+1 : "flip"
The "flip" can create some tegum products, as in the flip of a triangle creates a bicone, the tegum product of a circle and digon.
The part I want to focus on is the lathe around the center: When a shape spins into N+1, some of the panels are moving around in a circle, while the rest sit still and rotate. During an N+1 lathe, an N-dimensional shape can be cut in half by an N-1 dimensional shape ( or surface element for that matter ).
ICN5D wrote:[...]Now, lets consider the cylindrone: ||O>. Since we pointed out how the spin commutes, the sequence can also be ||>O, which suggests that spinning a square pyramid makes a cylindrone. How can this be??
ICN5D wrote:[...] Rotations will happen around an N-1 hyperplane. Quickfur pointed this out in the above posts. This is what I mean by " N-1 dividing plane". [...]
.Consider if you keep the base of the pyramid fixed, and rotate its apex around the plane of this base. The apex will trace out a circle, while the triangular faces will trace out triangular torus-shapes (with a zero-radius hole -- the cross section is a rhombus). There are 4 such torus shapes that connect the square base to the circle, so this shape is actually the square-circle tegum (the dual of a cubinder)
ICN5D wrote:[...]
Thanks for elaborating on my answer to the "spin of square pyramid" question. I may have asked it rhetorically, but it did deserve a little more explanation. And, you're right, there are MANY ways to spin it, depending on the orientation of the stationary square of rotation..Consider if you keep the base of the pyramid fixed, and rotate its apex around the plane of this base. The apex will trace out a circle, while the triangular faces will trace out triangular torus-shapes (with a zero-radius hole -- the cross section is a rhombus). There are 4 such torus shapes that connect the square base to the circle, so this shape is actually the square-circle tegum (the dual of a cubinder)
I guess I could call that the "flip" of the square pyramid. I hadn't really looked much into this motion, but that's interesting that it creates square-circle-tegum.
About this whole "spin operation" business: I can't really say that I created it, but more precisely discovered it, as a linear way to create cartesian products with circles. I'm inclined to call that a "circle product" ( n-cylinders ), where we can also have square-product ( n-diprisms ), and triangle-product ( n-trianglinders ).
The linear way to make a circle-product, with an N-D shape, is by extruding along N+1, then holding in place and rotating into N+2. By deriving the surface elements of many, many shapes that had spins, and searching for patterns, I soon found this phenomenon. In the single-spin cases, it was only one pair of elements on some axis that got the torus operation. The rest of the elements, after the spin, got the circle-product. The number of spins is directly proportional to the number of curved rolling side torii. This evolved into my definition of the spin, that originally got derived from cartesian products. So, hopefully it won't be so scary anymore
-Philip
ICN5D wrote:Yes, but that's just the spin of a prism. The spin of a triangular shape ( where we see >O and O> ) is more around the dividing n-cube that sits parallel to the last constructed axis of tapering. Sure there are more ways, but I am focusing on the specific one that turns triangles into cones, square pyramids into cylindrones, cones into sphones. That method is transferable to any tapertope. Even if the shape has both a prism and triangular perspective, like the triangle prism. Applying the rule will still make the triangle ends whip around into 4D, along the circular path, to make the triangle torus.
But, perhaps one has to be more stringent in choosing which of these element pairs will become the new torus. I guess we sort of have to "search" for it in a ways, if not familiar with a shape's surface elements. As in, not trying to connect one of the square panels into the torus, but identifying that the triangle pair should be the one. Of course, this is only according to the spin method I'm defining.
[...]
quickfur wrote:ICN5D wrote:But, perhaps one has to be more stringent in choosing which of these element pairs will become the new torus. I guess we sort of have to "search" for it in a ways, if not familiar with a shape's surface elements. As in, not trying to connect one of the square panels into the torus, but identifying that the triangle pair should be the one. Of course, this is only according to the spin method I'm defining.
[...]
If you have an unambiguous way of choosing the plane of rotation, then it works. But this may be more tricky than it first appears, as I point out above.
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