PWrong wrote:I've found a simple technique for counting rotatopes in each dimension. I'm not sure if someone's noticed this before, since I don't read the geometry forum very often, but here it is.
Let's call the process of "spherate" to make a surface r distant from an already existing structure. This gives a figure equivalent to replacing all the points by spheres of radius r from the point.
This suggests that we put a sphere at every point on the circle. This would give a solid torus, but it seems like an inelegent construction. Most points in the torus correspond to two unrelated points in the circle. It's also difficult to define the surface of a torus like this.For example, the three-dimensional torus is points in xyz equidistant from a circle in xy.
To make a general torus, you need a set of R's that don't clash with each other. Successive division by 1/3 does here.
If you do a radial inversion to swap inside to outside, the symbol is reversed. This is why symmetric symbols appear the same, while assymetic symbols become a different solid, but the surface is topologically the same.
This of course makes nonsense of the claim that surface has a genus.
PWrong wrote:I don't know much about topology.
It's actually quite simple: imagine you made a shape out of blutack. Now a shape is topologically equivalent to it, if you can make that shape from the original one without sticking parts of the shape together, or making holes in it.
so a 2-torus (topological notation for '3-d hollow torus', 2 after dimensionality inherent to object in question, regardless of the space of embedding) has 3 circles ('1-spheres' in topological notation?), not transformable into each other.
so analogically, 3-torus (geometrical 4d-torus), would have 3 2-spheres, not transformable topologically into each other ?
then that would be somehow connected to the fact that Dupin's cyclides are families of spheres tangent to 3 fixed 2-spheres? and torus is a kind of cyclide ?
somewhere i read, that a glome can be considered as a family of nested tori. how is that kind of foliation made, and according to what do different tori correspond to different whats of glome ?
Yes, it has 3 different circles. You can wrap around the larger circle, or around the small circle, or you can just draw on the surface without wrapping around anything. Only the last circle can be shrunk to a point.
http://mathworld.wolfram.com/Torus.html
'due to the similarities between toroids and twistors, mathematical topologies believed to act as a bridge between higher dimensional domains and 4D space/time reality, toroidal DNA can facilitate the cascade of higher dimensional information of consciousness into the electromagnetic (EM) domain. Since EM fields are well known to influence biological systems, they can carry the information of consciousness into the electro-chemical level. A mechanism is also proposed for the resonances between the quantum fields of consciousness and the energetic template of DNA. Such a mechanism is supported by recent experimental evidence for an energetic template of DNA using laser correlation spectroscopy (Poponin, 1998) and mathematical evidence that quantum fields can have a toroidal topology (Beltrami, 1889).'
two 2 toruses are joined with their surfaces, and their space between is the 4d-volume of 3-sphere.
information is not energy !
http://www.zynet.co.uk/imprint/Tucson/4 ... sciousness
:Quote:
'due to the similarities between toroids and twistors, mathematical topologies believed to act as a bridge between higher dimensional domains and 4D space/time reality, toroidal DNA can facilitate the cascade of higher dimensional information of consciousness into the electromagnetic (EM) domain. Since EM fields are well known to influence biological systems, they can carry the information of consciousness into the electro-chemical level. A mechanism is also proposed for the resonances between the quantum fields of consciousness and the energetic template of DNA. Such a mechanism is supported by recent experimental evidence for an energetic template of DNA using laser correlation spectroscopy (Poponin, 1998) and mathematical evidence that quantum fields can have a toroidal topology (Beltrami, 1889).'
DNA and consciousness has a lot to do with quantum mechanics, how can you state otherwise ?
PWrong wrote:There's a few shapes that I'm not sure about yet. For instance, is this an illegal shape? If it is, what kind of rule should we have to prevent it?
a(x,y), b(z,w), c(a,b)
If we only count the glome, three 4D torii and this beast among "toratopes", each cross-section is either one 3D toratope or two toratopes which differ either in one of their dimensions or in one coordinate of their centers.
n (T(i) + k(i))!
Sum ( Product (--------------))
k i=1 T(i)! k(i)!
PWrong wrote:To be honest, at first I was almost certain that the "tiger" wouldn't exist. It's essentially the product of a duocylinder and a glome, and I didn't think you could multiply two 4D objects and get another 4D object. But now I don't know what to think . Maybe I should be less judgemental of shapes with ugly formulas. I got the same parametric equations for the tiger as you did. It's the sum of the equations for a duocylinder and a glome.
Before we get carried away though, we should try to prove the existance or non-existance of this creature. I think it should be possible to find out whether or not it obeys my first rule, that every point on the object is unique. If we can find two different sets of vectors a, b, c, with the same sum, then the shape will be illegal.If we only count the glome, three 4D torii and this beast among "toratopes", each cross-section is either one 3D toratope or two toratopes which differ either in one of their dimensions or in one coordinate of their centers.
Actually, I count all the rotatopes as separate toratopes.
So I propose to only call "toratopes" those shapes which are smooth, i.e. which don't have any lower elements than their surfaces. Torinder doesn't belong here, since it has two 2D faces where its three separate faces meet.
However, there should be, indeed, a higher category combining rotatopes and toratopes and containing their various products. Perhaps "rotopes" might be a good name?
Ah, yes - the thing is that if we limit the prismatic product to SURFACES and ignore the inner volumes, then we get rotatopes which are smooth, but can have much lower dimension than the one they are embedded in. The simplest case here is product of two circles - the duocylinder margin. The tiger should be then gotten by "inflating" this margin much as you get torus by inflating the circle.
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