Hey all.
Yesterday, I began to realize a whole other area of 4D geometry that proves to be very interesting.
First of all, I was thinking of how the duocylinder's bounding manifolds may be thought of as 3D cylinders bent around in 4D until their ends meet. This is rather suggestive of the approach in topology of gluing faces together; so I wondered what would happen if we took a cube, twisted it around in the ZW plane, and glued its top face to its bottom face. The resulting manifold would have 4 cylinder-shaped holes formed by the +/- X and +/- Y faces of the cube. Filling in these holes with 4 cylinders, we get the cubinder.
Then I was reading up on the n-torus, and was trying to imagine what a 3-torus might look like based on what I've learned of 4D visualization so far. Is it the same shape as the chorage traced out by a 3-ball if it was rotated around a plane outside of it? It seems, from the Wikipedia page, that the 3-torus is rather the trace of a 2-torus rotated around a plane. This would imply that there are at least 2 different types of torii in 4D: a spherical torus, and a toroidal torus! The former has only one "doughnut hole", but the latter, if my intuition is correct, has two such holes. So the two shapes would be topologically distinct.
Not only so; if you take the cubinder (minus the 4 cylindrical "lids") and glue another pair of opposite faces together, you get a cylindrical torus: a shape with 2 torus-shaped cavities which, if covered with two torus-shaped lids, would bound a 4D volume. Is this a third type of torus possible in 4D? (Of course, if you glue the last pair of opposite faces together, you get the 3-torus.)
What I'm curious about, is if any of these shapes are topologically equivalent to the duocylinder. I suspect not, unless I missed something obvious, because the duocylinder has two bounding surfaces that are circular in two disjoint planes.
Of course, things start to get interesting if you make Moebius-strip-like twists with these topological constructions. I'm still trying to picture how the real projective plane would embed in 4D. I'd love to get a good handle on it. (Unfortunately, I'm not sure if I'm ready to deal with 2-manifolds in 4D yet; so far, I've limited myself to 3-manifolds because at least they are more restricted in degrees of freedom and so are easier to comprehend. Nevertheless, I've come to the conclusion that the Klein bottle is a misnomer: a 4D liquid would not be confined in a Klein bottle; it would just spill off its "sides", just like water spills off a Moebius strip. The container must be a closed 3-manifold in order to hold the liquid in place.)
Cubinders, torii, and topology
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Cubinders, torii, and topology
by quickfur » Tue Aug 09, 2005 4:51 am
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by wendy » Tue Aug 09, 2005 5:24 am
In four dimensions, holes come in two flavours, which i call ch and hc. In three dimensions, you only have hh holes.
One you get by taking a sphere-prism, and connecting top and bottom, rather like connecting a hose end to end. This hole is spanned by a hedrix outside, and a chorix inside, and so is hc. It's the kind of holes bridges make (ie the space around the bridge).
The other kind of hole is made by taking a sphere-prism, and rolling the top to meet the bottom, rather like taking off a sock. This hole is chorix-outside, hedrix inside. It's the same kind of hole that tunnels make.
If you do an inversion of a surface whith x hc and y ch holes, you get a surface with x ch and y hc holes. The genus of the solid (solids, rather than surfaces, have topological genus [kind], is then yHxC. In three dimensions, there is only one kind of hole H, so the genus of both the inside, and outside is gH.
The torus-shaped hole is actually two holes, one inside the other, these are 1 hc + 1 ch holes, so its genus is 1H1C.
You get some rather interesting holes, such as 3H1C, which occur when you take a page in 4d (remember, this is a 3d figure), and connect top to bottom, left to right, and up to down, torus-wise.
When you deal with holes in topological polytopes, you have to deal with hh holes as well. These occur in the faces.
There is actually a torus-product, where you can multiply polytopes to get a torus. This one reduces the dimension (by one for each added item) as one multiplies: so a product of polyhedra (3d things), gives a polyteron (5d thing). A product of three polygons gives a 4d thing, but the product is order-dependent: that is, abc, acb, bca, bac, cab, cba are six distinct shapes. In four dimensions, one can multiply polygons by polyhedra, for example, the twelfty-faced figures 'dodecahedron-decagon torus' (which is hc), and the 'decagon-dodecahedron torus', which is a ch figure.
In five dimensions, you have ht, th, cc holes, in six, you have also hp, tc, ct and ph holes, and so on.
The e-class holes (ie disjoint or cavity holes) make the process even more interesting, but one would not regard disjoint (ie ee, eh, ec, et, ep, ex) or cavities (ee, he, ce, te, pe, xe) as proper holes.
Composite holes, like hhh, usually degenerate into smaller holes, like hc + ch, but i have not looked at this rule as yet.
Wendy
One you get by taking a sphere-prism, and connecting top and bottom, rather like connecting a hose end to end. This hole is spanned by a hedrix outside, and a chorix inside, and so is hc. It's the kind of holes bridges make (ie the space around the bridge).
The other kind of hole is made by taking a sphere-prism, and rolling the top to meet the bottom, rather like taking off a sock. This hole is chorix-outside, hedrix inside. It's the same kind of hole that tunnels make.
If you do an inversion of a surface whith x hc and y ch holes, you get a surface with x ch and y hc holes. The genus of the solid (solids, rather than surfaces, have topological genus [kind], is then yHxC. In three dimensions, there is only one kind of hole H, so the genus of both the inside, and outside is gH.
The torus-shaped hole is actually two holes, one inside the other, these are 1 hc + 1 ch holes, so its genus is 1H1C.
You get some rather interesting holes, such as 3H1C, which occur when you take a page in 4d (remember, this is a 3d figure), and connect top to bottom, left to right, and up to down, torus-wise.
When you deal with holes in topological polytopes, you have to deal with hh holes as well. These occur in the faces.
There is actually a torus-product, where you can multiply polytopes to get a torus. This one reduces the dimension (by one for each added item) as one multiplies: so a product of polyhedra (3d things), gives a polyteron (5d thing). A product of three polygons gives a 4d thing, but the product is order-dependent: that is, abc, acb, bca, bac, cab, cba are six distinct shapes. In four dimensions, one can multiply polygons by polyhedra, for example, the twelfty-faced figures 'dodecahedron-decagon torus' (which is hc), and the 'decagon-dodecahedron torus', which is a ch figure.
In five dimensions, you have ht, th, cc holes, in six, you have also hp, tc, ct and ph holes, and so on.
The e-class holes (ie disjoint or cavity holes) make the process even more interesting, but one would not regard disjoint (ie ee, eh, ec, et, ep, ex) or cavities (ee, he, ce, te, pe, xe) as proper holes.
Composite holes, like hhh, usually degenerate into smaller holes, like hc + ch, but i have not looked at this rule as yet.
Wendy
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by Marek14 » Tue Aug 09, 2005 8:49 am
Could the holes have something to do with various types of cones?
In 3D we have a single type of cone. This cone divides class of one-part hyperboloids from two-part hyperboloids. One-part hyperboloid is, basically, a "hole" and a given hole in 3D space could be pierced through by a shape something topologically equivalent to one-part hyperboloid.
In 4D, there are two different types of cones -
x^2+y^2+z^2-w^2=0
and
x^2+y^2-z^2-w^2=0
The first type of cone can be made into one-part 4-hyperboloid
x^2+y^2+z^2-w^2=1, which could be put through certain 4D holes.
The second cone doesn't divide the 4-space into well-defined "inside" and "outside", though - it divides it in two equivalent subspaces. If we put positive or negative number instead of 0, the shape of the surface will be still the same.
If we define the origin [0,0,0,0] as being "inside" 4-hyperboloid
x^2+y^2-z^2-w^2=1,
then is there a kind of hole this hyperboloid could be sticked through?
In 3D we have a single type of cone. This cone divides class of one-part hyperboloids from two-part hyperboloids. One-part hyperboloid is, basically, a "hole" and a given hole in 3D space could be pierced through by a shape something topologically equivalent to one-part hyperboloid.
In 4D, there are two different types of cones -
x^2+y^2+z^2-w^2=0
and
x^2+y^2-z^2-w^2=0
The first type of cone can be made into one-part 4-hyperboloid
x^2+y^2+z^2-w^2=1, which could be put through certain 4D holes.
The second cone doesn't divide the 4-space into well-defined "inside" and "outside", though - it divides it in two equivalent subspaces. If we put positive or negative number instead of 0, the shape of the surface will be still the same.
If we define the origin [0,0,0,0] as being "inside" 4-hyperboloid
x^2+y^2-z^2-w^2=1,
then is there a kind of hole this hyperboloid could be sticked through?
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by quickfur » Tue Aug 09, 2005 2:53 pm
Marek14 wrote:[...]In 4D, there are two different types of cones -
x^2+y^2+z^2-w^2=0
and
x^2+y^2-z^2-w^2=0
Wait, are you referring to 2-manifolds here or 3-manifolds? A 2-manifold in 4D only fills a region equivalent to a string, and can only define the boundary of a zero-chorage (4D volume) shape. If my intuition is correct, you need 3-manifolds if you want to get surfaces that bound non-zero chorages, indeed, that can divide 4-space.
This is why it is inadequate to define the duocylinder as x<sup>2</sup>+y<sup>2</sup>=r<sup>2</sup>; z<sup>2</sup>+w<sup>2</sup>=r<sup>2</sup>. These pair of equations only define its ridge; you need inequalities in order to fully define its surface.
As to cones in 4D, I have not fully explored all possibilities, but I do know of at least two varieties: the spherical cone, which one obtains by stacking spheres with radii proportional to their displacement from the origin along the W axis; and the cylindrical cone (or conical cone), which one gets by stacking cones instead.
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by quickfur » Tue Aug 09, 2005 3:26 pm
quickfur wrote:[...]Wait, are you referring to 2-manifolds here or 3-manifolds? A 2-manifold in 4D only fills a region equivalent to a string, and can only define the boundary of a zero-chorage (4D volume) shape. If my intuition is correct, you need 3-manifolds if you want to get surfaces that bound non-zero chorages, indeed, that can divide 4-space.
Actually, I just realized that an equation in 4 variables does define a 3-manifold. Oops, my bad.
On that note, it seems that the spherical cone does correspond with the first equation. I'm not sure if the cylindrical cone corresponds with the second equation; maybe it does.
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by wendy » Wed Aug 10, 2005 2:25 am
The equation for the duocylinder is
r = max( rss(w,x), rss(y,z)).
where rss = root-sum-squares.
In relation to holes, a hole is a linkage between the inside and outside, and so requires complementry links (that add to N+1).
When you demonstrate a hole by a non-vanishing M-sphere-surface, the hole is actually defined by what it takes to be added to prevent the M-sphere-surface from forming: that is an N-M+1 manifold.
For normal connexions, we require N-M+1 and M to be both 2 or greater.
It is possible to have half-a-hole, such occur in eg multitopes and where the holing device runs to a point. In the first instance, internal walls prevent the formation of spheres, while the second one has an infinite region, that is discontinious around a point, for example.
For a normal torus, one can have two different circles, one that goes outside (through where the hub would go), and the other inside the tyre, where the tube would go. To prevent a circular tube going in, one would put up a wall inside the tyre, this is the inside hedron. The outside closure is like when you put a tyre on a hub. So a hole in 3d is actually a linkage of outside/inside of the type hh. Inversion swaps inside and outside, and so reverses the hole-symbol.
In four dimensions, a tunnel-like hole has an inside closure of a 3d manifold, and an outside closure of a 2d closure, so it's a hc hole. Inversion of this makes a ch hole, which is like what you get with a bridge, or pipe. (literally, ground going through the air). To stop the sphere from forming around the bridge, one needs a 2d manifold between the bridge and the ground. so it's hole form is a hc.
I'm not sure about the other thing, (the conics that Marek mentions), since they are not really "solids", and the hole is either ch or hc depending on what side of the surface one is on. That's the real point. Surfaces don't have a genus, solids do.
r = max( rss(w,x), rss(y,z)).
where rss = root-sum-squares.
In relation to holes, a hole is a linkage between the inside and outside, and so requires complementry links (that add to N+1).
When you demonstrate a hole by a non-vanishing M-sphere-surface, the hole is actually defined by what it takes to be added to prevent the M-sphere-surface from forming: that is an N-M+1 manifold.
For normal connexions, we require N-M+1 and M to be both 2 or greater.
It is possible to have half-a-hole, such occur in eg multitopes and where the holing device runs to a point. In the first instance, internal walls prevent the formation of spheres, while the second one has an infinite region, that is discontinious around a point, for example.
For a normal torus, one can have two different circles, one that goes outside (through where the hub would go), and the other inside the tyre, where the tube would go. To prevent a circular tube going in, one would put up a wall inside the tyre, this is the inside hedron. The outside closure is like when you put a tyre on a hub. So a hole in 3d is actually a linkage of outside/inside of the type hh. Inversion swaps inside and outside, and so reverses the hole-symbol.
In four dimensions, a tunnel-like hole has an inside closure of a 3d manifold, and an outside closure of a 2d closure, so it's a hc hole. Inversion of this makes a ch hole, which is like what you get with a bridge, or pipe. (literally, ground going through the air). To stop the sphere from forming around the bridge, one needs a 2d manifold between the bridge and the ground. so it's hole form is a hc.
I'm not sure about the other thing, (the conics that Marek mentions), since they are not really "solids", and the hole is either ch or hc depending on what side of the surface one is on. That's the real point. Surfaces don't have a genus, solids do.
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by quickfur » Wed Aug 10, 2005 5:50 am
wendy wrote:The equation for the duocylinder is
r = max( rss(w,x), rss(y,z)).
where rss = root-sum-squares.
Odd. This means that the duocylinder is, at least algebraically, a quartic (4th degree) surface.
[...]In four dimensions, a tunnel-like hole has an inside closure of a 3d manifold, and an outside closure of a 2d closure, so it's a hc hole.
Would this be the type of hole resulting from the detachment of one of the bounding manifolds of the duocylinder?
Inversion of this makes a ch hole, which is like what you get with a bridge, or pipe. (literally, ground going through the air). To stop the sphere from forming around the bridge, one needs a 2d manifold between the bridge and the ground. so it's hole form is a hc.
Would this be like a hole that results when you punch through a tetracube with a spherindrical spear?
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by wendy » Wed Aug 10, 2005 12:10 pm
The duocylinder surface comprises of two faces, each of which has an internal hole, ie each face is h. The margin then represents a surface of either a hh or hh solid (depending on which face is interior).
A single face of a duocylinder makes for a hh figure: but if the thing is given in 4d some substance, it becomes a hc figure, since the body needs a c to span it.
Running a sphere-line prism through a cube, makes for a ch hole. This is a tunnel [Chunnel = channel tunnel], and needs a c-manifold (3d) to span inside, and a 2d manifold (ie along the length of the tunnel, to the outer wall), to stop non-vanishing spheres forming.
The duocylinder is not a quartic, since the equation for it is an intersection of two conics, ie
max(rss(w,x), rss(y,z))
max is a quantum-function: that is, it changes cathostropically when one gets through points like 0,1,0,1.
A single face of a duocylinder makes for a hh figure: but if the thing is given in 4d some substance, it becomes a hc figure, since the body needs a c to span it.
Running a sphere-line prism through a cube, makes for a ch hole. This is a tunnel [Chunnel = channel tunnel], and needs a c-manifold (3d) to span inside, and a 2d manifold (ie along the length of the tunnel, to the outer wall), to stop non-vanishing spheres forming.
The duocylinder is not a quartic, since the equation for it is an intersection of two conics, ie
max(rss(w,x), rss(y,z))
max is a quantum-function: that is, it changes cathostropically when one gets through points like 0,1,0,1.
The dream you dream alone is only a dream
the dream we dream together is reality.
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the dream we dream together is reality.
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