types of 4-space

[thigle]

hey people ! what types(aspects) and types(meta-levels) of 4-space you think usually ?

i mean, usually, we have flat 4-space - E4 (or R4 ? ) in minds when talking about tetraspace.

but we can also have elliptic or hyperbolic 4-spaces with change of curvature parameters.

we can also have projective 4-space, which is a cool thing. it is orientable though, as it's dimension is even. (sentimental as i prefer paradox of the nonorientable)

or topologically, we can consider just general 4-manifold before it gets any metric.

what about 'non-euclidean projective' 4-space ?
or complexified 4-space ?

*

any additions, generalisations, or relevant specifications ? :?

any other suggestions ? can anyone clarify on the differences / sameness between these different species of the conceptual genus "4space" ?

[bsaucer]

There are three "geometries" in 2D: Elliptic (S2), Euclidean (E2), and hyperbolic (H2). Projective space is elliptic. "Spherical" space is a "double covering" of projectice space.

In 3D space, there are actually eight "geometries". Besides the elliptic (S3)(projective and spherical), Euclidean (E3), hyperbolic (H3), there are "product" geometries (S2 x E1, and H2 x E1), "twisted" geometries, like the "twisted E3" (Nil3) and "Twisted H2 x E1" (~(SL)2). Note the "twisted S2 x E1" is really S3, so it doesn't count. Finally, there's one called Sol3, which is hard to describe.

In 4D space, there are actually 19 geometries! Again, there's S4, E4, and H4. The real projective space (RP4) has S4 geometry, but there is a complex projective plane (CP2) which has two "complex dimensions" which is really four real dimensions. The CP2 is different from RP4, and has its own geometry. There's also a real hyperbolic space (RH4) and a complex hyperbolic plane (CH2).

Of the product geometries, there are many (any 3D geometry x E1, and any 2D geometry x any 2D geometry). There are also some "twisted" versions, and several "Sol4" types.

I'm not sure how this works, but there are also "fake" maniflds, which are not "smoothable, such as the "fake" CP2, and the fake "RP4". I speculate that they may be "non-orientable" in a "complex" sense, even though they may be orientable in a "real" sense.

[Marek14]

Is there any place I could learn more about all those kinds of geometries?

[thigle]

great, essential reply ! thanxsalot bsaucer ! that opens a whole new field of potential understanding and solves up some of my older misunderstanding and fuzzy conceptual mixtures.
btw, i wanna go to that place with Marek14 too ! where it's at ?

meanwhile, you say that # of "geometries" is as follow for different dimensionality of spaces:

2-space: 3: (S2, E2, H2)
3-space: 8: (S3, E3, H3), products(S2xE1, H2xE1), twisted(TwE3(Nil3),Tw(H2 x E1)(~(SL)2), & the most mysterious (Sol3)
4-space: 19: (S4(=RP4 ?), E4, H4), complexified (CP2, CH2), (RH4), products(any 3d geometry x E1, any 2D geometry x 2D geometry), twists(...), (Sol4)types

what is then this: (S3 x RP1)
and this: (S7 x RP1)

4 & 8 dim spaces/geometries ? what kind of ?

actually, i don't understand these 'product' geometries. twisted are not hard, but how does this product function ? what is a geometrical product ? the cross product ?

also, could you please try to elaborate a bit on those Sol-types of geometries ? never heard of those. at least give some general info, you got to now that you teased us

[bsaucer]

I'm not sure exactly what S3 x RP1, since RP1 is a circle. However, the manifold would probably have the product geometry S3 x E1, since the circle has E1 geometry.

There's a manifold where you take a hyper-spherinder and connect the ends. If you connect them the direct way, you get a S3 x S1 product manifold. If you flip one end "inside out" (like making a Klein bottle), you get what's called the "non-trivial bundle, and the "x" has a tilde on top of it. It has the same geometry.

A geometric manifold has the same "curvature" at all of its points. However, the curvature is not always "isotropic" (same in all directions). Not all manifolds are geometric, but they can probably be cut up into geometric pieces.

An example of a product geometry is H2 x E1. Think of a space where all horizontal planes are hyperbolic, and the vertical lines are all Euclidean lines. In 4D you could have H2 x E2, where the x-y planes are Euclidean, and the w-z planes are hyperbolic. There are compact manifolds with that geometry: The product of the torus with the double torus, T2 x (T2 # T2).

The product of the torus with the pseudosphere is another E2 x H2 manifold, but it's not compact.

The "weird" geometries include "Nil" and Sol" type geometries. These are not products. There are two ways to describe Nil3. One way is a bundle of lines over a plane. At each point in the "horizontal plane" there is a vertical line, but the lines are all shifed up or down with respect to the line at the origin. It's almost E3, except that it is "twisted". Another way to describe it is a bundle of E2 planes ofer the line, where each plane suffers a slight "shear" with respect to the neighboring plane. Cut out a "cube" and connect up the opposite faces (remember the shear), and you get the "twisted torus" which has this geometry. Water can flow in a circle downhill all the way around! There are some Escher drawings based on this geometry. Note: The "horizontal planes" in Nil geometry are not true planes in this geometry. I'm not sure there are any true planes.

The ~(SL2) (The Tilde is above the "SL"), is similar except the "horizontal plane" is hyperbolic.

These "twisted" 3D geometries are called "Hopf fibrations".

Sol3 geometry is even harder to describe. Think of it as a bundle of E2 planes over the line, where each plane has a "lorentz boost" with respect to the neighboring plane. You can stretch the plane along the x axis, and shrink it by the same factor along the y axis. As you travel along the z-axis, the x-y planes expand exponentially along the x-axis, and shrink exponentially along the y-axis. Or they could be stretched along the X = Y lne and shrunk along the X = -Y line.

You can make a compact Sol3 manifold by taking a "cube" in this geometry and somehow connecting up the opposite faces.

[thigle]

what is a tilde ?

where do you learned all this ? any recommended books ?

hopf fibrations, well, in another thread on these forums, we couldn't really figure out what a 'fibration' is. can you please clarify that a bit too ? i've seen some animations of 'hopf fibrations on the torus', it seems that the circles on the torus are skew nontrivial geodesics, and during the fibration their radius pulses between 0 when the circles collapse to points on the circle through the middle of the torus. but still, i don;t understand what 'fibrating' generally is.

also, ever heard of 'polar euclidean space' ?

[bsaucer]

A "Tilde" is the wavy line "~" character. In topology it means "universal covering" when placed over a symbol.

The 8 3D geometries are described in Thurston's "Geometrization Conjecture". Hillman describes the 19 4D geometries in some of his works. The material is difficult to understand. Jeff Weeks describes the 8 3D geometries in his "Shape of Space" without giving their official names.

He also describes a "bundle", also called a "fibration". The best way to view a bundle is to think of a "base" manifold, such as a torus. Then at each point on the base, construct a "fiber" manifold (such as a line or a circle) perpendicular to the base. Each fiber differs from its neighboring fiber by a "homeomorphism", which maps the fiber onto a copy of itself by mapping each point to another (or same) point in the fiber.

If all fibers are related by the "identity mapping" (all the same), then the bundle is a "product". A torus is a product of a circle and a circle (S1 x S1). A Klein bottle is a bundle of cicles over a circle with a flip.

I recommend reading Jeff Week's "The Shape of Space". He also has a video by the same name, but I recommend the book.

[PWrong]

There's a manifold where you take a hyper-spherinder and connect the ends. If you connect them the direct way, you get a S3 x S1 product manifold.

Is this the same as the surface of the circle*sphere toratope? If so, what's the geometry of the sphere*circle toratope? Is the "geometry product" commutative?

Are the geometries you describe the only ones available? And is there a way to find all the geometries in nD, for any n?

[bsaucer]

The S3 manifold is not the sphere but the glome. It's called S3 because the hypersurface itself is three-dimensional at each of its points. However, it can be embedded in R4. The circle is S1.

The S3 x S1 manifold can be described as a "hyper-hyper-doughnut", which is a 4-manifold embedded in 5-space. It's geometry is S3 x E1. The manifold has that geometry at each of its points.

n topology, a torus is the product of circles. a 4-torus (T4) is S1 x S1 x S1 x S1. It is Euclidean. I call S(n-1) x S1 an "n-doughnut".

Geometric products are commutative. The symbol is the "x", not the "*". It is NOT a vector cross product, though. It is more like a cartesian product.

I don't know how many dimensions there are in five dimensions. I'm not sure if anyone does. The isotropic geometries (Sn, En, Hn) are easy, and so are the products. Projective and hyperbolic spaces can be real, complex, quaternion, or octonion. (Octonion dimensions are limited to 2, which has 16 real dimensions). The real n-projective spaces always have Sn geometry. Then there are geometries based on non-product fiber bundles, but questions arise as to how many different ways each bundle can "twist" or "squeeze".

[thigle]

but still, what about polar euclidean space, aka 'counterspace' ? does it have anything to do with birational geometries ?

it seems counterspace is the space within the pole of euclidean space that it is 'counter' to. can then there be a next level of that ? a counter(counterspace), or counter(^2)space ?

also is the Sol3 the co-called twisted cubic ? prolly not, just bluffing on false intuitions.

[PWrong]

The S3 manifold is not the sphere but the glome. It's called S3 because the hypersurface itself is three-dimensional at each of its points. However, it can be embedded in R4. The circle is S1.

Of course, I forgot. So S3 x S1 is the circle*glome toratope

n topology, a torus is the product of circles. a 4-torus (T4) is S1 x S1 x S1 x S1. It is Euclidean. I call S(n-1) x S1 an "n-doughnut".

Have you seen any of the toratope threads? Toratopes are a kind of generalisation of cylinders and torii, that Marek14 and I discovered. There are 10 kinds of toratope in 4D, and 24 in 5D.

but still, what about polar euclidean space, aka 'counterspace' ? does it have anything to do with birational geometries ?

I've never heard of counterspace, but I would imagine that polar euclidean space is just ordinary flat space in polar coordinates.

I don't know how many dimensions there are in five dimensions. I'm not sure if anyone does. The isotropic geometries (Sn, En, Hn) are easy, and so are the products. Projective and hyperbolic spaces can be real, complex, quaternion, or octonion. (Octonion dimensions are limited to 2, which has 16 real dimensions). The real n-projective spaces always have Sn geometry. Then there are geometries based on non-product fiber bundles, but questions arise as to how many different ways each bundle can "twist" or "squeeze".

What if we restrict ourselves to the easy geometries then? How many isotropic and product geometries are there in 5D? Is there an easy way to tell whether something is a real (and unique) geometry? I know a geometry is supposed to have constant curvature at every point, and two shapes with the same curvature have the same geometry. For instance, the surface of a torus or duocylinder doesn't count, because it's the same as E2.

[thigle]

i just found this link to counterspace. it surely doesn't seem like ordinary flat space in polar coordinates. haven't had time to go through it fully, but much of what i read through was way beyond me.

[bsaucer]

Is there a "parallel postulate" for counterspace? Say: Given a point and a line not through the point, there is exactly one point on the line that is not collinear with the given point... While at the same time you have a postulate saying that any two lines intersect?

[bsaucer]

Every dimension has three isotropic geometries: Sn, En, and Hn, except that in 1D, the only geometry is E1. So in 5D, you have S5, E5, and H5.

The total number of geometries in dimensions 1 thru 4 are 1, 3, 8, and 19. So the total number of product geometries in 5D can be found by finding products of lower dimensional geometries. Let Gn be an arbitrary geometry in "n" dimensions. Then you can have G4 x E1, G3 x G2, or G2 x G2 x E1.

Note that if any two of them are Euclidean, they can be combined: E2 x E3 = E5. There are other such cases, such as Sol3 X E1 is called Sol4_m,m. Note: in 4D, there are several kinds of Sol4: Sol4_0, Sol4_1, and Sol4_m,n (an infinite family of geometries). m and n are positive integers. In the special case where m=n, you get the Sol3 x E1 product.

As for complex, quaternion, or octonion spaces, a complex manifold has an even number of real dimensions. A quaternion manifold has 4n real dimensions, and an octonion manifold has 8n real dimensions.

In each dimension n there is the RPn, and RHn. RP1 is the circle (S1), and RH1 is the line (E1).

In each even dimension 2n, there is the CPn, and CHn. CP1 is the sphere (S2), and CH1 is the plane (E2) (I think: Correct me if I'm wrong).

In each 4n dimension, there is the QPn anc QHn (Some use "H" instead of "Q" for "Hamilton"). QP1 is S4. QH1 is E4.

In 8 dimensions, the octonion spaces are OP1 = S8, and OH1 = E8. In 16 real dimensions you have OP2 and OH2. That's as far as you can go with octonion spaces!

Each of these projective and hyperbolic planes and higher spaces have their own geometry. Note that RPn has Sn geometry, since the n-sphere, Sn, is a "double-covering" of the RPn manifold.

[Marek14]

I was unable to find some more precise information for all these types dimensions. I guess the best way would be to find (or make) a sort of table depicting the various properties of the geometries so that each row would have an unique combination...

[bsaucer]

I've found my paperwork on the 19 geometries, which I will list here.

Compact geometries:
1. S^4
2. CP^2
3. S^2 x S^2

Solvable Lie geometries:
4. E^4
5. Nil^4
6. Nil^3 x E^1
7. Sol^4_0
8. Sol^4_1
9. Sol^4_m,n

Semisimple geometries:
10. RH^4
11. RH^2 x RH^2
12. CH^2
13. F^4

Mixed Geometries:
14. H^2 x E^2
15. ~SL(2,R) x E^1
16. H^3 x E^1
17. S^2 x E^2
18. S^2 x H^2
19 S^3 x E^1

These are described in "Four Manifolds, Geometries, and Knots" by J. A. Hillman.

[Marek14]

Thanks, I found the text on the web. I'll have a look.

Edit: well, I had a look, but it's a bit incomprehensible to me I suppose I'd need some more "geometrical" approach.

Take something easier, like S2xE1, for example. How do the straight lines look there? Some will be the great circles of S2, some will be straight lines of E1, but what about the rest? The best I'm able to visualize them is as sort of spirals, with two coordinates going in a great circle while the third steadily moves upwards. How would look the lines, planes, hyperplanes, in the various non-isotropic geometries?

[thigle]

does each of these spaces have its own algebra ?

[bsaucer]

You said:

Take something easier, like S2xE1, for example. How do the straight lines look there? Some will be the great circles of S2, some will be straight lines of E1, but what about the rest? The best I'm able to visualize them is as sort of spirals, with two coordinates going in a great circle while the third steadily moves upwards. How would look the lines, planes, hyperplanes, in the various non-isotropic geometries?

Lines are simply geodesics. The path continues within the space without twisting or curving to the left, right, up, down, etc. It might help to "flatten" out the space, like flattening out the torus to a square. Draw the straight line, and then curve up the space again. A line in a torus may be a spiral twisting around and through the hole. Some spaces cannot be completely flattened out, but they can sometimes be "unrolled" into a cylinder or spherinder, etc.

Planes are a bit tougher. To construct a "plane" in a 3-manifold, begin at a point, and consider its neighborhood. The neighborhood is "nearly Euclidean", so a plane can be constructed within the neighborhood. You can specify a line through the point, and construct the plane perpendicular to the line. This plane is "flat", not rounded like a bowl, or saddle shaped.

Now "extend" the surface beyond the neighborhood. Through the point, construct the set of lines (geodesics) within the tiny "plane". Extend the lines (as geodesics) as far as they will go. These lines form a surface, which contains the tiny "plane" in the neighborhood of the point.

Now you must "test" the surface. At each of the other points of the surface, the surface must be flat, not bowl-shaped or saddle-shaped. Also, the set of lines "geodesics" emanating from each point must remain IN the surface. They must not "leave" the surface like a tangent.

The test passes in isotropic spaces. For example, S3, the "planes" are S2. They may appear "round" to an outsider, but they are actually flat within the space. The concept of "flatness" does not emply zero curvature. For example, in H3, the horospheres have zero curvature, but are rounded. In S3, the Clifford surfaces have zero curvature, but are saddle shaped. The truly flat surfaces (planes) in S3 and H3 have positive and negative curvature, respectively.

In non-isotropic surfaces, the attempt to construct a "plane" at a point sometimes fails to remain flat everywhere, so the plane cannot be constructed.

[Marek14]

I thought that perhaps a plane could be constructed by taking three non-colinear points and the rule "Whenever two points are in a plane, so is the whole straight line containing them." How would that work out?

[bsaucer]

Your argument is based on two postulates found in Euclidean geometry:

1. Through any three non-collinear points there exists one, and only one plane.

2. If two points of the line lie in a plane, then all of the points of the line lie in the plane. (Axiom of the plane).

I'm not sure both of these postulates are true in all geometries. Suppose you start with three points. There are many surfaces that contain them. How can we establish that one of these surfaces (and only one) is a plane?

Suppose we have a surface containing the three points. Then cunstruct the three lines containing the points in pairs. These three lines either lie in the surface, or they don't. Assume that they do. So far, so good.

Now pick two more points in this same surface. Do the lines that contain them also lie in the surface? If not, then the surface is not a plane.

This must be proven for all pairs of points in the surface. I think one of these postulates must break down in some of these geometries. Either you can't always construct a plane through any three non-collinear points, or if there is a special surface that meets this requirement, it isn't "flat", so that lines completely lie within them. Suppose there are lines tangent to the surface at one point?

[PWrong]

Lines are simply geodesics.

geodesics are simple ?

According to that, a geodesic is a curve with locally minimizing length. Does that mean a "plane" is a surface with locally minimizing area?

[moonlord]

Erm, guys, when do you learn topology at that level? All we've done at school is E2 and some E3...

[thigle]

bsaucer asked:

Is there a "parallel postulate" for counterspace? Say: Given a point and a line not through the point, there is exactly one point on the line that is not collinear with the given point... While at the same time you have a postulate saying that any two lines intersect?

i forwarded the question to Nick Thomas, this is his reply:

...
Yes there is !

In counterspace there are no parallel planes, but there are parallel points (which follows from the duality involved). So the postulate is:

"Given any line and a plane not containing it, there is exactly one line in that plane which is polar-parallel to the given line"

What this means is that there is only one line in the plane which contains a plane both through the given line and the counterspace absolute point (polar to the plane at infinity).
It is easily seen as the absolute point and the given line determine a plane which intersects the given plane in just one line.
What may be confusing if you are not used to counterspace is that polar-parallel lines meet!
However this is simply polar to the fact that in Beltrami's version of the postulate the two ordinarily parallel lines share a common plane.

I hope this helps.

Best wishes,

Nick Thomas

[wendy]

i normally consider there to be one 4d space, of various curvatures.

[thigle]

sure. for dimensionality is a category of higher order than curvature , which is rather an attribute of dimensionality than its Kind.

[Batman3]

I am learning topology. There can be many different topologies on a set. A set could be something like the elements of tetraspace. I suppose a physical territory is a kind of set, the the goverment over the territory is a topology over the set and the citizens ruling the gov't are elements of the territory. The empty set would be a territory considered when it is not ruled by any elements. An element 'could' be God. Or not, if He chooses not. A proper subset of a gov't is not as bound by as many elements as the set. U.S. citizens are bound by the federal set and subsets.

That is how I am trying to interpret math now. I.e. in a moral vein.

[Batman3]

If a set such as 4d is described by more than 1 topology, it could be considered as having more than 1 'space'.

[wendy]

Real space is of indefinate curvature (ie it could be spheric / euclidean / hyperbolic). Therefore it must be possible to approach the geometric space from this aspect. Since the formulae that i use in circle drawing make no real presupposition of space (except that the circle-circumference is also that of an euclidean circle), the notion of S vs E vs H is really indeterminate.

I am not sure about the fibulation is uniquely four-dimension. The existance of spaces like CE3 and CE4 suggest that a corresponding fibulation must exist in both six and eight dimensions.

W

[thigle]

what is CE3 ? complexified euclidean ? each dim as complex value ?