Uninitiated

Higher-dimensional geometry (previously "Polyshapes").

Uninitiated

Postby moonlord » Sun Feb 05, 2006 7:55 pm

I'm sorry for such a question, but can you explain to me the types of spaces? I'm only familiar with euclidian... Thanks in advance.
moonlord
Tetronian
 
Posts: 605
Joined: Fri Dec 02, 2005 7:01 pm
Location: CT, RO, CE EU

Postby Marek14 » Mon Feb 06, 2006 7:55 am

The Euclidean space is the one which results from the ancient form of Euclid's Fifth Postulate. In Euclidean plane, for every straight line, and every point not lying on it, there is exactly one straight line which passes through the point and doesn't intersect the line. You get other two basic geometries by replacing this axiom with another.

If you claim that NO such line exists, i.e. that any two straight lines intersect, you get elliptic geometry. This is similar to geometry on the surface of the sphere, but since great circles on the sphere intersect in two points instead of just one, elliptic geometry considers any two antipodal points on the sphere identical. Sphere, basically, is made up of two identical copies of full elliptic space. We will use spherical geometry here, as it's better known and a bit simpler to think about.

If you claim that MORE THAN ONE such line exists, you get in the realm of hyperbolic geometry.

Some of the important properties of those spaces are:

1. Curvature

Euclidean space has zero curvature. Spherical space is positively curved, while hyperbolic is negatively curved. You can imagine this as taking a base straight line and constructing two perpendiculars in different points. In Euclidean space, those lines will be parallel. In spherical space, as you move along one of the lines, you'll find the other line to come closer and closer, until they intersect. In hyperbolic space, the other line starts to go away from yours, quickly disappearing in the distance. Intuitively, you might grasp the fact that spherical space is "smaller" than Euclidean (as it is finite, while Euclidean space is infinite), but also that hyperbolic space is "larger" than Euclidean.

2. Angular sum of triangles.

In Euclidean space, the sum of angles in a triangle is always pi. In spherical space, it's LARGER, in hyperbolic space it's SMALLER. But that's not the only interesting thing. The other interesting thing is that the area of the triangle is directly proportional to it's "excess" or "defect" - i.e. how much its angle sum deviates from pi. In both S and H spaces, very small triangles have the angle sum close to pi.
Maximum possible angle sum in S-space is 5*pi, at which point the triangle would comprise the whole sphere. In H-space, the minimum possible angle sum is zero. At this moment, the triangle has infinite sides, but still only a finite area. This also means that there are finite, compact areas in H-space that can't be covered with any triangle, no matter what size.

3. Demise of congruent triangles

Unlike Euclidean space where two triangles can have the same angles despite difference in side lengths, the angles determine the sides in S and H spaces. This means that many things we do daily (like painting a picture, or drawing floor plans) would be very hard to do in S-space, and almost impossible in H-space. It also means that there is "absolute scale" in these spaces.

4. Isocurves
"Isocurve" is simply any curve with constant curvature. In Euclidean space we have circle (with positive curvature) and straight line (with zero curvature). In other spaces, it is more complex.
The basic notion is that curvature of isocurve must be GREATER OR EQUAL to the curvature of the whole space. The isocurve is straight line if and only if it's curvature is EQUAL to the space curvature. This obviously holds in E-space, which has curvature zero.
Spherical space has positive curvature, so ONLY circles are accepted as isocurves. The straight lines are "great circles" with the same radius as the sphere.
In H-space, though, the space curvature is negative, and so we have three distinct classes of isocurves:

a) Circles with positive curvature
b) Horocycles with zero curvature
c) Pseudocycles with negative curvature

Straight line is then a special case of pseudocycle.

The corollary is that if we construct a two-dimensional analogue of the horocycle, the HOROSPHERE, we find that its surface has common Euclidean planar geometry. Living in H-space doesn't need you to abandon the Euclidean amenities completely!

This also ties to the "projectant's nightmare" in hyperbolic world. In spherical world, you can do your drawings in scale by drawing on surface of a smaller sphere than the all-space. In hyperbolic space, though, you would be forced to draw on pseudospherical surface, but the scale would be LARGER - in other words, there is no way to perfectly shrink a planar figure.

5. Tilings.
As you might know, Euclidean plane can be tiled with polygons in three different ways:
{3,6} - six triangles per vertex.
{4,4} - four squares per vertex.
{6,3} - three hexagons per vertex.

There are no other possibilities, since the inner angles of regular polygons are fixed. In S-space, they are not fixed, but we demand them to be larger than in E-space. Thus, we have five regular tilings, which correspond to five Platonic polyhedra:

{3,3}, {3,4}, {3,5}, {4,3}, {5,3}

In H-space we demand the inner angles to be smaller. Since there is no limit of how small they can be, there is an infinite family of regular tilings.

{3,7+}, {4,5+}, {5,4+}, {6,4+}, {7+,3+}
where n+ means "n or larger"

While 3D E-space can only be tiled with cubes (of all regular polyhedra), there are more possibilities for S and H spaces. The E-tiling is written as {4,3,4}, reading as "cubes ({4,3}), meeting by four at an edge, and by eight ({3,4}) at vertex. There are six regular tilings of S-space:

{3,3,3}, {3,3,4}, {3,3,5}, {3,4,3}, {4,3,3}, {5,3,3}

There are only four regular tilings of H-space if we demand that the cells must be finite in their extent:

{3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Postby bo198214 » Mon Feb 06, 2006 9:11 am

Wow, thanks a lot for this really informative introduction!

Does anyone know btw about the tilings 4d E-Space?
bo198214
Tetronian
 
Posts: 692
Joined: Tue Dec 06, 2005 11:03 pm
Location: Berlin - Germany

Postby wendy » Mon Feb 06, 2006 9:22 am

The regular tilings in 4D are as follows.

S4 {3,3,3,3}, {3,3,3,4}, {4,3,3,3}
E4 {4,3,3,4}, {3,3,4,3}, {3,4,3,3}
H4 {3,3,3,5}, {5,3,3,3}, {4,3,3,5}, {5,3,3,4}, {5,3,3,5}
H4 stars {5/2,5,3,3}, {3,3,5,5/2}, {5,5/2,5,3}, {3,5,5/2,5}
CE4 3{3}3{3}3{3}3{3}3

and that's it for the regular tilings in 4D
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby bo198214 » Mon Feb 06, 2006 11:54 am

Marek and Wendy, you are really some sweethearts! ;)
bo198214
Tetronian
 
Posts: 692
Joined: Tue Dec 06, 2005 11:03 pm
Location: Berlin - Germany

Postby thigle » Mon Feb 06, 2006 1:45 pm

yep. these people really are incredible. (in the good sense)

good to be here.
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm

Postby moonlord » Tue Feb 07, 2006 6:38 pm

Nice. Thank you. I fancy that you can represent a nS space in a (n+1)E and perhaps a nE in a (n+1)H exactly, but you cannot the other way round. May the inner exterior surface of a torus be thought of as a portion of a H space? If so, then what dimensional hyperbolic?

By 'inner exterior surface of a torus' I mean the surface on it's out side that lies in the C X d (cartesian product between the circle ON WHICH another circle moves to generate the torus and the symmetry axis of the torus).
moonlord
Tetronian
 
Posts: 605
Joined: Fri Dec 02, 2005 7:01 pm
Location: CT, RO, CE EU

Postby Marek14 » Wed Feb 08, 2006 7:11 am

IIRC, the parts of torus on the surface of its inner side do indeed have hyperbolic geometry.

That being said, complete H2 can't be inserted in E3 without deformation. It would be like fitting Euclidean plane onto sphere - it's just too large.

There are methods to show hyperbolic features with sufficiently deformed systems, however. For example, the Poincare disc.

Draw a circle in the plane, and decree a hyperbolic plane to be the set of all points inside of the disc. Then, decree that a line between two points will be a circular arc perpendicular to that boundary circle (or "horizon"). This gives you working model of hyperbolic geometry, most notably one where angles will agree (lengths, of course, will not).
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Postby wendy » Wed Feb 08, 2006 8:03 am

You can represent any curvature (eg spheric, euclidean, hyperbolic), in any other curvature. It is this particular observation that implies that if you can't construct it in euclidean space, you can't do it in H or S either.

It all depends on what you call "straight".

If for a given circle, an isocurve is straight if it crosses the given circle at diametric points, the geometry is spheric. or glomos

If for a given point, an isocurve is straight if it contains it, then it is straignt, gives the Euclidean geometry or horos

If for a given circle, an isocurve is straight if it crosses it at right angles, the resultant geometry is hyperbolic or bollos.

One can model this by the embedded inversive sphere, where a straight isocurve on the sphere is represented as a plane in the embedding space containing a given point.

Imagine you have a sphere I2 embedded in E3. On this one has circles of any size, which are the result of planes E2 intersecting I2 in I1. One then defines "straight" as the intersection of I2 with an E2 containing a point U.

The resulting geometry derived is euclidean, sphric, or hyperbolic as U is on, in the interior, or at the exterior of the sphere.

When U actually coinsides with the centre if I2, then figure is then an isometric or natural representation of that space. For this reason, we can place U at the centres of a sphere or horosphere, to get natural representations of the spheric and euclidean geometries in those geometries that support spheres or horospheres.

The interior of a torus is probably hyperbolic, but in practice, the full torus has an overall geometry typically euclidean in nature.

W
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby moonlord » Wed Feb 08, 2006 1:02 pm

wendy wrote:If for a given circle, an isocurve is straight if it crosses the given circle at diametric points, the geometry is spheric. or glomos

If for a given point, an isocurve is straight if it contains it, then it is straignt, gives the Euclidean geometry or horos

If for a given circle, an isocurve is straight if it crosses it at right angles, the resultant geometry is hyperbolic or bollos.


You lost me here.
1. You have a circle and a curve. You know the curve is a isocurve and it crosses the circle at diametric points. Therefore you are in spheric geometry and the curve is called 'straight'. Is glomos another name to it or did you mean something else (perhaps, glomar geometry)?
2. You have a point and a curve. You know the curve is a isocurve and it contains the point. Therefore you are in euclidean geometry and the curve is called 'straight'. How about horos? What does it mean?
3. You have a circle and a curve. You know the curve is a isocurve and it crosses the circle at right angles. Therefore you are in hyperbolic geometry. Is the isocurve a horocircle, then? What about bollos?

Is what I wrote correct? I personally don't think so...

wendy wrote:One can model this by the embedded inversive sphere, where a straight isocurve on the sphere is represented as a plane in the embedding space containing a given point.

Imagine you have a sphere I2 embedded in E3. On this one has circles of any size, which are the result of planes E2 intersecting I2 in I1. One then defines "straight" as the intersection of I2 with an E2 containing a point U.

The resulting geometry derived is euclidean, sphric, or hyperbolic as U is on, in the interior, or at the exterior of the sphere.

When U actually coinsides with the centre if I2, then figure is then an isometric or natural representation of that space. For this reason, we can place U at the centres of a sphere or horosphere, to get natural representations of the spheric and euclidean geometries in those geometries that support spheres or horospheres.


I don't think I really understood this. Maybe you can explain simpler. Maybe I just need to give it up a few years... It just seems horrendously complicated...

When did you learn these things, by the way? Was it highschool? Maybe I just have to study more before I may start non-euclideans.

Marek14 wrote:IIRC


What does IIRC mean :lol: ?
moonlord
Tetronian
 
Posts: 605
Joined: Fri Dec 02, 2005 7:01 pm
Location: CT, RO, CE EU

Postby Marek14 » Wed Feb 08, 2006 6:39 pm

IIRC means "If I remember correctly", IIRC.
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Postby moonlord » Wed Feb 08, 2006 8:21 pm

Thanks! I didn't know that...
moonlord
Tetronian
 
Posts: 605
Joined: Fri Dec 02, 2005 7:01 pm
Location: CT, RO, CE EU

Postby wendy » Thu Feb 09, 2006 8:28 am

The names "glomos", "horos" and "bollos" are the PG equivalents to the spheric, euclidean and hyperbolic spaces.

The inversion-space is simply a conformal model of all three geometries, but i realised they could be combined into one by using a point representing straight (ie U). The three geometries are thus united, as they should.

The point and circle given in the models are the absolutes of these geometries. In practice, they can be implemented by way of a point in an embedding space.

You don't learn these things at school, i suppose. You just use some good old common sense and clear thinking, and all is wonderfully clear.

glomos is derived from glomo- stem, which designates a positive curvature sphere. horos comes from horo- eg horosphere. bollos is from the stem bollo- (negative curvature), which is from hyperbolic, in much the same way that omnibus gives "bus" frall . The PG is a fairly large language project, and its words are largely meaning-derivable from its stems.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby thigle » Thu Feb 09, 2006 12:16 pm

wendy said:
The inversion-space is simply a conformal model of all three geometries, but i realised they could be combined into one by using a point representing straight (ie U). The three geometries are thus united, as they should.

i cannot resist to ask: wendy, did you notice the counterspace issue in the types of 4-space thread ? please look at NickThomas' answer that I pasted there.

do you see any parallels between your inversion-space model and the space/countespace idea ?
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm

Postby wendy » Fri Feb 10, 2006 8:09 am

I had a look at it. I saw in counterspace nothing that i understood, including inversion-space. Inversionspace is simply a common unity of the conformal and projective maps.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby thigle » Fri Feb 10, 2006 5:22 pm

ok. i had a look at conformal and projective maps through google and similar, but was unable to understand what these are, apart from remembering few properties of them, like what they distort what not etc.

inversion space, like the space where the mappings occur ? :?
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm

Postby wendy » Sat Feb 11, 2006 8:31 am

inversion space was discovered by Mobius as the horizon of the hyperbolic space. In this space, one takes a sphere as all-space, and treats every circle as straight. A straight line is then defined by three points.

One can remove one of these three points, by proposing that some point-like condition exists, viz lines pass through a point (euclidean), or at right angles to a circle (hyperbolic), or at diametric points (spheric). Because the circles do not change, the models are "conformal".

One can then realise these pointlike conditions, by making the circles on the sphere (I1), being the intersection of the sphere (I2), and some plane in an embedding space (E2). The pointlike conditions become points either at the E3 centre of the circle (spheric), or the apex of tangents to the circle (hyperbolic), or the surface point (euclidean).

Inversion is itself a conformal (or angle preserving) transform that maps the point (r, theta) onto (1/r, theta). It preserves euclidean geometry and angle.

Because we can set the condition of straight so that a straight I1 is any intersection of E2 containing U, and I2, we then create on I2 one of the three models (stereographic, inversive, poincare) of geometries that preserve angle.

One can then take a plane in E2, called M2. M2 is the projective plane (or map). We map the straight lines I2 onto M2. This maps straight onto straight, since there is for every straight line (M1) on M2, a plane that contains all of M1 and the point U. This leads directly to the projective model, and one can derive the projective plane, the gnomic projection and the beltrami-klein projections, as U is on, inside or outside the sphere.

If U falls at the centre of I2, the projection is not distorted, and great circles represent lines (since they are the intersect of planes through the centre). It follows that if U falls at the surface of any embedded isocurve of greater curvature, then the conformal mapping is also geodesic.

W
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby thigle » Sat Feb 11, 2006 3:05 pm

a straight line is then defined by three points.

i dont see these 3. these 3 are ... ?

pointlike conditions become points at..., ... the apex of tangents to the circle (hyperbolic), ...

that is where ? nowhere here, somewhere there ? at the ideal point at infinity where each tangents 'starts' from ? or just anywhere outside the circle ?

so in the middle (& at the centre) of circle - glomos,
at,on,'within' the circle - horos,
outside of circle & infinitely far away (idealy far) - bollos

i still cannot run these models in my imagination fully, but at least you let me understand how curvature works, what that is.

but then, what is the point U ? one of those 3 points you mention ? the centre of projection, the inversion-point ? it seems so, yes. or the ORIGIN (of what space?) ?

but i cannot follow fully the following:
... take a plane in E2, called M2...the projective plane (or map). We map the straight lines I2 onto M2.
This maps straight onto straight, since there is for every straight line (M1) on M2, a plane that contains all of M1 and the point U. This leads directly to the projective model, and one can derive the projective plane, the gnomic projection and the beltrami-klein projections, as U is on, inside or outside the sphere.

the boldFonted I2, shouldn't it have 'of' before itself, or isn't it a mistyped I1 - circle/line ? just that that way it would give me more sense.

mainly, where does M2 come from ? is that the ideal plane at infinity ? what is it ?

i would really like to understand this as well as i am able of.
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm

Postby wendy » Sun Feb 12, 2006 9:44 am

In inversion-space, every isocurve (circle, straight line), is straight. One can for any given pair of points A, B, pass through it circles that contain both points, from the one where AB is the diameter, and any larger.

In fact, one can create a circle to pass through a given point C, which means to uniquely define an isocurve, one needs three points, A B C.

You can replace a point by some other condition that is "point-like". For example, it is always possible to pass a circle through A, B, and D, where D is a condition meaning that it is diametric points on a given circle (d). Or you could use D to mean that it intersects the d-circle at right angles. You can even use the "diametric" or "orthogonal" conditions freely, as if they were alternatives to fixed points. In this sense, they are pointlike.

When you embed the inversion-sphere in euclidean space, then one can make the circle I1 the intersection of E2 and I2 (ie a plane crossing through a sphere). The "diametric" and "orthogonal" conditions become real points. The diametric point now lies in the centre of the circle, as it appears in the E2 that makes I1. The "orthogonal" condition makes the apex of a cone of tangents to the circle. The radial lengths from the centre of the sphere, to the diametric and orthogonal points, multiply together give the same as the radius square.

When you have "diametric" [interior] and "orthogonal" [exterior] points, you can readily see that any three points in E3 define an E2, which may, or may not, intersect I2. If they do, they will usually do so at a circle I1.

One can from this space, and these point-like conditions, define a particular point U, that any "straight" line arises from the intersection of a plane containing U, and the sphere I2. When you do this, for the several locations of U, one can get the various conformal projections of the E2, H2, S2. If one then defines a second point V, one can show that there is a conformal mapping of the space defined by U to that defined by V. For example, if U is outside the sphere, and V is anywhere, the projection that comes is the beltrami-klein. If U and V coinside, the projection is that of nature (ie straight lines = straight lines).

The plane M2 is used to derive the projective geometry from this model. It is a plane that does not contain U, and one intersets planes that contain U (ie straight lines), onto M. ie E2 intersects M2 at a line M1. The geometry produced by M1 does not preserve angles, but preserves straightness. Euclidean parallelism is then represented by the existance two lines M1a, M1b, that cross the line M1u at the same point. The line E2u is the tangent to the sphere at U, this intersects M2 at M1u.

And one derives therefrom all of the notions of the projective geometry thence.

W
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby thigle » Sun Feb 12, 2006 6:18 pm

100 % clear now. thanxalot :lol:
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm

Postby moonlord » Mon Feb 13, 2006 6:20 pm

Uh, and do these geometries help in any way or are they just 'purely mathematical'? :)
moonlord
Tetronian
 
Posts: 605
Joined: Fri Dec 02, 2005 7:01 pm
Location: CT, RO, CE EU

Postby wendy » Tue Feb 14, 2006 8:58 am

In more ways than you can imagine, are they practical.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby moonlord » Tue Feb 14, 2006 5:51 pm

I assume they are just as useful to the 'real' world just as the complex numbers have proven to become. I also assume I will one day have enough knowledge to understand and, perhaps, use them...
moonlord
Tetronian
 
Posts: 605
Joined: Fri Dec 02, 2005 7:01 pm
Location: CT, RO, CE EU

Postby wendy » Wed Feb 15, 2006 6:43 am

Some things are useful for knowing that they exist. For an incorriagable hacker, one has at one's finger tips thousands of little factlets, and one takes lessons learnt at one point, and apply them at another. It is in this way that things come together.

W
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia


Return to Other Geometry

Who is online

Users browsing this forum: No registered users and 12 guests

cron