geometries & algebras

Higher-dimensional geometry (previously "Polyshapes").

geometries & algebras

Postby thigle » Sat Sep 23, 2006 9:44 pm

can anyone help me with this train of thought ?

if reals are real line and complex numbers are complex plane, then what are quaternions (geometrically)? wendy says these correspond to great arrows.

the unit circle in reals is +-1.
in complex plane the unit circle is a circle of radius 1, that maps as equator of the rieman's sphere touching it at [0,0i] and having the other pole as infinity. is that so ?
now for unit circle of quaternions, that is a unit sphere actually, what is it ?
in other words, what shape can be a unit sphere of quaternions mapped onto one-to-one, like the complex plane maps onto rieman sphere

considering all possible rotations in 3-space, each rotation can be determined by a unit quaternion, real part for the amount of rotation and 3 imaginaries for the axis. the points of a star through a point thus stand for different rotations through that point in 3space.
but a rotation around an axis in positive direction by amount a, is equal to rotation around that axis in the other direction by amount -a.
so the unit sphere of quaternions is double cover for group of possible rotations in 3 space.

so how do all the quaternions map onto what ? i still wonder.
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Postby wendy » Sun Sep 24, 2006 7:41 am

The quarterions map onto four dimensions.

There are certain integer-systems, some of which are infinitely dense (like the pentagonal numbers x+yø).

The group of 8 form {4,3,3,4}

The group of 24 form {3,3,4,3}

The group of 48 form the tiling 5BB, a tiling of octagonny.

The group of 120 form the tiling 5V, exampled by {5,3,3,5/2}.

The units of these correspond to the vertices of the {3,3,4}, {3,4,3}, dual {3,4,3}s, and {3,3,5} respectively.

The only other integer system that has a corresponding tiling occurs in eight dimensions, where E8 is the unit system of the octonions.
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Postby PWrong » Sun Sep 24, 2006 2:19 pm

in complex plane the unit circle is a circle of radius 1, that maps as equator of the rieman's sphere touching it at [0,0i] and having the other pole as infinity. is that so ?

I think that's right.

now for unit circle of quaternions, that is a unit sphere actually, what is it ?

It's more of a unit glome.

considering all possible rotations in 3-space, each rotation can be determined by a unit quaternion, real part for the amount of rotation and 3 imaginaries for the axis. the points of a star through a point thus stand for different rotations through that point in 3space.

I don't think so. To express a rotation in 3-space you only need two angles defining the axis, and a third angle for the amount of rotation. I think there may be other operations that you could represent with a quartonian. Like you could rotate around an axis, and also scale the object along the axis. But I don't know if that would be useful. You can perform much more general actions with a 3x3 matrix.
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Postby thigle » Sun Sep 24, 2006 7:19 pm

i suppose you might be right about the unit glome.
as for the rotations, it's commonly used that way in computer graphics. each quaternion is a vector - directed magnitude. direction for axis, lenght for amount of rotation.

what more general actions can you perform with 3x3 matrix ?

anyway, i wanna know that geometric mapping of quaternionic unit sphere or glome onto whatever.
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Postby thigle » Sun Oct 15, 2006 11:58 am

so the incomplete analogy stand like this now:

complex numbers map fully onto plane which maps onto rieman sphere.
quaternions map onto 4-space (as wendy reminded me of hamilton's finding) which map onto what??? a glome ?

i really wanna have this clear.

to wendy: you're still lightmiles ahead, each of those you mention is like a dark forest for my childish imagination yet. thanx anyway.
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Postby PWrong » Sun Oct 15, 2006 1:11 pm

complex numbers map fully onto plane which maps onto rieman sphere.
quaternions map onto 4-space (as wendy reminded me of hamilton's finding) which map onto what??? a glome ?


Actually, since complex numbers are 2D and map onto a 2-sphere, and quaternions are 4D, they should really map onto a 4-sphere, which is a 5D sphere. I don't know if they actually do though.

what more general actions can you perform with 3x3 matrix ?

Any linear map. Including rotations, reflections, scaling, shears, and any combination of these. You can't do translations in 3D with a 3x3 matrix, but you can with a 4x4 matrix.
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Postby thigle » Sun Oct 15, 2006 3:17 pm

complex numbers map fully onto plane which maps onto rieman sphere.
quaternions map onto 4-space (as wendy reminded me of hamilton's finding) which map onto what??? a glome ?

Actually, since complex numbers are 2D and map onto a 2-sphere, and quaternions are 4D, they should really map onto a 4-sphere, which is a 5D sphere. I don't know if they actually do though.

rieman's sphere isn't just simple 2-sphere, is it. it has infinite value of parameters at its pole, so one can consider it as family of parallel lines in projective space and their vanishing point.
similarily, if quats map onto 4-sphere, this should be a special case of it, with metric such that there is something with infinite values of its parameters. an ideal line or plane or space or what at infinity ? here my analogical reason lets me down.
so what is the 'pole' of 4-sphere (5d sphere), when all 4 parameters run infinite ?
what metric does it have (similar to riemann's metric on 2-sphere) ?
for rieman's sphere and complex plane, there is stereographic projection as mapping between those two. can anyone describe similar construction for quaternions & 4-sphere ? is

[as an off topic to the main theme of this thread:
matrices... still foggy landscape for me.
what kind of mapping is a linear map? what does it preserve ? lenghts&angles ?
why cannot one do translations with 3x3 matrix and can with 4x4 ?
what if one considers translations as rotations around ideal point at infinity ?]
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Postby PWrong » Thu Oct 19, 2006 10:17 am

In 3D we have 3 axes, which intersect the Riemann sphere at 6 points. At these intersections we put the numbers 1, i, -1, -i, 0 and infinity. We have 0 at the south pole and infinity at the north pole. The others all lie on the equator.

In 5D, there are 5 axes, and 10 points. These are:
1, i, j, k, -1, -i, -j, -k, 0, infinity.

what metric does it have (similar to riemann's metric on 2-sphere) ?
for rieman's sphere and complex plane, there is stereographic projection as mapping between those two. can anyone describe similar construction for quaternions & 4-sphere ? is
I don't know about the metric, or how to map between the two.

[as an off topic to the main theme of this thread:
matrices... still foggy landscape for me.
what kind of mapping is a linear map? what does it preserve ? lenghts&angles ?

A linear map takes lines to lines (or sometimes a point), and the origin to the origin. It usually doesn't preserve lengths or angles. If you start with a square, you end up with a parallelogram.

In 2D, if you have a point represented by (x,y), you can multiply it by a matrix like this:
Code: Select all
[a b] (x)  =  (ax+by)
[c d] (y)  =  (cx+dy)

This gives you a new point, (ax + by, cx + dy).

You can also multiply a matrix by another matrix. That's just like as doing one linear map, then the other.

why cannot one do translations with 3x3 matrix and can with 4x4 ?
what if one considers translations as rotations around ideal point at infinity ?]

A translation isn't linear, because it doesn't take the origin to the origin. You can represent a 3D translation with a 4x4 matrix using a trick. I'll show you how to do it in 2D.

first extend your vector to (x,y,1). Then multiply it by:
[1 0 a]
[0 1 b]
[0 0 1]
This gives you (x+a, y+b, 1). Throw away the 1 at the end, and you get (x,y) translated by (a,b).

what if one considers translations as rotations around ideal point at infinity ?

I don't know how you could express that in real notation.
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Postby pat » Thu Oct 19, 2006 1:16 pm

PWrong wrote:A linear map takes lines to lines (or sometimes a point), and the origin to the origin.

In particular, for scalars a and b, and vectors x and y, a linear map L obeys: L( ax + by ) = aL(x) + bL(y).

So, for example, because ( x + -x ) is the origin, L( x + -x ) = L(x) + -L(x) = the origin. And, if we define a line parametrically by giving a point x on the line and a direction y from that point as p(t) = x + ty, we see that: L( x + ty ) = L(x) + tL(y) is also a parametrically-defined line.
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