Generalised Hopf coordinates

Discussion of shapes with curves and holes in various dimensions.

Generalised Hopf coordinates

Postby PWrong » Tue Dec 09, 2014 3:26 am

We know that the 3-sphere can be expressed with two different sets of parametric equations. There's the standard one:
sin a sin b cos c
sin a sin b sin c
sin a cos b
cos a,

and one based on the Hopf fibration:
cos a cos b
cos a sin b
sin a cos c
sin a sin c

Clearly a similar thing can be done with higher dimensional spheres, as well as rotatopes and toratopes that involve these spheres. The Hopf coordinates can be viewed as a special case of the tiger with both minor radii equal to zero.

Here's a conjecture. There is a unique set of Hopf spherical coordinates for each nD toratope that does not contain a lonely '1'. So for example, in 6D we have Hopf coordinates based on the following toratopes:
(IIIIII)
((IIII)(II))
(((II)(II))(II))
((III)(III))
((II)(II)(II))

The (((II)(II))(II)) Hopf coordinates in 6D are:
cos a cos b cos c
cos a cos b sin c
cos a sin b cos d
cos a sin b sin d
sin a cos e
sin a sin e
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Re: Generalised Hopf coordinates

Postby Marek14 » Wed Dec 10, 2014 5:54 pm

Basically, this is about branching trees once again. In 2D, we have
sin a
cos a

In 3D, we have to "split" one of them:
sin a sin b
sin a cos b
cos a
If we split the second, "cos a", we'd get something equivalent, of course. This is related to normal torus ((II)I). The parametric equations of torus have this as minor diameter terms, added to circle with major diameter terms.

In 4D, then, we have two non-equivalent types of lines, leading to two distinct sets:
sin a sin b sin c
sin a sin b cos c
sin a cos b
cos a
(((II)I)I)

sin a sin b
sin a cos b
cos a sin c
cos a cos c
((II)(II))

So this method leads to a different conjecture: there is a set for every toratope that contains only binary parentheses (i.e. that has no parentheses enclosing three or more terms).

In 5D, there are 3 such toratopes: ((((II)I)I)I), (((II)(II))I) and (((II)I)(II)), corresponding to 3 sets of coordinates:
sin a sin b sin c sin d
sin a sin b sin c cos d
sin a sin b cos c
sin a cos b
cos a

sin a sin b sin c
sin a sin b cos c
sin a cos b sin d
sin a cos b cos d
cos a

sin a sin b sin c
sin a sin b cos c
sin a cos b
cos a sin d
cos a cos d

6D has 6 such toratopes: (((((II)I)I)I)I), ((((II)(II))I)I), ((((II)I)(II))I), ((((II)I)I)(II)), (((II)(II))(II)) and (((II)I)((II)I))
sin a sin b sin c sin d sin e
sin a sin b sin c sin d cos e
sin a sin b sin c cos d
sin a sin b cos c
sin a cos b
cos a

sin a sin b sin c sin d
sin a sin b sin c cos d
sin a sin b cos c sin e
sin a sin b cos c cos e
sin a cos b
cos a

sin a sin b sin c sin d
sin a sin b sin c cos d
sin a sin b cos c
sin a cos b sin e
sin a cos b cos e
cos a

sin a sin b sin c sin d
sin a sin b sin c cos d
sin a sin b cos c
sin a cos b
cos a sin e
cos a cos e

sin a sin b sin c
sin a sin b cos c
sin a cos b sin d
sin a cos b cos d
cos a sin e
cos a cos e

sin a sin b sin c
sin a sin b cos c
sin a cos b
cos a sin d sin e
cos a sin d cos e
cos a cos d
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Re: Generalised Hopf coordinates

Postby Marek14 » Wed Dec 10, 2014 6:03 pm

Also, note that tiger can't have "both minor radii equal to zero" for the simple reason that it has only one minor radius.

Now, parametric equations of nonbinary toratopes (like spheres) seem to involve "fake binaries"; additional parentheses can be added until the result is binary.

So, if a torus has equations
x = R sin b + r sin a sin b
y = R cos b + r sin a cos b
z = r cos b

the equations of torisphere ((III)I) are
x = R sin b sin c + r sin a sin b sin c
y = R sin b cos c + r sin a sin b cos c
z = R cos b + r sin a cos b
w = r cos a
The only possibility is to add fake parentheses to change it to the form (([II]I)I).

On the other hand, a spheritorus ((II)II) has two distinct options:
([(II)I]I):
x = R sin c + r sin a sin b sin c
y = R cos c + r sin a sin b cos c
z = r sin a cos b
w = r cos a
((II)[II]):
x = R sin b + r sin a sin b
y = R cos b + r sin a cos b
z = r cos a sin c
w = r cos a cos c
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Re: Generalised Hopf coordinates

Postby PWrong » Thu Dec 11, 2014 1:10 am

So this method leads to a different conjecture: there is a set for every toratope that contains only binary parentheses (i.e. that has no parentheses enclosing three or more terms).

Yes, this makes a lot more sense.

Also, note that tiger can't have "both minor radii equal to zero" for the simple reason that it has only one minor radius.

Good catch, I meant major radii.

So there's more options than I thought even in 4D. With these new square brackets, how do we write things in general? I guess we write regular spherical coordinates as ([II]I)? Would it make sense to say that [...] implies a radius of zero while (...) implies a nonzero radius? With that interpretation, I guess you could say that [II]I is simply a line segment on the z axis.

On the other hand, a spheritorus ((II)II) has two distinct options:
([(II)I]I):
x = R sin c + r sin a sin b sin c
y = R cos c + r sin a sin b cos c
z = r sin a cos b
w = r cos a
((II)[II]):
x = R sin b + r sin a sin b
y = R cos b + r sin a cos b
z = r cos a sin c
w = r cos a cos c


This is interesting. These are two unique sets of parametric equations. I wonder if they correspond to two unique orthogonal coordinate systems similar to toroidal coordinates? Also, ((II)[II]) is the same as tigric coordinates with only one of the major radii equal to zero. It's a bit surprising that this would describe a spheritorus, but maybe not that surprising.

3D coordinate systems:
III
(II)I
([II]I)
((II)I)

4D coordinate systems:
IIII - Cartesian
(II)II - cubindrical
(II)(II) - duocylindrical
([II]I)I - spherindrical
((II)I)I - torindrical

([[II]I]I) - Hyperspherical
([II][II]) - Hopf
([(II)I]I) - Spheritoroidal?
((II)[II]) - Spheritoroidal 2?
(([II]I)I) - Torispheroidal?
((II)(II)) - Tigric
(((II)I)I) - Ditoroidal
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Re: Generalised Hopf coordinates

Postby PWrong » Thu Dec 11, 2014 2:01 am

There are some interesting connections between these things.

Here the coordinate systems use '.' instead of 'I'. Dashed vectors indicate that a system is made by starting with the toratope and changing some radii to zero. Solid vectors indicate that this is a set of parametric equations for the toratope. So for example, ((..)[..]) is a coordinate system for the spheritorus so it has a solid line from the spheritorus, but it's made from tigric coordinates with one radius set to zero so it has a dashed line from the tiger.
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Re: Generalised Hopf coordinates

Postby PWrong » Tue Dec 16, 2014 4:45 am

I think the easiest way to get the toratopes is to do the brackets first and then "colour" them () or [].

The number of Hopf parametrisations of an n-sphere is given by the Wedderburn–Etherington numbers, and they can be expressed as "Otter trees". The sequence is 1, 1, 1, 2, 3, 6, 11, 23, 46, 98.

5D:
((((II)I)I)I)
(((II)(II))I)
(((II)I)(II))

Now colour them. I've included duplicates up to rotation but I've labelled them as such.
((((II)I)I)I)
((([II]I)I)I)
(([(II)I]I)I)
([((II)I)I]I)
(([[II]I]I)I)
([([II]I)I]I)
([[(II)I]I]I)
([[[II]I]I]I)

(((II)(II))I)
([(II)(II)]I)
(([II](II))I)
(((II)[II])I) (duplicate)
([[II](II)]I)
([(II)[II]]I) (duplicate)
(([II][II])I)
([[II][II]]I)

(((II)I)(II))
([(II)I](II))
(([II]I)(II))
(((II)I)[II])
([[II]I](II))
([(II)I][II])
(([II]I)[II])
([[II]I][II])

So there are 3 * 2^3 of these coordinate systems including rotational duplicates. Without the duplicates that's 22. That's also not including the open toratopes, so we double the number to get everything.

In n dimensions we have up to 2^{n-1} WEn toratopic coordinate systems, including duplicates.
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