Dissecting the 3 sphere

Discussion of shapes with curves and holes in various dimensions.

Dissecting the 3 sphere

Postby Secret » Sun Mar 06, 2011 1:16 pm

EDIT: I've switched to telstra and is now possible to upload the pics to imageshack
And I've replaced your image links with direct links because imageshack is ridiculously full of ads. ~Keiji
A imagined story:
===============================================================================
...One day when I was eating I noticed an orange. This orange is bigger than other oranges. In fact it is more spherical than other oranges.
I grab a knife and slice it open...

Image
===============================================================================
This is a sphere.

Image

Looks 3D? Think again, it's in fact the 3 sphere (projection)
Rotate the sphere a bit, and the w axis is now visible

Image

Viewing another 3 sphere edge on, results in the usual 2 sphere view plus the 3rd curvature collasped into the classical dounut shape projection (Area near the two thick vertical lines)

Image

The 3quator and the poles of the 3 sphere.

Image

>>>
A 3 sphere is the 4D analogue of the 2 sphere (the common 3D ball we all familiar with)

To slice the 3 sphere, apply dimension analogy:
-1 sphere (point)
A point can be sliced by emptiness into emptiness along emptiness, with emptiness as cross sections (emptiness corresponds to empty (-1) space required to make some maths complete)
0 sphere (line segment)
A line can be sliced into half along its mid point, with points as cross sections
1 sphere (circle)
A circle can be sliced into half along the diameter, with line segments of length d as cross sections
A maximum of 2 orthogonal cuts can be made to dice the circle into quarters
The quarters has the shape of a right angled triangle but with one side a curve (the surface of the circle)
2 sphere
A 2 sphere can be sliced into half along any great circles. And the cross sections are circles
A maximum of 3 orthogonal cuts can be made to dice the 2 sphere into 1/8 identical chunks
The chunks has the shape of a right angled tetrahedral shape with one curved face (the surface of the 2 sphere)
3 sphere
Similarly a 3 sphere can be sliced into half along any great 2 spheres.
Here are the various hemi 3 spheres obtained when one slice along the great 2 spheres in the x, z, y and w direction respectively

Image

Similarly a maximum of 4 orthogonal cuts SHOULD be made to dice the 3 spheres into identical 1/16 chunks
The chunks (which the projection is not found elsewhere) are a pentachoroid composed of 4 cells of the same shape as the 1/8 chunks of the 2 sphere plus a "inflated tetrahedron" cell at the front (the surface of the 3 sphere)

Here are the 3-chunks

Image

Another way to slice it is to
(for convenience, 2 spheres are referred as simply 'sphere' in this section)
1. Treat the 3 sphere (projection) as one sphere (squished into a spheriod in the projection) and stick it into the equator of another sphere so that the equator and one of the great circles of the "flattened" sphere intersects. (Similar to how one draw a sphere on paper, one draws a circle then mark the equator on the circle. depth thus can be inferred from the resulting image for those who can perceive 3D)
2. Choose an arbitary direction to slice, then mark 3 pairs of points anywhere on the unsquished sphere in the projection (note the points can be placed somewhere in 4D, which is done by moving the 3quator to a desired portion of of the large sphere, then mark the point anywhere on the resulting 3chord).
*3. 3 straight lines can be drawn from each pair of points and should be concurrent (intersecting at a single point). This marks the direction of the slicing plane and the poles of the spherical cross section to be obtained.
4. Connect any 4 coplanar points (lying on the same plane) to form an oval in the projection. Then choose any one pair of points directly opposite to each other on the oval and use it along with the remaining two points (which should also be directly opposite to each other) to draw another oval.
5. The two ovals (circles) should intersect at two points forming two intersecting great circles of a sphere (wireframe). Fill in the wireframe to obtain the cross sectional sphere.
6. Choose the portion of the projection to the left or right of the cross sectional sphere to discard.
7. Now the remaining portion in the projection along with the cross sectional sphere should yield the projection of the sliced chunk of the 3 sphere
(The above can be generalised to any dimensions.A n sphere requires n pairs of points forming concurrent lines to define the slicing plane. n-1 ovals can be drawn on the projection to form the n-1 great circles wireframe of the cross sectional n-1 sphere)
*It's meaingless if one cannot check whether the lines are concurrent USING THE PROJECTION ALONE and/or the above method cannot be generalized to any objects and surfaces. The slicing plane is undefined if at least one line is not concurrent
But I don't know how to develop a prove mathematically


With the 3 sphere visualized, you can easily construct the polychorons (4 polytopes) on a 3 sphere using anagolus rules in constructing their 3D conterparts on a 2 sphere.

e.g. An octahedron can be constructed by locating the 3 pairs of polear points along the 3 axes, then joint the four points on the equator to form a square and connect the 4 vetices of the equatorial square to the remaining two poles above and below the square respectively.
Similarly a 16 cell can be constructed by locating the 4 pairs of polar points on the 3 sphere. Joint the 6 points on the 3quator to form an octahedron. Then connect the 6 vertices of the 3quatorial octahedron to the remaining 2 poles ana and kata the octahedron respectively.

Here ilustrates a 16 cell drawn this way with the 3 sphere removed

Image

These are 4 sphere and 5 sphere respectively, projected successively from its dimension to the 2D screen.

Image
Image

NOTE: I might be entirely wrong, cause I'm not expereinced enough to do the above mathematically nor to prove them mathematically
NOTE 2: The above ideas are result on my limited understanding of 4D plus TONS OF PAPERWORK
P.S. Imageshack doesn't allow me to upload for some reason ("no files were uploaded" error), thus i've to use facebook. IF someone manage to get into the album, mind help me rehost the pics to a more accessible place? (cause the virginbroadband does not allow me to upload them besides facebook)
P.S. 2: The facebook account above is my dummy account (old and unused anymore), thus its security and privacy is set to the lowest
P.S. 3: Due to the above problems, it took about 2 hours to get this post done
Last edited by Secret on Sat Mar 12, 2011 11:32 am, edited 2 times in total.
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Re: Dissecting the 3 sphere

Postby Keiji » Fri Mar 11, 2011 7:27 pm

Virgin doesn't stop you uploading things outside of facebook :glare:

Try one of the following:
http://ompldr.org/
http://sadpanda.us/
http://imgur.com/
http://www.tinypic.com/
http://imageupload.org/
http://www.postimage.org/
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Re: Dissecting the 3 sphere

Postby Secret » Sat Mar 12, 2011 11:34 am

thanks for the advice
[offtopic] I'm quite a lazy person and only image shack can allow multiple pics to be uploaded at once. It seems i've to be patient at difficult times and use other hosting website to upload the pics (actually I can barely upload the pics in other sites including facebook while i was still using virginbroadband) -_-[/offtopic]

Btw I've switched to telstra thus it is now possible to put the pics on imageshack
Post also edited
So it should now be possible to investigate this idea to see whether it will work
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Re: Dissecting the 3 sphere

Postby Keiji » Sat Mar 12, 2011 11:56 am

Your post might be long, but the idea seems quite simple: choose unit vectors w, x, y, z in 3D such that z points up and w, x and y point at the vertices of an equilateral triangle centred at the origin and perpendicular to z - in other words, a trigonal bipyramid projection.

Now, I thought we already projected polychora like that - but on checking the wiki, apparently not! I do think it'd be useful to have such projections around.

As for your "3quator" thing, though, we discussed equators in another topic and there's actually two different ways of doing them. You can indeed use two point poles and a sweep of spheres between them (so including a spherical equator), or alternatively you can use two circular poles and a sweep of torii between them (so including a toric equator). It turns out that the latter way makes far more sense when dealing with planets.
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Re: Dissecting the 3 sphere

Postby Secret » Sat Mar 12, 2011 12:28 pm

Keiji wrote:Your post might be long, but the idea seems quite simple: choose unit vectors w, x, y, z in 3D such that z points up and w, x and y point at the vertices of an equilateral triangle centred at the origin and perpendicular to z - in other words, a trigonal bipyramid projection.

Now, I thought we already projected polychora like that - but on checking the wiki, apparently not! I do think it'd be useful to have such projections around.

that's what I like to call a "global projection" or "outside projection"

the aim of this projection is to allow all the cells of the shape to be obsurbed in the least amount yet does not porduce confusing pictures as the parallel projection, basically it's a mix of parallel projection and perspective projection.
i think this type of projection allows people to see the shape more completely than either the two aformentioned projections, and is the projection i used to analyse 4D shapes on my paperwork
This projection also allows me to render the 3 sphere without the 3rd curvature collasped into the torus like view, thus make it easier to understand and to visualize its 3D surface (on paper)
(It's what I think a 4D being will actually see)

Keiji wrote:As for your "3quator" thing, though, we discussed equators in another topic and there's actually two different ways of doing them. You can indeed use two point poles and a sweep of spheres between them (so including a spherical equator), or alternatively you can use two circular poles and a sweep of torii between them (so including a toric equator). It turns out that the latter way makes far more sense when dealing with planets.


When I checked wikipedia about 7 weeks ago, I noticed there is a (mathematical) way to divide the 3 sphere into two identical 2-torus, but I still yet to visualize this sort of slicing.
(due to the fact I still can't make a trigonal bypyramid/global projection of the duocyclinder/clifford torus, or in other words visualize the geometry)
though i understand how is that possible (by comparing it with a perspective projection of the tesseract)
as for the planet thing, I need some time to digest that

P.S. Below is the illustration of the sphere slicing algorithm (not sure if that works mathematically though)

Image

Edit: As what I understand in the planets post, dimension analogy breaks down, so does that mean the alogorithm is invalid? If yes then I might need help from quickfur to get that duocyclinder to visualize as it is one of the few 4D objects I can't visualize in trigonal bipyramid projections. I might also need to learn some concepcial analogy in 4D in order to continue on the visualization

each time I saw that duocyclinder view (similar when one look at a torus from the side), I will be confused. It's one of the few things that get into my way of visualizing 4D
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Re: Dissecting the 3 sphere

Postby Keiji » Sat Mar 12, 2011 5:03 pm

I've drawn a bipyramidal projection of the aerochoron:

Image

The leftmost image shows the outer sphere, together with the equatorial cell which was condensed to a flat hexagon in the projection.
The middle image just shows the edges by themselves.
The rightmost image shows the aerochoron with all faces coloured, so you can see how many faces overlap at each point in the projection.
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Re: Dissecting the 3 sphere

Postby Secret » Sun Mar 13, 2011 1:41 am

[status] 1 question unanswered [/status]

After checking wikipedia, I noticed that the bipyramid projections are actually a slanted version of the face first projections of the polychrons.

Still they are much clearer than other projections as the cells never got flattened completely or with the ana (4D frontmost) cell blened to the surroundings/w axis not visible (as in the case of cell first projections (e.g. the big cube is blended within the other cubic cells in a perspective projections of a tesseract)) in them nor having some vertices, edges hidden behind another vertex, edge. (as in the case of the vertex first projection of the 16 cell)
The bend between ana/kata and one of the 3D directions is also more visible

IMO, I can see the shapes hollowed out for a longer period of time than when seeing other projections (except the perspective projection of the tesseract, where both bipyramid and perspective also works)

P.S. I prefer the old system (-cell) for the regular polychorons as they are more descriptive
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Re: Dissecting the 3 sphere

Postby Keiji » Sun Mar 13, 2011 9:16 am

Secret wrote:[status] 1 question unanswered [/status]


And what would that be?

Secret wrote:P.S. I prefer the old system (-cell) for the regular polychorons as they are more descriptive


Naively, they might seem that way.
The point of the Tamfang names is to keep the dimensional analogs.

With the standard names, the cross polytopes are octahedron, hexadecachoron, icosidodecateron, tricositetrapeton, etc. Where's the pattern?
With the Tamfang names, the cross polytopes are aerohedron, aerochoron, aeroteron, aeropeton, etc. Shorter and simpler.
You don't really need to know how many facets there are in a particular cross polytope, because the n-dimensional cross polytope always has 2n facets!

Also, would you kindly stop quoting the last post in its entirety? Doing so is unnecessary and is an eyesore, I've been having to edit them out of your posts. Quoting is useful for when you're quoting part of a post, or a post that isn't the last one before yours.
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Re: Dissecting the 3 sphere

Postby Secret » Sun Mar 13, 2011 10:16 am

my way of quoting post is to tell users what sections I am responding to,thus it will be easier for them to keep track of the subject I'm responding to (for discussion or for research purpose)
although I admit I'm too lazy thus I quote the whole thing (even with it split into sections). I'll shorten my quotes and focus on the important points to be responded in my future posts.
I'm sorry this gives you an eyesore (I'm unaware of this until now as none of the other users had addressed this issue before)
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Re: Dissecting the 3 sphere

Postby Keiji » Sun Mar 13, 2011 6:27 pm

Most people ignore it, I'm a pedantic idealist.

Yes, quotes are to tell people what you are responding to - however if you don't start with a quote, it's assumed that you're responding to the last post, hence why there's no need to ever quote the last post in full.
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Re: Dissecting the 3 sphere

Postby Secret » Mon Mar 14, 2011 3:29 am

According to my experience, at least pwrong, wendy and quick fur do care, yup they are one of the few.

Main topic:
Btw what software do you use to generate the above projections?

P.S. Currently working on visualizing the toric 3quator of the 3 sphere.
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Re: Dissecting the 3 sphere

Postby Keiji » Mon Mar 14, 2011 4:27 pm

I didn't generate them, I drew them in Inkscape.

When I have some time though, I may decide to write a program to truly generate these projections.
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Re: Dissecting the 3 sphere

Postby Marek14 » Sat Nov 12, 2011 6:19 pm

When on holiday this year, I was thinking about the two basic coordinate systems on 3-sphere, and how to visualize the lines of constant coordinates if I imagine a solid ball as one half of a 3-sphere (the same way you can project a half of sphere onto a disc).

The two systems of coordinates stem from two sets of parametric equations for the 3-sphere:

x = r * cos a * cos b * cos c
y = r * cos a * cos b * sin c
z = r * cos a * sin b
w = r * sin a

and

x = r * cos a * cos b
y = r * cos a * sin b
z = r * sin a * cos c
w = r * sin a * sin c

Try it :) It's fun.
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Re: Dissecting the 3 sphere

Postby quickfur » Sat Nov 12, 2011 6:42 pm

Hey Marek,

I'm trying to think of a way of cutting up the surface of the 3-sphere into cubes or mostly cube-like shapes, so that you can somewhat see the curvature on its surface. But currently my polytope viewer doesn't handle curved shapes, so I need to come up with some way of approximating it with a polytope with cube-like facets. What do you think would be the best way to generate the coordinates of this polytope? I'm afraid that if I simply sample those equations at regular intervals, I would just end up with simplex facets instead of (mostly) cubical facets.
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Re: Dissecting the 3 sphere

Postby Marek14 » Sat Nov 12, 2011 7:42 pm

quickfur wrote:Hey Marek,

I'm trying to think of a way of cutting up the surface of the 3-sphere into cubes or mostly cube-like shapes, so that you can somewhat see the curvature on its surface. But currently my polytope viewer doesn't handle curved shapes, so I need to come up with some way of approximating it with a polytope with cube-like facets. What do you think would be the best way to generate the coordinates of this polytope? I'm afraid that if I simply sample those equations at regular intervals, I would just end up with simplex facets instead of (mostly) cubical facets.


If you take values (a, a+da), (b, b+db), and (c, c + dc) and compute values for all their combinations, then you can connect the eight points into a tiny cuboid. Would that work?
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Re: Dissecting the 3 sphere

Postby quickfur » Sat Nov 12, 2011 8:45 pm

Hmm. That's worth a try. I'll check it out.
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