Wireframe spheres [Split from "Quickfur's renders"]

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Fri Nov 18, 2011 7:42 pm

Alright, after 2 days of struggling with a toroidal subdivision of the 3-sphere that doesn't seem to work, I finally discovered the bug in my subdivision program that was breaking the lattice enumeration algo (it was generating some vertices twice, causing the lattice enumeration algo to produce edges with 3 vertices, etc.). And so behold! The toroidal subdivision of the 3-sphere:

Image

Since I know that if you only see the wireframe you wouldn't be able to tell (not easily anyway) where the tori are, I've painstakingly picked out two of them and colored them in yellow and magenta, respectively. Each of these represent a duocylindrical layer of the 3-sphere (well, at least the ridge of the duocylinder; the bounding manifolds are obviously inside the 3-sphere, not on its surface). If you look carefully, you can see a third duocylindrical torus between these two, projected to a slightly inflated cylinder. The yellow and magenta duocylinders, of course, are cartesian products of circles with unequal radii.

Also, you will notice a vertical line running through the 3D image; that is one of the 2 ring-poles in this subdivision of the 3-sphere. So first, you start with that ring, then the next layer is the yellow torus, then the torus between the yellow and magenta tori, then the magenta torus, and then the other ring-pole (not visible here 'cos i'm using perspective projection, and it projects to the far side of the 3-sphere, just barely on the limb -- but you can imagine it as the circle running through the center of the magenta torus).

If you regard the yellow torus's ringpole as "north" and the magenta torus's ringpole as "south", then the yellow region would be the "northern torosphere" (toroidal analog of hemisphere), and the magenta region would be the "southern torosphere", and the unmarked torus in the middle would be the equator (toroquator?), which corresponds with the ridge of a duocylinder with equal radii. The cyan regions around it would be the "torotropics".

You can see that to cross from the northern torosphere to the southern torosphere, you must pass the toroquator.

Also note that the angle between the northern ringpole and the southern ringpole from the center of the 3-sphere is 90°, so the torospheres are not antipodal, but rather orthogonal. If you walk from the northern ringpole to the southern ringpole, you have only covered 1/4 of the corresponding great circle on the 3-sphere (whereas on earth, walking from north pole to south covers 1/2 of a great circle).

Now, the radial lines that you see emanating from the northern ringpole to the magenta region are geodesics connecting the two ringpoles (i.e., they are the shortest path to get from any given point on the northern ringpole to the southern ringpole -- there are many such paths, each of which gets you to a different point on the southern ringpole). All of these paths are exactly 1/4 of the 3-sphere's great circles.

If you walk along one of these geodesics for a full circle, you will encounter each ringpole twice: first you start from point A in the first ringpole, pass the toroquator, then point B in the second ringpole, then pass the toroquator again, then reach point C in the first ringpole (which is antipodal to point A), then pass the toroquator the 3rd time, and reach point D in the second ringpole (which is antipodal to point B), then pass the toroquator again the last time, before you arrive back at point A where you started.

This also means that to get from point A to its antipode (on the same ringpole), you can either walk along the ringpole itself, or you can take a grand tour down to the other ringpole and back -- both are exactly the same distance. :)

I'm telling you, Keiji, this is an awesome setting for creating a truly interesting 4D culture/story. :)
Last edited by quickfur on Tue Nov 22, 2011 5:42 pm, edited 1 time in total.
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Marek14 » Fri Nov 18, 2011 8:29 pm

Fill the 3-sphere with water, add some currents along the ringpoles, and you can have an interesting story even with 3D creatures :)
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Keiji » Fri Nov 18, 2011 9:44 pm

quickfur wrote:Epic post


Excellent work, quickfur! I can finally visualize this now thanks to your image and explanation.

The topic on 4D planets described the concept of time- and climate-zones. On this projection, I'd say you'd travel through the cycle of time-zones by starting in the south toropole and travelling horizontally (as would appear on our screens, anyway), whereas to travel through the cycle of climate-zones, you'd start in the north toropole and travel vertically (which would look like a sine wave up and down the line that it's projected to). Travelling in the remaining axis - in/out of the "sphere" - would not change your position in either zone, but would instead change how much you were affected by time and climate (much like if one travels to the north or south pole on a 3D planet, you get half a year of day followed by half a year of night).

Of course, you wouldn't have to reside in those toropoles to travel through the time- and climate-zones, that premise was only there so I could demostrate the directions necessary.

Watching the planet from afar would be interesting: much like how we have strips of day and night that appear to move around our planet as it rotates on its axis, someone observing a 4D planet would also see (if they had a good enough telescope, anyway!) strips of blossoming flora appearing to move around in the other direction (of climate-zones) as it orbited its star. Similarly, 4D farmers and markets would want to cooperate globally, so that in each month, the climate-zone where food was harvested would cycle around, and then be sent from there to the rest of the planet. :D

(Of course, this post assumes a planet can orbit a star, but we'll ignore the lack of stable orbits for now...)
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Fri Nov 18, 2011 11:18 pm

Marek14 wrote:Fill the 3-sphere with water, add some currents along the ringpoles, and you can have an interesting story even with 3D creatures :)

True! Or if our universe were the surface of this 3-sphere, ... :)
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Sat Nov 19, 2011 2:46 am

Keiji wrote:
quickfur wrote:Epic post


Excellent work, quickfur! I can finally visualize this now thanks to your image and explanation.

Thanks! *bows*

The topic on 4D planets described the concept of time- and climate-zones. [...]

I have just given some thought to this, and it turns out that the rotational ringpoles may actually be more accurately understood as the equivalent of a 3D planet's circular equator and a ring version of a 3D planet's poles. Here's why:

First, as we've discussed before, any leftover angular momentum of the 4D planet after whatever process that formed it will eventually redistribute itself into a double rotation. Now, we're assuming that the planet has a perfectly circular orbit around its star (just for argument's sake, we'll ignore the instability of this orbit). It seems reasonable to assume that the initial angular momentum of the planet should come from a plane rotation roughly the same as the orbital plane. Furthermore, it seems reasonable to assume that as this initial plane rotation redistributes its energy into a double rotation, it will more-or-less retain its orientation, so that when it reaches equillibrium in a double rotation, the two component rotations will have one very close to the orbital plane, and the other in the complementary, orthogonal plane.

For convenience, let's say the ringpole in the magenta region corresponds with the orbital plane, and the ringpole in the yellow region with the orthogonal plane.

Now, let's say the star is shining from the same direction as the 4D camera is looking, so the "day" half of the planet is exactly what we can see in the projection image, and the "night" half is the other half of the planet that lies on the far side. The center of the projection, then, is the closest point on the planet to the star; 4D beings standing on that location will see the star directly overhead. Because of the rotation of the yellow ringpole, the star will be overhead once per rotation, i.e., there is a full day/night cycle.

Now, the boundary between day and night is exactly the projection envelope, which is spherical. Furthermore, the magenta ringpole lies right on this envelope. This means that it's a region of perpetual sunset: the star at noon is at its lowest point in this region. Or, if we allow the rotational plane of this ringpole to be slightly off from the orbital plane, half the year it will be day, and half the year it will be night -- exactly analogous to the arctic circle on Earth. As you move from this ringpole towards the yellow ringpole, the height of the star at noon increases, until it passes directly overhead at the yellow ringpole.

Therefore, the magenta ringpole is more like a 3D planet's poles, and the yellow ringpole is more like a 3D planet's equator.

Furthermore, at points outside the yellow ringpole, the path of the star across the day sky is spiral due to the rotation of the magenta ringpole, although it won't be as exaggerated as a coil of spring, because the period of the spiral is equal to the period of a day, so you will only ever see it spiral by a half-circle at the most.
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Sat Nov 19, 2011 2:53 am

In terms of climate, then, the yellow ringpole would be tropical, the rotational "equator" (which perhaps is more accurately termed "the tropic" -- there is only one, unlike the two on Earth) would be the boundary between tropical and temperate, and the magenta ringpole would be arctic.

This also means that the 4D planet will have an ice ring around it -- the analog of 3D planets' icecaps.

Isn't 4D fun? :)
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Sat Nov 19, 2011 2:29 pm

Weirder configurations are possible, if we relax the assumption that one of the rotational planes matches (or is very close to matching) the orbital plane.

For example, if for some reason the orbital plane is orthogonal to both ringpoles, that is, it projects to a horizontal line across the projection image (i.e., if the magenta ringpole corresponds to XY rotation and the yellow ringpole to WZ rotation, and the orbital plane is the XW plane), then the following will happen:

At the beginning of the year, which we'll assume for no good reason has the star shining from the right of the projection image, thus illuminating the right half of the projection, the yellow ringpole would have arctic climate, as it will remain on the boundary between day and night, while the magenta ringpole will experience a full day/night cycle. However, as the year progresses, the illumination from the star migrates from the right half of the projection to the left, and in 1/4 of the year, it will have migrated to the center of the projection (albeit with a different W coordinate from the planet -- they don't collide) so it will illuminate the near half of the planet, corresponding with what is visible in this projection, so now the magenta ringpole has become arctic and the yellow ringpole is tropic. Then after another 1/4 of the year, the planet is now illuminated from the left, so the magenta ringpole has become tropic again and the yellow ringpole arctic. And so they swap climate types every 1/2 year.

A planet in this configuration will not have permanent ice caps, since the amount of heat it receives from the star is evenly distributed over its surface if averaged over a year. It will have global seasons, though; no matter where on the planet you live, 1/4 of the year it will be arctic (sun will barely rise above the horizon, if at all), and 1/4 of the year it will be tropic (sun will pass overhead at noon). So you will have 8 seasons: winter (arctic), spring (transition), summer (tropic), fall (transition), then 2nd winter, 2nd spring, 2nd summer, and 2nd fall. If the planet's orientation is not fully aligned with the orbital plane, the first 4 seasons will be distinct from the 2nd 4 seasons (e.g., 1st winter could be perpetual evening, with sun spinning above horizon, and 2nd winter could be perpetual night, with sun spinning just below horizon). And these 8 seasons would be global throughout the entire planet.
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Mrrl » Sat Nov 19, 2011 8:36 pm

I'm not sure that you will have polar day of polar night somewhere of the planet. After a half of the day every point on the planet will be on the opposite side of the sphere (is it right?), so almost everywhere we'll have equal day and night duration.
And there are two inclination angles of the rotational axes. For example, sun may pass the equatorial ring twice a year, or be at the same angle of this ring, or travel between latitudes L1 and L2... But I didn't calucate climate pictures for these schemes yet.
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Sat Nov 19, 2011 9:24 pm

Mrrl wrote:I'm not sure that you will have polar day of polar night somewhere of the planet. After a half of the day every point on the planet will be on the opposite side of the sphere (is it right?), so almost everywhere we'll have equal day and night duration. [...]

OK, let's be concrete to avoid ambiguities. Let's say the planet is orbiting the star in the WX plane, and the planet is rotating on its own origin with a double rotation in the WX and YZ planes. (And just to be clear, I'm writing vectors as (w,x,y,z), NOT (x,y,z,w).)

For convenience, let's say the star is at the origin (0,0,0,0), and the planet is at (R,0,0,0) where R is the orbital radius. So relative to the planet, the star appears in the direction (-R,0,0,0).

Now let's take a point on the planet, say (R,0,0,0)+(r,0,0,0) where r is the planet's radius. If you stand on this point, the star appears overhead. But because the planet is doing double rotation in WX and YZ planes, after 1/4 of a day, your position is now (R,0,0,0)+(0,r,0,0). So now the star appears on the horizon. After another 1/4 of a day, your position is (R,0,0,0)+(-r,0,0,0), so now it's midnight, the star is on the opposite side of the planet. So we have established that (R,0,0,0)+(r,0,0,0) has a full day/night cycle.

Now let's take another point on the planet: (R,0,0,0)+(0,0,r,0). If you stand on this point, the star appears on the horizon. But because the planet is rotating also in the YZ plane, after 1/4 of the day, your position is now (R,0,0,0)+(0,0,0,r). But if you calculate the angle with the star, you'll see that the star is still on the horizon! After another 1/4 of the day, your new position is (R,0,0,0)+(0,0,-r,0). The star is still on the horizon. After yet another 1/4 day, your position is (R,0,0,0)+(0,0,0,-r). So basically the star remains on the horizon for the whole day.

OK, so let's say it's 1/4 of a year later. The planet has rotated around the star and now its new position is (0,R,0,0). Now let's see what is happening with our two reference points. The first point is now (0,R,0,0)+(r,0,0,0): now the star is on the horizon. After 1/4 of a day, this point rotates to (0,R,0,0)+(0,r,0,0): the star is now overhead. Then after another 1/4 day, the point rotates to (0,R,0,0)+(-r,0,0,0). The star is back on the horizon. After another 1/4 day, it's now at (0,R,0,0)+(0,-r,0,0). It's now midnight. So the first point still undergoes a full day/night cycle.

What about the second point? It's now at (0,R,0,0)+(0,0,r,0): the star is on the horizon. After 1/4 day, it rotates to (0,R,0,0)+(0,0,0,r): again, the star is on the horizon. After yet another 1/4 day, it rotates to (0,R,0,0)+(0,0,-r,0): the star is still on the horizon. Another 1/4 day, and it's now at (0,R,0,0)+(0,0,0,-r): the star remains on the horizon.

We can repeat this analysis for another 1/4 year later, etc., and you'll see that the first point always undergoes a full day/night cycle, but the second point always has the star on its horizon. Of course, this assumes perfect alignment of the orbital plane with the rotational plane. If there is a slightly offset, then you'll get polar day/polar night for the second point, and for the first point you still get full day/night cycle (just with varying lengths of day/night).
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Mrrl » Sun Nov 20, 2011 12:23 am

I prefer geocentric coodrinates for this. So the origin is at the planet center, and point on the surface has coordinates
w=r*cos(a)*cos(t)
x=r*cos(a)*sin(t)
y=r*sin(a)*cos(t+b)
z=r*sin(a)*sin(t+b)
Here a is point latitude, and b is phase shift parameter. We don't specify third parameter (longitude) because it is connected with the origin of t.
The star goes by circle
ws=w1*cos(T)+w2*sin(T)
xs=x1*cos(T)+x2*sin(T)
ys=y1*cos(T)+y2*sin(T)
zs=z1*cos(T)+z2*sin(T)
where vectors v1=(w1,x1,y1,z1) and v2=(w2,x2,y2,z2) have length R>>r and are orthogonal.
When star is moving near the equatorial plane of the planet, we can get v1=(R,0,p,0) and v2=(0,R,q,s), where p,q,s are small and w2=-p*q/R is very small.
Take a point on the polar ring. For it a=pi/2, so w=x=0, and we may assume b=0.
At T=0 the star has coordinates S0t=(R,0,p,0), point is P0t=(0,0,cos(t),sin(t)) and there is a daytime when p*cos(t)>0, i.e. from 6am to 6pm if p>0 and from 6pm to 6am if p<0.
After 1/4 of year we have T=pi/2, star is at (0,R,q,s), and daytime is when q*cos(t)+s*sin(t)>0. Values of t satisfying this condition depend on q and s parameters, but again they cover half of the day. Still no polar night...
Actually, you may take any point (a,b), any star orbit (v1,v2) and any time of year T - and star will be over the horizon when A*cos(t)+B*sin(t)>0 for some A,B. So you will get either moving of star along the horizon sphere, or half-day daytime and half-day nighttime.
What is interesting, in this time of planet rotation (with exactly equal equatorial and polar periods) we have 1D trace of any point on the surface and 1D trace of star projection. That means that there is only 2D (or sometimes 1D) region where the star may be exactly overhead sometime. I can't imagine the shape of this region (it depends on the parameters of star orbit), but the planet has climatic zones - and they are separated not only by latitudes, but by phase shift parameters as well.
And if periods are slightly different, say, 24h and 24h 0m 1s, then the planet has very slow drift of these zones! They make full cycle around the planet in 86400 days = 236 years :) May be, 2 or 4 times less - but it doesn't matter :)
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Keiji » Tue Nov 22, 2011 12:13 am

quickfur, do you mind if I add all this amazing information about glome subdivisions to the wiki?

If that's okay, I'd appreciate if you could upload the images yourself (using :AddFile and "files from other websites") so that they get the right authorship information and you can choose a license for them. :)
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Tue Nov 22, 2011 12:34 am

Keiji wrote:quickfur, do you mind if I add all this amazing information about glome subdivisions to the wiki?

I would be flattered if you did! :)

If that's okay, I'd appreciate if you could upload the images yourself (using :AddFile and "files from other websites") so that they get the right authorship information and you can choose a license for them. :)

In fact, I'll be happy to move these images there. Right now they are temporary files on my website, not part of my revision tracking system, so if for whatever reason i have to reinstall my website, they will be gone forever. And I can't say I'm very pleased about dangling image links from here, 'cos that detracts a lot from old discussions. :(

This might take some time, tho... currently my brain cpu power is being hogged by a background process trying to calculate the cells of the CRF polytope with 4 square antiprisms. :P
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Keiji » Tue Nov 22, 2011 12:43 am

quickfur wrote:currently my brain cpu power is being hogged by a background process trying to calculate the cells of the CRF polytope with 4 square antiprisms. :P


If it even exists ;)

I'll wait until the images show up, and then re-read the topic and write up either a big section in the Glome article, or a separate page on 4D planets, or both.
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Tue Nov 22, 2011 5:44 pm

Keiji wrote:
quickfur wrote:currently my brain cpu power is being hogged by a background process trying to calculate the cells of the CRF polytope with 4 square antiprisms. :P


If it even exists ;)

Unfortunately you're right, it doesn't exist. Or at least, it does exist but is non-CRF because some of the tetrahedra are non-uniform.


I'll wait until the images show up, and then re-read the topic and write up either a big section in the Glome article, or a separate page on 4D planets, or both.

OK I've uploaded all the images in this topic.

I highly suggest a separate page on 4D planets, although the subdivision stuff can go in the glome page. After all, the 4D planets stuff is pure speculation, and besides, not very realistic speculation given that stable orbits don't exist (or can't possibly maintain themselves without some kind of fiat).
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Marek14 » Tue Nov 22, 2011 5:51 pm

quickfur wrote:
Keiji wrote:
quickfur wrote:currently my brain cpu power is being hogged by a background process trying to calculate the cells of the CRF polytope with 4 square antiprisms. :P


If it even exists ;)

Unfortunately you're right, it doesn't exist. Or at least, it does exist but is non-CRF because some of the tetrahedra are non-uniform.


I'll wait until the images show up, and then re-read the topic and write up either a big section in the Glome article, or a separate page on 4D planets, or both.

OK I've uploaded all the images in this topic.

I highly suggest a separate page on 4D planets, although the subdivision stuff can go in the glome page. After all, the 4D planets stuff is pure speculation, and besides, not very realistic speculation given that stable orbits don't exist (or can't possibly maintain themselves without some kind of fiat).


Stable orbits could exist if the 4D realm was just a "slice", i.e. if it was squeezed between two hyperplanes :) That would work, wouldn't it?
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Tue Nov 22, 2011 6:35 pm

Marek14 wrote:
quickfur wrote:[...]
I highly suggest a separate page on 4D planets, although the subdivision stuff can go in the glome page. After all, the 4D planets stuff is pure speculation, and besides, not very realistic speculation given that stable orbits don't exist (or can't possibly maintain themselves without some kind of fiat).


Stable orbits could exist if the 4D realm was just a "slice", i.e. if it was squeezed between two hyperplanes :) That would work, wouldn't it?

Not really, unless you're saying that the gravitational force is confined to a hyperplane, and so its flux decreases as an inverse square instead of an inverse cube. Besides, if you're confined between two hyperplanes, you're back in 3D. :P
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Marek14 » Tue Nov 22, 2011 6:44 pm

quickfur wrote:
Marek14 wrote:
quickfur wrote:[...]
I highly suggest a separate page on 4D planets, although the subdivision stuff can go in the glome page. After all, the 4D planets stuff is pure speculation, and besides, not very realistic speculation given that stable orbits don't exist (or can't possibly maintain themselves without some kind of fiat).


Stable orbits could exist if the 4D realm was just a "slice", i.e. if it was squeezed between two hyperplanes :) That would work, wouldn't it?

Not really, unless you're saying that the gravitational force is confined to a hyperplane, and so its flux decreases as an inverse square instead of an inverse cube. Besides, if you're confined between two hyperplanes, you're back in 3D. :P


Yes, that's what I'm saying. And there's a difference between a 3D and narrow 4D. The planet could still be 4D.

Another solution might be if the star was an infinitely-long spherinder (or a spherinder of finite length, if one dimension was circular). That would make the flux inverse square as well, wouldn't it? ;)
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Tue Nov 22, 2011 7:08 pm

Marek14 wrote:[...]Yes, that's what I'm saying. And there's a difference between a 3D and narrow 4D. The planet could still be 4D.

But in that case what happens to gravity at the confining hyperplanes? Does it get reflected? Absorbed? And would space have a discontinuity there?

Another solution might be if the star was an infinitely-long spherinder (or a spherinder of finite length, if one dimension was circular). That would make the flux inverse square as well, wouldn't it? ;)

But if the star was a spherinder, wouldn't it eventually collapse into a 3-sphere under the force of its own gravity?

Also, even though gravity would be inverse square, it won't result in a periodic orbit in general, because the planet will just spiral along the length of the spherinder with a constant displacement along the spherindrical axis. EDIT: The projection of the planet's movement would be an ellipse, but its actual motion would be aperiodic.
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Marek14 » Tue Nov 22, 2011 7:14 pm

quickfur wrote:
Marek14 wrote:[...]Yes, that's what I'm saying. And there's a difference between a 3D and narrow 4D. The planet could still be 4D.

But in that case what happens to gravity at the confining hyperplanes? Does it get reflected? Absorbed? And would space have a discontinuity there?

Another solution might be if the star was an infinitely-long spherinder (or a spherinder of finite length, if one dimension was circular). That would make the flux inverse square as well, wouldn't it? ;)

But if the star was a spherinder, wouldn't it eventually collapse into a 3-sphere under the force of its own gravity?

Also, even though gravity would be inverse square, it won't result in a periodic orbit in general, because the planet will just spiral along the length of the spherinder with a constant displacement along the spherindrical axis. EDIT: The projection of the planet's movement would be an ellipse, but its actual motion would be aperiodic.


It wouldn't collapse since gravity would be the same at each slice. And the aperiodic motion of the planet wouldn't be important - the important thing is whether the conditions on the planet would be stable, and the distance of planet from sun would be bounded.

Of course, not taking into account the energy output of the infinitely long star :)

For some reason, this reminds me of one of the more crazy fantastic world I thought of: a torus-shaped star with planets as smaller toruses. The star passes through the central hole of the planets, each planet passes through the holes of its moons, etc... Gravity is hacked to be perpendicular to the surfaces, of course :) Different planets have wildly differing diameters, so they can pass through each other.
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Tue Nov 22, 2011 7:41 pm

Marek14 wrote:[...]It wouldn't collapse since gravity would be the same at each slice. And the aperiodic motion of the planet wouldn't be important - the important thing is whether the conditions on the planet would be stable, and the distance of planet from sun would be bounded.

This can only be possible if the star is an actual infinitely-long star. Any finite-length star will eventually collapse into a 3-sphere because its ends will feel gravity towards its center.

Of course, not taking into account the energy output of the infinitely long star :)

Well, there will be other consequences in a universe that permits infinitely large objects. :) What if there is another infinitely long star that isn't 100% parallel to this one? The resulting anisotropy will distort the two stars so that they bend towards each other. They may either collide, or oscillate around each other causing a really strange shifting gravitational field. :P

For some reason, this reminds me of one of the more crazy fantastic world I thought of: a torus-shaped star with planets as smaller toruses. The star passes through the central hole of the planets, each planet passes through the holes of its moons, etc... Gravity is hacked to be perpendicular to the surfaces, of course :) Different planets have wildly differing diameters, so they can pass through each other.

Heh, that's pretty crazy. :)
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Marek14 » Tue Nov 22, 2011 7:58 pm

quickfur wrote:
Marek14 wrote:[...]It wouldn't collapse since gravity would be the same at each slice. And the aperiodic motion of the planet wouldn't be important - the important thing is whether the conditions on the planet would be stable, and the distance of planet from sun would be bounded.

This can only be possible if the star is an actual infinitely-long star. Any finite-length star will eventually collapse into a 3-sphere because its ends will feel gravity towards its center.


Not true. Can be done with a finite star, if the length dimension is finite and circular as well, so there are no ends.
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Tue Nov 22, 2011 8:03 pm

Oh, you mean like a torus-shaped star?
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Mrrl » Tue Nov 22, 2011 8:05 pm

quickfur wrote:
Marek14 wrote:[...]It wouldn't collapse since gravity would be the same at each slice. And the aperiodic motion of the planet wouldn't be important - the important thing is whether the conditions on the planet would be stable, and the distance of planet from sun would be bounded.

This can only be possible if the star is an actual infinitely-long star. Any finite-length star will eventually collapse into a 3-sphere because its ends will feel gravity towards its center.

I thought that you talk about the finite length of star when there is the circlular dimension. Such star will have no center, but unfortunately it will collapse to one or several 3-spherical stars (and the same is true for infinite spherinder in R4): the smallest irregularity in mass distribution will create centers of gravity on the line along the star, and parts of star will condense around centers... I don't know if it is true for toroidal star (that spins quickly around the plane), but probably yes: eventually we'll get a set of stars following the same circular orbit...
But we know that orbit is not stable. So stars may start to fall to the center of system - and what we get in result? They have huge angular momentum - resulting star should spin very quickly, and gravitational forces may be not enough to keep its parts in the center... will it expand to the initial toroidal shape again?
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Tue Nov 22, 2011 8:13 pm

All of this just means that we need to forget about our 3D-centric stars and planets type of system, and imagine a truly native 4D system. :)
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Keiji » Tue Nov 22, 2011 8:17 pm

Or forget about conventional gravity. My particle system allows for planet-star orbits, though they would form a wave pattern rather than a perfect ellipse:

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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Mrrl » Tue Nov 22, 2011 8:18 pm

quickfur wrote:All of this just means that we need to forget about our 3D-centric stars and planets type of system, and imagine a truly native 4D system. :)
Yes, one without planetary systems, without all day-and-night stuff, with the life on the surfaces of cold stars :)
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Tue Nov 22, 2011 8:58 pm

Mrrl wrote:
quickfur wrote:All of this just means that we need to forget about our 3D-centric stars and planets type of system, and imagine a truly native 4D system. :)
Yes, one without planetary systems, without all day-and-night stuff, with the life on the surfaces of cold stars :)

Not necessarily.

Years ago I invented an entire universe in which a completely different set of physical laws work.

This universe actually consists of 3 sections, but the 1st two are unimportant for the present discussion. The important part is the 3rd section, which consists of crystallized space quanta that form a 3D brane the inhabitants call "space". (There are actually 3 distinct branes, but they are more-or-less equivalent so I won't bore you with them here.) Within this space are a bunch of mostly irregular-shaped chunks of matter called "continents", on which the inhabitants live. These fragments are irregularly shaped, and gravity on the surface is not caused by mass but by fields of force that are, generally speaking, perpendicular to the surface. These fields can have sharp transitions, dividing gravity into zones of different directions, so when you walk from one zone to another, a sudden change in vertical perspective happens, which can be very disorienting.

But anyway. The sky, as perceived by the inhabitants on these "continents", does not have any stars or anything like that; instead, there are matter flows shaped by a certain kind of force that forms river-like structures, "sinks" which are punctures in the brane where matter leaks out into the 2nd section (which is a higher-dimensional area), and "sources" from which matter explodes as they enter into the 3D brane from the 2nd section of the universe. The continents were originally formed by condensation of cooled matter ejected from these sources, the the reason they are (mostly) irregular is because these sources and sinks come and go, so the original spherical shell of matter condensed from a source gets broken up over time.

Also, motion in this universe does not follow the inertial law; motion consumes energy, so these continents settle into a fixed place after a while.

Anyway, I shan't bore you with more details; there are various reasons why this universe behaves in this peculiar way, but that's unimportant. The point is that there's no need to limit our imagination to what we see in the 3D world -- after all, 4D has already proven itself to be very different by having no stable orbits! As you can see, in my fictional universe there's no such thing as planets, stars, or orbits, but that doesn't mean everything exists in a dark cold place.
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Keiji » Tue Nov 22, 2011 9:13 pm

I'm curious to learn more of this alternate universe of yours. :)
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby quickfur » Tue Nov 22, 2011 11:13 pm

Keiji wrote:I'm curious to learn more of this alternate universe of yours. :)

Well... er.... There's a reason I specifically mentioned it was "many years ago". ;) It was the result of accumulated story fragments, speculations, (and too many fantasy rpgs), dating back to my childhood, shoehorned together, and then retroactively hammered into a more-or-less self-consistent system complete with its own cosmological origin story. It even sports an invented "natural" language that defies almost all natural language "rules". But the core part of it was made during a time in my life when I just wanted to be different, so there are many things in it that are there just because they deliberately break conventional assumptions about universes. Over time, these little quirks got a bit tiresome and I stopped working on it actively years ago. In fact, at one point, in a fit of dissatisfaction with the whole thing I destroyed most of the notes I had on it (although there may still be backup copies somewhere... never had to reason to look for them since). Since that time I've worked on various other less mind-bending projects (like visualizing 4D, har har har), and other saner universes.

One of which, in fact, might interest us here, and that's an idea for a basis of chemistry that is based on geometrical symmetry rather than on emergent physics laws (which are extremely hard to come up with -- the slightest change can make the difference between an unworkable system, a workable but totally boring system, and a system that's "just right"). Unfortunately, I haven't actually worked it out in detail yet, but perhaps the underlying idea might be useful to you creative minds out there.

Before I start, I need to warn you that i'll be using terms like "energy" and "frequency" in a loose, analogical sense only; they may not (and probably do not) correspond to how real world physics defines them. Complaints about apparent contradictions arising from interpreting these words in the physics sense will be conveniently ignored. :P

The basic premise is that "energy" comes with resonant frequencies, and that symmetry amplifies this resonance. Matter that has the right shape will amplify this resonance, whereas matter that has the wrong shape will cause "destructive interference" that makes them very unstable. Thus, matter "prefers" to be in highly-symmetrical forms. For example, an equilateral triangle is preferred over an isosceles triangle, because there is a 3-fold "amplification" of energy due to its 3-fold symmetry, whereas an isosceles triangle lacks this symmetry. A scalene triangle doesn't even have the reflective symmetry an isosceles triangle has, so it's even less preferred.

Since matter is a form of energy, it is only stable when it has a highly symmetrical form. So shapes like regular polytopes are highly preferred, and correspond with "stable atoms". Highly symmetrical shapes also interact better with other forms of energy (they can "capture" ambient energy of matching frequencies via "resonance"). Shapes with less symmetry like uniform polytopes can exist, but are slightly less stable. The less symmetric it is, the less stable the matter. Now of course higher-dimensional polytopes are more complex than polygons, in that their symmetry has many aspects. For example, a rhombic dodecahedron's faces lack a high degree of symmetry, so that reduces its stability. But it is also face-transitive, which increases its stability. Similarly, Johnson polyhedra lose a lot of "resonance" because of their irregular shape, but their regular faces give them the ability to balance out part of their instability. There are also other stabilizing factors like edge lengths being equal -- a rhombic dodecahedron loses stability due to non-transitivity of its edges, but they are also equal length, which means some resonance is still there. And so forth.

This system isn't restricted to 3D, of course; it is only required that the shape of matter match its ambient space, so you can have 4D polytopes in a 4D space. In fact, 4D may turn out to be much more interesting than 3D, because there are relatively few highly-symmetrical shapes in 3D.

Now when these polytopes interact with each other, you have the complex interplay of these different aspects of symmetry. I haven't worked out how this interaction will work yet, but there will be shape changes involved, and what shape changes are more likely to happen depends on how well they maximize symmetry.

Not only does symmetry play a role at the microscopic level; at the macroscopic level you also have "shadow symmetry" -- a macroscopic object in the shape of a regular polytope will also resonate to energy frequencies corresponding to that polytope (albeit to a much lesser extent), even if it's made of atoms of unrelated shape. This resonance is sorta like a weak "echo" of the strong symmetry interactions at the microscopic level. As a result, macroscopic interactions are also influenced by symmetry. For example, a circular orbit is preferred over a spiral orbit because the circular orbit "resonates" better, and this resonance makes it self-stabilizing. (Now you know where i was going with all this. :P) Of course, a spherical orbit will be even better, but unfortunately a planet can't be in multiple places at the same time so it will just have to settle for a circular orbit. But circles aren't the only possibility... if you have 12 planets, for example, there is preferential treatment for them to form a icosahedral configuration around the star. Deviations from this icosahedral configuration break the symmetry and are thus "disliked"; the "symmetry force", if you can call it that, will try to pull the off-alignment planet back to its resonant position.

Anyway, I haven't worked any of this out in great detail, but it will be food for thought for you geniuses, i'm sure. :P
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Re: Wireframe spheres [Split from "Quickfur's renders"]

Postby Keiji » Tue Nov 22, 2011 11:27 pm

Hm, that's quite inspiring!

What if there were no matter, just "fractals" of atoms in arrangements?

You could have an icosahedron, where each vertex of the icosahedron was a smaller dodecahedron pointing to the adjacent icosahedron vertices.

And then each vertex of the dodecahedra would be more icosahedra.

But you need not have only "perfect" fractals like these, you could take subsets of the icosahedron's vertices, say three groups of four in tetrahedral symmetry, and one group could be truncated dodecahedra or something, another group could be pentagonal bipyramids...

When two shapes react they could rearrange their vertices to make a new shape (or more than one)... Of course, that'd cause the components at the vertices to rearrange themselves too, and there'd be a chain reaction going down the fractal tree until it became too insignificant to worry about.
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