4D Rotational Projection

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4D Rotational Projection

Postby gonegahgah » Sat Dec 06, 2014 1:25 pm

Is duosphere the correct term for a 2d circle->3d sphere->4d? equivalent?
Here's an animation image of what a glome 'duosphere' might look like at a certain distance and height from yourself using rotational projection. I haven't allowed for perspective.

Image This shows the resultant image of the glome with the 3D player tipping their head into the 4th dimension in a repetitive back and forth motion. The blue image is the current part of the glome in the 3Der's current view and the green misted part of the glome is outside the current 3D viewed frame.

The blue sphere section can be touched directly by the 3Der but the green misty area (remainder of the glome) can not unless that part of the glome object is rotated into the current 3D view exclusive to the other angles.
This is I'll try to do an animation one day of rotating through the sections of the glome object into the current 3D view.
[Thanks ICN for the correct term]
Update: Not so much that the glome is rotated into view but that the viewer rotates their view around into the 4th dimension seeing different angles of the glome
Last edited by gonegahgah on Fri Dec 12, 2014 12:44 pm, edited 8 times in total.
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Re: 4D Rotational Projection

Postby ICN5D » Sat Dec 06, 2014 5:45 pm

I believe the word you're looking for is Glome. This is what you get with bisecting rotation of 3D sphere. It's the 4D sphere, B4 and/or S3.
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Re: 4D Rotational Projection

Postby ICN5D » Sun Dec 07, 2014 12:47 am

Here's a good way to visualize the glome. We'll compare the moving 2D slice of a sphere, to the moving 3D slice of the glome.


The 3D sphere being reduced to only two of its dimensions. This is what a 2D being will see: only a circle. By sliding the sphere in and out of the 2D plane, we'll get a circle that expands and contracts
Image


The 4D glome being reduced to only three of its dimensions. This is what a 3D being will see : only a sphere. By sliding the glome in and out of the 3D plane, we'll get a sphere that expands and contracts
Image

I'm afraid rotations won't do it justice, much like a sphere. It's completely rounded in all 4 directions, and will always be a constant sphere.
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Re: 4D Rotational Projection

Postby gonegahgah » Sun Dec 07, 2014 11:28 am

Hi ICN,

Rotational Projection provides a means of showing the whole 4D environment at once while depicting the current 3D view.
The rotational aspect provides a place to project too rather than a motion itself.
All angles off into the 4th dimension are rotated up or down in our 3D space allowing for the depiction of the entire 360° of sideways space.

A similar thing could be done in 2D to depict our world where angles from the viewer into the 3D space are simply rotated up because they don't have a sideways to depict them.
This allows them in the same fashion to represent our 360° of surround. Fortunately our 3D world provides us with a bit more visual information when we do this.

When I animate it it should be a bit clearer. It becomes a combination of the slice method you've shown along with the rotated projection into the 4th dimension shown as a misted shape.
This allows the proposed player to get a feel of the object that they are dealing with.

In my image you can see the present 3D frame and you can see the bulk of the object in rotated projection.
When you rotate your view in one direction in the 4th dimension you see the bulk move down while the current frame depicts the part of the object that is in the present 4th dimension viewing angle.
When you rotate yourself in the other direction you see the bulk instead move up with the sphere growing or shrinking as it matches the current frame.
Rotating the whole 360° one way sees the bulk disappear up over your head, around to behind you, under you and finally returning to being before you.
Rotating it just 180° along the 4th dimension just puts it behind you and turning around in your present 3D frame 180° allows you to see the inverted 3D shape that has been described before.

One of the troubles of straight projection is that it doesn't provide a place to put the whole 4D environment but rotational projection does.
It should also provide the means to align with objects in more meaningful ways which might become more obvious when we look at a tesseract in rotated projection.
As you say a glome looks the same no matter where you move around it in the current frame but tesseracts can appear differently depending upon how you look at it.

But the main thing is that in rotational projection you see the glome like a football rather than a sphere because those parts off in the 4th dimension are projected upwards or downwards.
The amount of rotation is based upon their angle and distance from the viewer into the 4th dimension which as I discovered encompasses a full 360° of sideways.
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Re: 4D Rotational Projection

Postby Hugh » Mon Dec 08, 2014 5:18 pm

gonegahgah wrote:Here's an image of what a glome might look like at a certain distance and height from yourself using rotational projection. I haven't allowed for perspective.

Image

The blue sphere section can be touched by the 3Der but the green misty area (remainder of the glome) can not unless that part of the glome is rotated into the current 3D view exclusive to the other angles.
I'll try to do an animation one day of rotating through the sections of the glome into the current 3D view.

Update: Not so much that the glome is rotated into view but that the viewer rotates their view around into the 4th dimension seeing different angles of the glome


This is fascinating gonegahgah! I'm looking forward to further animation of rotational 3D viewpoints, within 4D.

When I look at the picture above, it shows a 3D "slice" of what we can see of our higher dimensional universe, and the misty area as being the "missing mass" or "dark matter" that we are looking for. :)
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Re: 4D Rotational Projection

Postby gonegahgah » Tue Dec 09, 2014 5:19 am

[The original text that was here - except the following - was deleted as the first post has been edited to contain the animation that was to be added]

Hi Hugh. The main problem with connecting 4D space and "dark matter" is that the gravitational forces would have an extreme and awkward space curvature if we had 4D space.
But, keep thinking about it...
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Re: 4D Rotational Projection

Postby gonegahgah » Fri Dec 12, 2014 1:00 pm

Turning your head into the 4th dimension for a glome is probably the simplest example to do.
A tesseract will be much more interesting to look at. I've noticed already that thinking about a tesseract highlights the two types of movement that I think are important.
Just as we can view the base of a cube (that's sitting on it's base) along one to two edges the same applies to a tesseract.
We can view the base of a tesseract along one to three of what we would call edges. These extend vertically to form what we consider to be the volume of the faces we see.
For us 3Der's we are a little more limited on what we can view of those faces.
First we have to consider that, whereas we can only turn our head along left and right to observe the base of a cube in our world, we can angle our head a whole 360° of angles to follow the base of a tesseract.
But, not only can we tilt out head we can also spin in a circle to change our orientation. It makes it quite interesting to depict a tesseract using rotational projection.
I would rather have it be interactive as this is really what you need to appreciate the experience.

It did give rise to the the thought that I believe that 4D creatures would actually develop legs that not only propel them forward but that also are designed to allow them to change their orientation easily.
So they would need multi-perambulating legs that can run in cicles as well as forwards. Interesting thought.

Added 27/12/2014:
Here is a depiction of a tesseract in rotated projection. There are certainly things I want to improve in the depiction.

Image

The important things it shows presently are that whereas we can see only three faces of a cube at any one time; using rotated projection you can see four faces of the tesseract at once: The three front cube faces: blue, pink and red and the top cube face: green. Most of these are shown ghosted off into the 4th dimension.

The things I would like to fix are: 1. I would like for each object its parts that are in the ana direction to be behind the object's current frame and parts that are in the kata direction to be in front of the object's current frame, 2. Full projection rather than stepped projection as was utilised here, 3. Less transparency for the denser parts in the 4th directions, 4. None of those black outlines.

As I mentioned this rotated projection style should more readily allows us to understand how the cube faces of a tesseract work.
Also a fully interactive model should give us a great feel for a tesseract allowing us to orient ourselves around it in ways we find more comfortable.
Last edited by gonegahgah on Sat Dec 27, 2014 11:21 am, edited 1 time in total.
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Re: 4D Rotational Projection

Postby Secret » Fri Dec 12, 2014 3:17 pm

If I understand the rotational projection properly... (see pics)
I strongly encourage you to make a gif of looking at a tesseract vertex on, cause my preliminary insight/hindsight using what I learnt form quickfur and others seemed to suggest this view have the most structural info about the tesseract

I have not checked whether my insights are correct through because my mathematica had just expired and it takes time to renew its license
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Re: 4D Rotational Projection

Postby gonegahgah » Fri Dec 12, 2014 9:43 pm

Nice interpretation and depiction Secret. :)

While for a 2Der we could depict a cube sitting on its base, or sitting on an edge, or sitting on a point, most usually in our world cubes rest on a base.
We can depict a tesseract sitting on its base (a cube), on a face, on an edge, or on a point; so they have one other resting (or balancing/supported) orientation than we do.
While it certainly would be interesting to view those other orientations in rotated projection I'm thinking about a simple world first.

So for the moment I'm only looking at the first 3 orientations you depicted for us which would be the different possible views of the normal resting position for a tesseract.

[Added the following]
Though we are considering resting tesseracts only at the moment it needs to emphasized that this doesn't preclude viewing from a multiple of orientations that would otherwise not seem resting in our 3D world.
For example, the following depicts an acceptable resting base for a tesseract as being observed by the 3D observer; though they would see only one of the rotated slices in their frame.
The other rotated frames indicated will appear misted to the 3D observer.

Image

As Secret has already depicted, each of these seen frames would then extrude vertically - in their rotated frame and at the height of our tesseract - to form the overall rotated projection that the 3D observer would see.
The red slice is the base of the current 3D slice of the observer which can change as they tilt or rotate their head into the 4th dimension to the other frames.
This frame extends vertically to form the currently viewed frame - which for the resting tesseract itself extends up for its height.
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Re: 4D Rotational Projection

Postby Prashantkrishnan » Sat Jan 17, 2015 5:53 am

The rotational projection, from the animation of the glome, looks like a method to watch the realmic cross-section of the glome in blue as it passes through our realm and the remaining part projected in green. But the one of the tesseract is stationary and I am not able to make out which part of it lies in our realm.
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Re: 4D Rotational Projection

Postby gonegahgah » Fri Jan 30, 2015 11:46 pm

Hi Prashantkrishnan,

Here is a depiction that is a little closer to what I want:

Image

Notice that it looks very similar to a combination of general projection and slice depiction.
The difference for rotational projection is that everything has somewhere to project to.
Note you could just as easily use sideways to project into; as well as the shown up and down; or in fact any other angle diagonally.
Both sideways and up-down would be useful at different times.
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