Looking at a 3D object from 4D

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Offline marcm200

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« on: December 07, 2019, 01:22:03 PM »
I was wondering: An ant living in a plane, looking onto the Mandelbrot set from its exterior (I mentioned this earlier) sees only a wall of points, some closer some farther, maybe shaded according to curvature.

We as 3D observers see the Mset in its totality from the 3D world looking onto a 2D object.

But we see only the surface of a 3D Juliabulb, shaded, but more or less just points closer and farther away.

If one could look onto the bulb from a 4D point - would I then see the front side of the Juliabulb, its rear side and all its interior (so not just slices thereof) at once?

Considering how different a line and the Mandelbrot set are: Does that mean the 3D Juliaset actually would look very different as well?

What would the ant be able to comprehend when moving along the wall it sees? Would it get an idea of all the bulbs and filaments there?

Can this difference of what a 2D object looks like in its plane of existence or a dimension higher be expressed mathematically?




Linkback: https://fractalforums.org/off-topic/29/looking-at-a-3d-object-from-4d/3222/

Offline 3DickUlus

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« Reply #1 on: December 07, 2019, 11:09:16 PM »
I think this offers some insight...
https://en.wikipedia.org/wiki/Tesseract
Quote
In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.
Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells.
The tesseract is one of the six convex regular 4-polytopes.

What we view in 3D of the 4D realm is equivalent to the 2D shadow of a 3D object.
A 3D representation of a 4D object is it's "shadow"... makes me think that a hologram could be the shadow of 4D(X,Y,Z,time) ??? It appears 3D but you can't touch the objects any more than you can touch the 2D shadow of a 3D object.

re:op the math for this might be similar to projecting a shadow of a 3D object, add 1D to object (edit:and light), add 1D to shadow ??? object = quaternion, shadow = triplex, of course this is all speculation on my part as a math dummy  :clown:
Fragmentarium is not a toy, it is a very versatile tool that can be used to make toys ;)

https://en.wikibooks.org/wiki/Fractals/fragmentarium

Offline tavis

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« Reply #2 on: December 20, 2019, 11:54:21 PM »
The Mandelbrot set is closely related to a circle, so that's how I will take a stab at this. A circle, mathematically, can be considered a sphere if we define a sphere in a general sense as being the set of all points that are equidistant from the center. In technical terms, a 2D circle is considered an S1 sphere, because it is made out of a 1-dimensional border (line). A 3D ball is considered an S2 sphere because it has a 2d surface. Likewise, the S3 sphere is 4D and so on.

We can't observe 4D objects directly but as 3DickUlus pointed out, we can project down to the next lower dimension. So, for example, you've probably seen the World map where the globe is projected downward onto a circular shape, with the North Pole at the center. The South Pole becomes a singularity that explodes, thus Antarctica surrounds the entire perimeter.  Mathematically, this is known as stereographic projection of an S2 sphere onto an S1 sphere. An interesting property of this projection is that even though the continents are distorted, the perfect circles of latitude on the globe are still perfect circles on the map.



It is possible to do the same thing with higher dimensions. Circles on an S3 sphere (4 dimensional) can be stereographically projected down onto an S2 sphere (3 dimensional) where we can see it. The result is a distorted view of the 4D sphere but as before these circles maintain their circular nature, and they are known as "fibers." The collection of these fibers is known as a Hopf fibration. The collection of fiber circles that are mapped to a single line of latitude on the S2 sphere form a torus. So the Hopf fibration can be thought of as nested tori. The circles are known as Villarceau circles  https://en.wikipedia.org/wiki/Villarceau_circles and although in the Hopf fibration there are an infinite number of these circles forming an infinite number of nested tori, you can choose any two circles and they will be interlocked, but no two circles overlap. Here is a great video that demonstrates the relationship between the fibers and the S2 sphere. A single point on the S2 sphere corresponds to a single circle. Notice that the singularity occurs at the north pole, where the Hopf fiber becomes an infinitely big circle and so just looks like a straight line. At the South pole, the torus that is formed collapses down into a circle with a radius of 1. It seems to be a really cool object but back to the original question - I honestly have no idea how to apply this to fractals in a practical way... maybe someone else here can help make that connection.

« Last Edit: December 21, 2019, 04:25:20 AM by tavis »
Check out the Mandelmap poster a detailed map of the Mandelbrot set in the style of a vintage map.

Offline marcm200

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« Reply #3 on: December 21, 2019, 01:14:14 PM »
That's a great video!

Would a stereographic projection of the classical 2D Mset's border into 1D look like a symmetrical line of different gray shades (if that's reflecting the distance to the observer)? As a 2D ant I would only see at most one half of the boundary, so the projection gives me more information?
(I think my brain just twisted :)

Offline tavis

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« Reply #4 on: December 21, 2019, 08:09:13 PM »
I don't think you can use stereographic projection for lower dimensions. An S1 sphere (2D circle) cannot be stereographically mapped to an S0 sphere because an S0 sphere is just the points -1 and 1 on the number line and there is no way to preserve the circular property. I think what you are talking about is more of an orthographic projection. The math is much simpler for orthographic projections, but more information is lost. Think about the globe example - with stereographic projection, every continent is visible, even though there is some distortion. This is in contrast to an orthographic projection from the top of the globe, where you would not see anything past the equator. Have you ever seen the movie (or read the book) Flatland? It is very similar to what you are talking about. They live in flat 2d land and all the characters are regular polygons. They see eachother as lines that change in value.


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