visualizing hyper-spin, about a 4D axis ??

[Widdekind]

imagine a 3D flat space, embedded in a 4D flat space

imagine a 3D object, in the 3D "hyper-3-plane" space

imagine a linear vector through the 4D flat space, from the center of the 3D object in the 3D "hyper-plane"…

hyper-dimensionally "out", to higher hyper-dimensional altitude, in the 4D space

i.e. the center of the 3D object = (0,0,0 | 0)

and the 4D linear vector = (0,0,0 | 1)

with its "tail" at the center of the object (0,0,0 | 0)

Now, please ponder spinning the 3D object, about the 4D linear vector, threading through its center, "out" to higher hyper altitude

in analogy, a 2D object, in a 2D flat-land, rotating about a 3D linear vector, from its center, "out" to higher 3D altitude…

would be rotating around, through the 2D perpendicular to the 3D vertical vector…

subject to the constraint, that all perpendicular distances were kept constant…

leaving (2-1)=1 degree-of-freedom to spin around in, which 1 DoF = azimuthal angle

so, if the 2D flat-land object were rotating with constant frequency, then it would be spinning around through azimuthal angle, periodically:

[math]\theta = 2 \pi f t[/math]

so, by analogy, a 3D object, in 3D space-land, spinning around a 4D axis threading through its center…

would be spinning through the 3D "xyz" perpendicular to the 4th hyper-dimension "w"…

subject to the constraint, of keeping constant all distances from that hyper axis…

so leaving (3-1)=2 DoF to spin around in, which 2 DoF = surface of a sphere of [math]4 \pi[/math] steradians…

i.e. a 3D object, spinning periodically about a 4D axis, would be seen "from the side" in 3D space-land, to be periodically spinning around, through the [math]4 \pi[/math] steradians of the surface of the sphere, i.e. of solid angle, centered at its center

i.e. from 2D to 3D:

[math]\theta(t) = 2 \pi f t[/math]

[math]\longrightarrow[/math]

[math]d\Omega(t) = 4 \pi f t[/math]

Question: is the above a correct extrapolation, from 2D "flatland", to 3D "spaceland" ? Would a 3D object in spaceland, spinning about a 4D axis threading through its center, be seen, to be spinning and twirling and tumbling around, through the 2 DoF of polar-angle and azimuthal-angle, coordinatizing the [math]4 \pi[/math] steradians of solid angle, about its center ?

[zeon23445]

I'll answer this with another sence of mathematics. Let's say you have a program, like any other, confounded to be only 2D in whatever shape it makes.

 

Let's say, that we decided to make a 3D plotting program that plotted points on the screen and assigned it a global position (according to absolute size on screen compared to position) and a size. The size, determines a plane. So we scale it to 3x3 and we make sure our clients camera (our individual view of this 3D world) is centered at this scaled objects axis. Once we rotate the view (again on that objects axis) we see how the object moves in its space.

 

Now imagine this object being overlayed with a circle (that doesn't respond to 3Dmvements.) This circle, at all times gets the magnitude of all the corner vertexes, and resizes it self from the 3D bricks axis and makes sure to always be a fraction of a unit bigger than the whole brick from our point of view. If this 'circle' was invisible we would always see the brick. If it was a full circle with no transparency, then we would never see the brick as the circle resizes itself to cover up the brick.

 

That's for a 3D perception in a 2D constricted space. Aka, the computer Program, and your monitor.

 

The next logical thing is a 4D perception, in a 3D constricted space. Imagine your entire 4D object being covered by a Sphere.

The sphere will only stay the same in size, as you rotate around the average XYZ position (which means the average of all the bozons, and particles and atoms XYZ position, through all the points in time of its coherence in the 3D dimension, compared to the global position based on the complete center of the of the universe, expressed through a 4 Dimensional additive) this is represented as the w axis, and the size of the sphere changes as you move through the axis (which is time [w axis]). So, the radius of the 2D object in the program is a scalar interpretted through a program to resize it, and the 3D objects radius (expressed from a 4D perception (Aka Time/ w axis)) is expressed as a vector, as it has a direction. This vector, is the Azimuthal angle.

 

This reply was to merely break down this whole idea to give you a broader perspective on what this is. I hope I helped.

Excuse me it isn't THE Azimuthal angle, it is expressed by an Azimuthal Angle

In the 4D space it would be a steradian, and would move depending on the size of the sphere. So as time goes by, if there's decay in a 3D object, this steradian might become smaller, if not it might stay the same, and if there's a hypothetical reverse decay where matter is simultaniously added on, then the steradian might increase in size. (Excuse me if I made a spelling error)