N-Dimensional Geometry

[SaxonViolence]

Well Friends, we all know that Time is the 4th Dimension.

 

When I say that Time isn't a "True" Spatial Dimension, Physicists start trying to cut me off and tell me that I haven't understood.

 

When I try to say that Time Is a True Spatial Dimension, I get the same "Tut-Tutting" and comments that I don't fully understand.

 

Lets forget about Time Momentarily.

 

The 4th Dimension that Abbot talked about in "Flatland" had no relation to Time and the book predates Einstein.

 

So although we may not be able to fully visualize Higher Dimensions—pure Abstract Geometrical Dimensions, we should be able to Axiomatically Deduce much about them.....

 

And some Geometers have.

 

This is my question:

 

Imagine a Circle. In 2-D, a Circle is merely a Circle.

 

In 3-D a Circle can be Rotated to give a Sphere, but Rotated another way, it becomes a Torus.

 

A Circle can also give rise to both Right Cylinders and Oblique Cylinders.....

 

And its only cheating a little to say that a Cone is Generated from a Circle.Well everyone has heard of the Hyper-Sphere....

 

But surely with all of 4-D to work with, we can come up with all sorts of Rotated Spheres, Toruses, Cylinders and Cones.....

 

And there is probably room to Rotate a 2-D Circle in 4-D to get a few more Regular Solids.

 

And it seems as we add Dimensions, the number of possible Regular solids expands exponentially.

 

What Branch of Mathematics studies Puzzles like this?

 

Just what would I need to know to study things like this myself?

 

And does anyone know the answer—How many ways can you Rotate a Sphere in 4-D to Generate Unique Figures?

 

 

 

Saxon Violence

[Aethelwulf]

Well Friends, we all know that Time is the 4th Dimension.

 

When I say that Time isn't a "True" Spatial Dimension, Physicists start trying to cut me off and tell me that I haven't understood.

 

When I try to say that Time Is a True Spatial Dimension, I get the same "Tut-Tutting" and comments that I don't fully understand.

 

 

 

Oh... time is a spatial dimension, it is what is called the imaginary space dimension. It is still an imaginary leg off your usual three dimensional space triangle.

[sanctus]

In my understanding, by going 4-D you can generate any 3-D (and 2-D) shape you want, you just put the stuff you don't want into the fourth dimension. Over-simplified: put all but one point of the circle into the fourth dimension and then move it around to "draw" any 3-D shape you want.

[CraigD]

So although we may not be able to fully visualize Higher Dimensions—pure Abstract Geometrical Dimensions, we should be able to Axiomatically Deduce much about them.....

...

What Branch of Mathematics studies Puzzles like this?

Descriptions of branches of math are pretty fluid, but most mathematicians would still call the study of geometric objects in more than 3 dimensions “geometry”.

 

Geometry, IMO, is best understood as “point set” and “constructed” geometry involving countable numbers of points and traditional figure constructing methods, such as producing circles given a point and a radius. It’s often useful to approach things from the perspective of the continuity of functions implied by them, an approach usually called topology.

 

Just what would I need to know to study things like this myself?

To simply be able to describe a collection of points, connected or disconnected, in any number of dimensions, a good starting place is an introduction to linear algebra. Here’s a very brief one:

• A point in n-dimensional space can be represented as an n-tuple, and as a 1 by n matrix.
   
  
• A 1 by n matrix can be transformed into another 1 x n matrix by multiplying it be a n by n matrix. Here’s an example showing how, in 4-D:
  [math]
  \begin{bmatrix}
  a & b & c & d \\
  e & f & g & h \\
  i & j & k & l \\
  m & n & o & p
  \end{bmatrix}
  \begin{bmatrix}
  w \\
  x \\
  y \\
  z
  \end{bmatrix}
  =
  \begin{bmatrix}
  aw+bx+cy+dz \\
  ew+fx+gy+hz \\
  iw+jx+ky+lz \\
  mw+nx+oy+pz
  \end{bmatrix}
  [/math]
   
  
• Rigidly “connected” collections of n-tuples can be “rotated” by multiplying them by special n x n “rotation matrixes” like this example, one of 6 possible 4-D rotation:
  [math]
  \begin{bmatrix}
  \cos \theta & 0 & \sin \theta & 0 \\
  0 & 1 & 0 & 0 \\
  -\sin \theta & 0 & \cos \theta & 0 \\
  0 & 0 & 0 & 1 \\
  \end{bmatrix}
  [/math]
  

 

And does anyone know the answer—How many ways can you Rotate a Sphere in 4-D to Generate Unique Figures?

A sphere in any number of dimensions can be defined by an n-tuple P and a radius (a single number) R. Traditionally, it’s called a line in 1-D, a circle in 2-D, a sphere in 3-D, and a hypersphere in 4 or more-D, but the computational mechanics of it are the same in any number of dimensions: it consists of all points S where distance(P,S)=R.

 

Along with a point, a single sphere in 3 or 4 or more-D is about the worst choice for generating interesting 2 or 3-D figures, because no matter how you rotate or translate it, its 2-D shadow remains a circle, it’s 3-D “shadow” a sphere.

 

N-dimensional cubes – “hypercubes” are good for generating interesting 2 and 3-D figures using the “shadow” method. See my Shadows on n-cubes album for a few.

 

When I say that Time isn't a "True" Spatial Dimension, Physicists start trying to cut me off and tell me that I haven't understood.

 

When I try to say that Time Is a True Spatial Dimension, I get the same "Tut-Tutting" and comments that I don't fully understand.

LOL! Yeah, talking to specialists when you don’t grasp the rudiments of what they’re talking about can be frustrating!

 

What physicists mean by equating the time dimension with the 3 usual spatial dimensions is that, similarly to the examples I gave above, there is a way to rotate objects in 3 space+1 time dimensions through their time dimension. The details are ... well, detailed – here’s a decent starting place if you want to pursue them.

 

What geometers mean when they distinguish the time dimension from the spatial dimension is that the transformation matrix used to rotate through it isn’t the same as the ones for rotating in spatial dimensions only.

 

In my experience, hands-on experience with all this arithmetic mechanics is the surest way to get past confusion of this sort, and not get “tut-tutted” so much by math and physics folk.