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A project I'm working on

lord_nightrose
(Web Page)

Anybody have any idea how to convert between 4D and 3D points?? I already have something to convert 3D to 2D... once I can do that extra step I will have a perfectlyworking 4D engine!!


4 November 2002, 20:45 GMT








Re: Re: A project I'm working on

lord_nightrose
(Web Page)

No, the fourth dimension is NOT time. It hasn't even been proven that time is linear, so how could it be a dimension? any given dimesion can only be
positive, negative, or zero.
and no, 3D games are not 4D, or you would be able to move *back* in time in them, according to your theory.
anyway, here's a brief rundown on 4D geometry.
0D space is a point. It has no size.
In 1D space, the only figure you can have is a line, formed by taking a point and moving it along a dimension to another point. It has one edge, which is
1dimensional, and 2 points, which are 0dimensional 'faces'.
In 2D space, the 'generic' figure is a square  four line segments connected at right angles. You can form it by taking a line and moving it along a second
dimension. It has four zerodimensional points, four onedimensional edges, as well as a twodimensional edge (a face).
In 3D space, the cube is the 'generic' figure. It is formed by taking a square and moving it along a third dimension. It has eight zerodimensional points,
eight onedimensional edges, six twodimensional faces, and one threedimensional face (the cube itself). Since the only way to represent a 3D object on a
flat surface is to project it from some angle, this is equivalent to saying that you are drawing the 2D shadow of a 3D object.
<end part 1>


5 November 2002, 09:21 GMT








Re: Re: A project I'm working on

lord_nightrose
(Web Page)

Notice the pattern yet? In a generic figure in ndimensional space, the number of edges equals the number of points, which equals 2^n, and there are 2*n
faces, with each face being a (n1)D figure. Thus, in 4D space, a generic figure (which is commonly called the 'hypercube', or 'tesseract') has 16 points, 16
edges, and 8 faces. These faces, as can be expected, are (n1)D, which means they are 3D. The eight faces of a hypercube are cubes themselves. To represent a
4D object, the only thing we can do is to draw the 2D shadow of the 3D shadow of the object, since we can't physically construct 4D objects.
The appearance of a 4D object can be quite confusing. Example #1:
http://www.cogsci.indiana.edu/ farg/harry/phy/4dmat09.gif
sorry about the big image.
The sphere in that picture is NOT changing size, or technically even shape. It is a 4D sphere, moving along the 4th dimension. You might call this the result
of 4dimensional 'perspective', of a sort: when you watch someone pass by you on the street, they may appear to be shaped or even sized differently, but this
is only because of the 3dimensional distance between the two of you. Similarly, a 4D object appears to change shape depending on your 4D distance from it.
The only problem is that when you watch it in 3D space, it seems almost undeniable that it *is* changing shape.
(note: if you visit the web site I got that image from, you'll notice they constantly refer to the 4th dimension as 'time'. however, they don't really mean
it *is* time. they simply used time as a way to introduce a fourth variable. it would be the same as using four embedded 'for' loops, say with variables
w,x,y, and z, to represent the position of a 4D object. time is simply a convenient way for mathematicians to take 3D snapshots of a 4D object as it moves
through 4D space.)
<end part 2>


5 November 2002, 09:22 GMT








Re: Re: A project I'm working on

lord_nightrose
(Web Page)

Another way to visualize why it seems to change shape is this:
When a line intersects with 0D space  a point  the entire intersection is that single point.
When a plane intersects with 1D space  a line  the entire intersection is that line.
When a cube intersects with 2D space  a plane, or square, or whatever  that intersection can be a square or a triangle (maybe something else, but it's 2:50
AM and I'm damn tired).
When a hypercube intersects with 3D space  any 3dimensional object or area  it can be assumed that there are an immense number of 3D shapes  in most
cases, an infinite number  which its intersection can form. these shapes don't even have to follow the laws of what we consider reality. for example, a 4D
tube could intersect with 3D space at one end, and the opening of the tube could hang in midair. when you go to the other side of the hole  which may be
infinitely deep, or whatever you'd call the fourth dimension  there might not even *be* another side. it could just be empty air.
OK. I know I said this would be brief, but I'm rambling. The main points I'm trying to make:
1. time is *not* the fourth dimension
2. representing 4D objects on a 2D plane is possible, but it requires complex mathematical formulas
3. I'm tired
Hope this clears evertyhing up.
lord_nightrose
P.S. For a better explanation than the one I've given, check out http://home.san.rr.com/ slawek/Tata/Math/4DGeom.html .
P.P.S. http://www.cs.mu.oz.au/ ~amb/4d/ has a huge library of 4D and ND related links. Check it out. There's java!! ooo!
<end part 3>


5 November 2002, 09:22 GMT


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