perspective

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perspective

Postby Upsilon » Fri May 27, 2005 1:51 pm

how can we understand 4D. How can you draw a 4D shape on a 2D plane? Since we are 3D and live in a 3D world is it possible for us to visualize this other dimension.
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Postby pat » Fri May 27, 2005 2:07 pm

How can we visualize something that's 3D unless that thing is made of a semi-transparent material.

How can you visualize a house? You look at it from a few angles. Then, you form a picture in your mind of how those angles fit together. But, you never really saw all of the sides at once.
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Postby wendy » Sat May 28, 2005 7:29 am

One can quite easily, and usefully, project four dimensions onto two dimensions. I often look at higher dimensional things by heavy dimension reduction.

In essence, a point (x,y,z) contains x, y and z coordinates, which can be presented as numbers.

A thing in (w,x,y,z) evidently projects as well onto (x,y) or (w,x) or (x,z) planes. Many of the uniform and regular figures in four dimensions project quite elegantly into (w,x) + (y,z) planes, where these are in some cases the same design [ie like the cube, plan = elevation].

The 24-choron, on the other hand does not (rather like the hexagonal, the view down perpendiculars is different).

The vertices of this is x3x o3o + x3o q3o + o3x o3q

What this means, is that we have the 24-choron breaking down into three duoprisms, which present thenmselves as one base on one axis, and the other base on the other axis.

x3x o3o = hexagon * point
x3o q3o = triangle ^ * triangle (r2) ^
o3x o3q = triangle v * triangle (r2) V

The left plot consists of a hexagon, with an inscribed star-of-david. The outer hexagon-vertices are "red", while the inner hexagon is alternately white and blue. In the alternate projection, we see the central dot as red, and the the outer hexagon being alternately white and blue vertices.

In the first projection, we see the edges as

outer perimeter
star of david on outer hexagon (makes three edges each)
star of david on inner hexagon (makes one edge each)

You can then see the faces (octahedra), as a central hexagon [which is what one gets face first] and outer rectangles [squares shortende by prospective] representing vertex-first views.

On the second projection, one sees the octahedra as two triangles (agani forshortened by prospective) and the outer six edges as "directly on the edge). When you draw a cube face-first, you see faces become lines.

Most of the other figures look identical projected down both axies.

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they are just like 3d , they are just taking a break^2....

Postby Flawless » Mon May 30, 2005 9:15 am

time is the only thing keeping us from seeing 4d things so stop it with a camera. one picture will catch a single layer of every 4d thing in the way from your face to the back of your head in a straight line ... thats right all the way around the world... so u have to know what your looking for build a 3d modle of what you wanna pull from tetraspace then you have to have some kind of identifier .... eyes are usually good .... take a picture and use it for the skin of your 3d modle :)))))

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Postby wendy » Mon May 30, 2005 11:00 pm

One should read my comment on time as the fourth dimension, in the hyperspace thing at:

http://www.geocities.com/os2fan2/gloss/index.html

There, i construct a peice of space-time with 2d space. Essentially, you lay peices of paper rather like a cartoon animation. The thing ends up a whole lot of paper, with no particular time. Let's face it, there's a sheet of paper representing 3:30, and another one representing 3:31.

We can no more talk to these characters than we can talk to the animations we draw in the sides of books.

In the other hand, this is -not- what we see as 4d.

Some of us have been as high as seven dimensions, so there you go.
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peek-a-boo!

Postby Flawless » Tue May 31, 2005 1:52 pm

you might catch some tetraspace stuff with this bad boy .... or you may just get arrested for spying on your neighbors. Remember what your looking for though thin slices of 3d objects. the sphere allows you to put some of those layers together to give 4d things more 3d volume.

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