Six 2D-view names of 4D-shape?

Higher-dimensional geometry (previously "Polyshapes").

Six 2D-view names of 4D-shape?

Postby begvend » Mon Sep 28, 2009 6:46 pm

Three 2D-projections defined a 3D-shape.
One on a xy plane (front or rear views).
One on a yz plane (right or left views).
And one on a xz plane (above and below views).

There are four 3D-shadows of a 4D-shape.
One in Xyz real space (front or rear shadows).
One in Xyw real space (right or left shadows).
One in xzw real space (above and below shadows).
And one in yzw real space (ana or kata shadows).

Finally six 2D-projections defined a 4D-shape.
One on a xy plane (front or rear views).
One on a yz plane (right or left views).
And one on a xz plane (above and below views).
But what are the 3 other names of 2D-projection of a 4D-shape?
On a xw-plane, on a yw-plane and on a zw-plane.

Many thanks for your comments.
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Re: Six 2D-view names of 4D-shape?

Postby wendy » Tue Sep 29, 2009 8:58 am

The names like front/back etc do not correspond to x/y/z axis. Some folk regard z as height, and so xy is above/below and yz is front/back and xz is left/right.

It depends on how ye regard 4d.

One could have U/D (up/down, ie gravity) and F/B (forward/backwards, ie motion), but the other two axies do not correspond to any difference we know (except across). Compression in U/D do not include these directions, not do those of F/B.
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Re: Six 2D-view names of 4D-shape?

Postby Keiji » Fri Oct 02, 2009 9:26 am

I generally regard z as frontal and y as vertical, leaving other dimensions for laterals.

In 3D, there are the 2D projections front/back [!z] [xy], top/bottom [!y] [xz] and left/right [!x] [yz].
In 4D, there are the 3D projections front/back [!z] [wxy], top/bottom [!y] [wxz], left/right [!x] [wyz] and ana/kata [!w] [xyz].
For 2D projections from 4D, there'd be six, but each projection has four sides! These are:

front-top/front-bottom/back-top/back-bottom [!zy] [wx] <-- not comparable to any 3D -> 2D view; it only shows laterals
front-left/front-right/back-left/back-right [!zx] [wy] <-- comparable to front/back view; it shows vertical and one lateral
front-ana/front-kata/back-ana/back-kata [!zw] [xy] <-- comparable to front/back view; it shows vertical and one lateral
top-left/top-right/bottom-left/bottom-right [!yx] [wz] <-- comparable to top/bottom view; it shows frontal and one lateral
top-ana/top-kata/bottom-ana/bottom-kana [!yw] [xz] <-- comparable to top/bottom view; it shows frontal and one lateral
left-ana/left-kata/right-ana/right-kata [!xw] [yz] <-- comparable to left/right view; it shows frontal and vertical
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