by **wendy** » Mon Jun 11, 2007 7:37 am

A change of axies, in the style of x -> y, y -> -x corresponds to a rotation of 90 degrees in the xy axies. This applies to all pairs of axies.

One should note that there's nothing special about the axis system: we don't live in a coordinate graph. It's just a convenient overlay for understanding the appropriate dimension: kind of like the map grid.

It is just easier to explain things in terms of cartesian coordinates (eg xyz axis) then it is to non-euclidean coordinates, largely because people have wrangled with graph paper in the past, and are thence more familiar with it.

Unlike three dimensions, the fourth dimension leads to a 'false friend' in the coordinate system. This is the notion that because the three axies in 3d are quite easy to set up common to all viewers, it applies in higher dimensions too: this is not so.

One axis is set by gravity: it appears as up/down. The second axis is partially set by time: front/back. Since we are creatures that move against time, this provides an arrow perpendicular to up/down.

There is nothing to set any further axies. In three dimensions, the across space is a line perpendicular to up and forward, and one can set left/right. In four dimensions, the space perpendicular to up/down (eg z), and front/back (eg y), is not simply x, but two axies (wx).

The effect is to imagine a room as a map of a floor of a 4d room, (ie wxy space), and then suppose that our height is y, and the floor is xy. The issue here then is that there is no special way you can place a clock on the floor based on zy axies, that will make individual clocks align. You can still have clockwise/anticlockwise, but this does not correspond to left/right of 3d.

The dream you dream alone is only a dream

the dream we dream together is reality.