-ve area

Completely random blog… was asked by a tutee yesterday what 2D and 1D meant… we then got onto 0D… this morning I got to thinking about –1D and then to an issue highlighted years ago by a friend at school that of –ve area. Well its easy: if distance is a vector rather than just scalar then a 2D cuboid with one negative side has negative area. In the quadrants of a Cartesian graph a cuboid with one corner at the origin has positive area in the top-right and bottom-left but negative area in the top-left and bottom-right quadrants. It can be seen that a –ve area is a reflection of its original, while rotation does not affect the area.

That is reflection through a line. Through a point is like the focusing of a image with a lens. Both sides are negated so the image is facing actually the right way up and is positive area just rotated a half circle. This is why a negative in a camera is a copy of the scene even though upside down and the wrong way around.

In 3D it’s the same. A reflection of a cube in the plane negates the dimension parallel to the plane’s normal. The volume becomes –ve. Through a line it is +ve and just a rotation of the original and through a point (if we could ever photograph in 3D) it is a negative image.

Now this makes me wonder how many orientations there are in a cuboid! In a rectangle with corners labelled clockwise there is only the negative area with corners labelled anti-clockwise. In a cuboid it is the same – obvious since you can only negate in one way. All the faces also change their clockwise/anticlockwise orientations also. Anticlockwise as we know from bearings is negative clockwise.

So –1D is simply a vector going in the opposite direction. –2D is a negative area etc. each is a reflection (a mirror image) of its 1D, 2D etc counterparts. Alice thus entered –3D when she went through the looking glass. One could imagine that time might go backwards in a -4D enhanced mirror in a 5D world.