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Orthogonality and vocabulary. In the familiar three-dimensional space of daily life, there are three coordinate axes —usually labeled x, y, and z —with each axis orthogonal (i.e. perpendicular) to the other two. The six cardinal directions in this space can be called up, down, east, west, north, and south.
What exactly is the 4th dimension? Let’s break down spatial dimensions into what we know. We can describe a point in 2-dimensional space with two numbers x and y, visualizing an object in the xy plane, and a point in 3D space with 3 numbers in the xyz coordinate system.
Four-dimensional rotations are of two types: simple rotations and double rotations. Simple rotations. A simple rotation R about a rotation centre O leaves an entire plane A through O (axis-plane) fixed. Every plane B that is completely orthogonal to A intersects A in a certain point P.
Don't forget that you aren't seeing a real 4-dimensional XYZW coordinate system - just a 3D projection of it. In four dimensions, all four axes should make a right angle with every other axis, but that's impossible in our lame 3D universe.
Each coordinate gives the position on one of the four orthogonal axes. These coordinates are typically labelled (x, y, z, w), and so will I. Setting one coordinate constant describes a hyperplane (3-dimensional affine subspace) orthogonal to the corresponding axis; for example, setting w = 0 gives the x - y - z space.
Comparatively, 4-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled w. To describe the two additional cardinal directions, Charles Howard Hinton coined the terms ana and kata, from the Greek words meaning "up toward" and "down from", respectively.
With respect to this choice of coordinate axes, any point is completely determined by the number quadruple (x, y, u, v), and the point is the sixteenth vertex of a parallelotope with one vertex at the origin. In this way we obtain a coordinate system for four-dimensional space.
The space described by these 4 dimensions is called 4-dimensional space, or 4D space for short. In a 4D world, there is another directional axis which is perpendicular to the X, Y, and Z axes. We shall label this axis W , and call the direction along this axis the fourth direction .
Drawing a picture of a four dimensional tesseract in a three dimensional space is straightforward. We take two of its faces--two cubes--and connect the corners. There are several ways of doing the drawing that corresponds to looking at the tesseract from different angles.
Four-dimensional geometry is Euclidean geometry extended into one additional dimension. The prefix "hyper-" is usually used to refer to the four- (and higher-) dimensional analogs of three-dimensional objects, e.g., hypercube, hyperplane, hypersphere. n-dimensional polyhedra are called polytopes.