The four-dimensional cube
The eight corners of a cube in three dimensions have coordinates of the form
(x, y, z) where each of the x, y, z are either 0 or 1. The cube has six faces each
of which is a square and there are 2 × 2 × 2 = 8 corners. What about a four-
dimensional cube? It will have coordinates of the form (x, y, z, w) where each of
the x, y, z and w are either 0 or 1. So there are 2 × 2 × 2 × 2 = 16 possible
corners for the four-dimensional cube, and eight faces, each of which is a cube.
We cannot actually see this four-dimensional cube but we can create an artist’s
impression of it on this sheet of paper. This shows a projection of the four-
dimensional cube which exists in the mathematician’s imagination. The cubic
faces can just about be perceived.
A mathematical space of many dimensions is quite a common occurrence for
pure mathematicians. No claim is made for its actual existence though it may be
assumed to exist in an ideal Platonic world. In the great problem of the
classification of groups, for instance (see page 155), the ‘monster group’ is a way
of measuring symmetry in a mathematical space of 196,883 dimensions. We
cannot ‘see’ this space in the same way as we can in the ordinary three-
dimensional space, but it can still be imagined and dealt with in a precise way by
modern algebra.
The mathematician’s concern for dimension is entirely separate from the
meaning the physicist attaches to dimensional analysis. The common units of
physics are measured in terms of mass M, length L, and time T. So, using their
dimensional analysis a physicist can check whether equations make sense since
both sides of an equation must have the same dimensions.
It is no good having force = velocity. A dimensional analysis gives velocity as
metres per second so it has dimension of length divided by time or L/T, which
we write as LT−1. Force is mass times acceleration, and as acceleration is metres
per second per second, the net result is that force will have dimensions MLT−2.
Coordinated people
Human beings themselves are many dimensioned things. A human being has many more
‘coordinates’ than three. We could use (a, b, c, d, e, f, g, h), for age, height, weight, gender, shoe
size, eye colour, hair colour, nationality, and so on. In place of geometrical points we might have
people. If we limit ourselves to this eight-dimensional ‘space’ of people, John Doe might have
coordinates like (43 years, 165 cm, 83 kg, male, 9, blue, blond, Danish) and Mary Smith’s
coordinates might be (26 years, 157 cm, 56 kg, female, 4, brown, brunette, British).