24 Dimension
Leonardo da Vinci wrote in his notebook: ‘The science of painting begins with the point,
then comes the line, the plane comes third, and the fourth the body in its vesture of
planes.’ In da Vinci’s hierarchy, the point has dimension zero, the line is one-dimensional,
the plane is two-dimensional and space is three-dimensional. What could be more
obvious? It is the way the point, line, plane and solid geometry had been propagated by
the Greek geometer Euclid, and Leonardo was following Euclid’s presentation.
That physical space is three-dimensional has been the view for millennia. In
physical space we can move out of this page along the x-axis, or across it
horizontally along the y-axis or vertically up the z-axis, or any combination of
these. Relative to the origin (where the three axes meet) every point has a set of
spatial coordinates specified by values of x, y and z and written in the form (x, y,
z).
The space of three dimensions
A cube plainly has these three dimensions and so does everything else which
has solidity. At school we are normally taught the geometry of the plane which is
two-dimensional and we then move up to three dimensions – to ‘solid geometry’
- and stop there.
Around the beginning of the 19th century, mathematicians began to dabble in