SCULPTURES BY ORTHOGONAL INTERSECTION IN THREE SPATIAL DIMENSIONS
THE DISCOVERY OF THE THIRD WAY
I recall with great pleasure a discussion with my mathematician friend Beppe Pea over a pizza one evening. At that time I had created my Columns of Thought and I was tormented by the idea that I had always cut my cubes and parallelepipeds in two spatial dimensions only. The third was absolute taboo. Untouchable. What excited me was the idea - for instance - of cutting a wooden cube using the so-called method of the columns of thought, but from all three dimensions which, until proved otherwise (leaving aside Einstein's extraordinary space-time reality) are X, Y and Z, so that a sphere can be seen within.
We had a lengthy discussion about the feasibility of these three successive cuts and I clearly remember the conclusion we reached - the third cut would undoubtedly destroy the work already done.
Shortly afterwards I managed to make a column of thought cut in three dimensions but I reached the conclusion (provisional in the light of subsequent developments) that it was feasible provided that the drawing on the three faces of the cube or parallelepiped was not a closed figure, and that anyway it always touched the four corners of each face.
In effect, if you look at the column of thought I am talking about from all three dimensions, you always 'see' an X shape.
It took me nearly two years to realise that a block could be cut in all three dimensions.