Now imagine the whole diagram being rotated out…

June 24, 2010

Now imagine the whole diagram being rotated out of the page about an axis that connects the centres of the two circles. The line will sweep through a plane, and the circles will sweep through spheres.
So in three dimensions inversion turns an infinite plane into a finite sphere in a one-to-one mapping of every point. This is where computer graphics comes to the geometer’s aid.
Drawing three-dimensional objects is quite difficult, but for computers the task is almost as easy as drawing a two dimensional object.
The computer simply projects the object onto a plane, in an act of transformational geometry called perspective.
Figure 8 shows a computer’s rendering of a plane and its image (a sphere) after inversion.
Each star on the plane corresponds to its inverted image on the sphere; you can see how the angles are preserved under the transformation.
There are various ways of mapping a plane on to a sphere (or vice versa) but there is no way of doing this without producing some kind of distortion or another; Mercator’s projection of the globe on to a flat map, leads to a diminutive Africa and an exaggerated Greenland.
Inversion has the advantage of preserving circles and angles, and hence proportion locally if not globally.
The upper pole of the sphere is the centre of the inversion and the increasingly small stars that approach it correspond to the increasingly large stars lying at greater and greater distances on the plane.
To produce this picture a computer program included a specification for both the position of the eye (the view-point) and the plane on to which the three-dimensional configuration is projected.
It also had to solve the problem of “hidden lines”: how do you tell the computer not to draw those lines that the eye would not be able to see?
Computer graphics can produce an image in a variety of ways, on a video screen, for example, or directly on to film.
For Figure 8, I used a plotter, a mechanical drawing board whose moving pen is controlled by the electronic signals coming from the computer.
Watching it in action is like seeing some fantastic automatic sewing machine, dropping and lifting its pen to draw and terminate the thousands of lines with a speed that belies its great accuracy.
Having created a plane pattern of tessellating hexagons and hexagrams (stars) and having programmed the computer to invert it, I watched the pen race round the board and wondered, would it draw anything sensible? Would it draw a sphere?
I knew it must in theory, but to see it happen was still a lovely surprise.
From its beginnings more than a century ago, modern mathematics has aimed at a total abstraction in the interests of rigour and distillation of its logic.
One consequence of this has been a distrust of geometric intuition and the downgrading of geometry by educationalists over the past 50 years or so.
This demise of the oldest discipline in the world is clearly absurd and ripe for serious reconsideration.
To begin with, geometry has more value than simply as a means of acquiring a grasp of mathematical concepts.
The applications of a disciplined spatial intuition to art and design, and to the study of natural morphologies in every conceivable science is so great that perhaps we might think of geometry as a semi autonomous department of mathematics with different as well as overlapping purposes to abstract mathematics.

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