Conor Lyons's Blog

By reinterpreting these number pairs as points in the plane…

By reinterpreting these number pairs as points in the plane and connecting them with lines we are drawing a transformation of the original square.
The rule for changing the numbers (x y) into new numbers () can be virtually anything, as simple or as complicated as we like.
Geometry disciplines our approach by analysing the variety of possible transformations and starting with the simplest and moving on to take each new complication in turn. It is in this way that we are able to develop an appreciation of spatial order.
Figure 1 shows several of the most elementary transformations together with the rules that generate them.
The formula, , for example, generates a translation : all points in the square are moved to new locations three units to the right.
A translation is a transformation that alters only the location of the square (or whatever) without altering its shape, its size or its orientation.
Transformations that leave shape and size unaltered are obviously the most basic, and geometers call them isometries . As well as translation,rotation , and mirror reflection are isometries.
Both of these alter orientation; the latter is also sense-altering and turning a left hand into a right hand.
Obviously, one transformation followed by another together constitute a transformation, and following one isometry by another isometry produces an isometry.
However, following one reflection by another reflection constitutes a rotation about the intersection of the mirrors, unless the mirrors happen to be parallel, in which case the product is a translation (Figure 2).
Even in this case we can consider the translation to be a special case of rotation about a centre infinitely far away.
Another class of transformations contains the similarities and transformations that alter size, but not shape.
If we feed the following rule into our computer, and then the consequent transformation makes the polygon, or whatever and three times bigger without altering any of its relative proportions. Like congruence the concept of similarity is as old as Euclid.
Yet it is currently playing a vital role in contemporary geometry, as in the work on self-similar fractals by Benoit Mandelbrot, at IBM’s research centre at Yorktown Heights in New York (Figure 3a).
More simply and the similarity transformations can be seen to describe the growth and form of mollusc shells (Figure 3b).
Then there are the affrine transformations, which destroy shape and size but maintain straightness and proportions in any line, as well as parallels; and projective transformations which destroy shape and size, proportions and parallels, maintaining only straightness, as in perspective.
But it is with inversion that we find the first really unfamiliar transformation.
To describe its rule most clearly it is best to use polar coordinates to locate points in the plane.
Instead of defining position in terms of x-across and y-up, we describe each point by its distance r and direction q from the origin point, O (Figure 4).
q is in fact the angle between a line from O to the point and the horizontal axis to the right of O (the “3 o’clock” direction).
Once again, each point is specified by an ordered number pair (r and q), and any manipulation of these numbers corresponds to a geometric transformation.
Inversion is the transformation that takes (r and q) to according to the rule: . What does this mean?
We have selected one point, O, in the plane as our reference origin, from which the distance and direction (r and q) of every other point is measured.
The rule tells us, or the computer, where to move every point in the plane by computing the new location .
It is a “one-to-one” rule, providing a unique image (new location) for each point.

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