By Tracy Kompelien
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Title: 2-D Shapes Are at the back of the Drapes!
Author: Kompelien, Tracy
Publisher: Abdo Group
Publication Date: 2006/09/01
Number of Pages: 24
Binding style: LIBRARY
Library of Congress: 2006012570
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Extra resources for 2-D Shapes Are Behind the Drapes!
221]. 2. AN OUTLINE OF THE AXIOMATIC APPROACH From the ordinary Euclidean plane we can derive the inversive plane by regarding a straight line as a kind of circle, namely a circle that passes through a special point called the point at infinity. This extra point, which is added to the Euclidean plane to make the inversive plane, enables us to declare, without any exception, that Any three distinct points lie on just one circle. 1) If the three given points happen to be collinear, the "circle" is a straight line.
S. M. COXETER Notice that the definition of a mid-circle belongs to inversive geometry, although we have used Euclidean methods to develop its properties. The advantage of allowing ourselves such freedom is well illustrated by the following discussion of "orthocyclic point pairs", in which we first use a strictly inversive method and then, for comparison, the Euclidean method. Two point pairs, LL' and MM', are said to be orthocyclic [1, p. 100] if there is a circle through Land L' that inverts Minto M'.
She would then write a precisely analogous problem, say call on the next student, and so on. I repeat that two 45· minute periods were devoted to this activity. What she did not do: she did not consider contrast cases, such as she did not relate this fact to other parts of mathematics (for example, by the simple observation that we are adding the number of factors); she did not use an appropriate mathematical language to write this as a generalization and she did not consider implications or extensions of this idea (for example, to consider exponents that are not positive integers).