visual1 Visual Explanations in Mathematics
1. What are visual explanations?
Perhaps the most famous and certainly one of the oldest visual explanations in mathematics is this visual proof of the Pythagorean theorem.
The theorem states that the yellow square (``the square on the hypothenuse'') is equal to the green square plus the blue square; this should become clear upon contemplation of the two compound squares on the right.
This proof is unusual in its brevity and its complete appropriateness to the problem. Pictures and diagrams are also used in non-geometrical parts of mathematics, mostly for psychological reasons: harnessing our ability to reason ``visually'' with the elements of a diagram in order to assist our more purely logical or analytical thought processes.
Some diagrams are more useful than others. Edward R. Tufte has written three books analyzing the effectiveness of figures and diagrams in communicating ideas. His latest,Visual Explanations -Images and Quantities, Evidence and Narrative (Graphics Press, Cheshire CT 1997) was reviewed in the January 1999 AMS Notices by Bill Casselman of the University of British Columbia. Casselman focusses on the applications of Tufte's ideas to explanations in mathematics, and distills from the book a set of rules designed to make graphics contribute most effectively to the communication of mathematics.
Tufte's Rules, after Casselman, abridged:
- Tone down secondary elements of a picture: layer the figure to produce a visual hierarchy.
- Replace coded labels in the figure by direct ones.
- Produce emphasis by using the smallest possible effective distinctions.
- Eliminate all unnecessary parts of a figure.
- Use small multiples: numerous repetitions of a single figure with slight variations.
- Make the graphics carry a story.
Casselman gives a sample application of these rules to a visual proof of the incommensurability of side and diagonal in a regular pentagon. This proposition is of additional interest because the ratio of these two lengths is exactly the Golden Mean. In this column we will follow an adaptation of Casselman's argument to the web, using Casselman's own figures (thanks, Bill!)
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