Foundations & Tilings in Perspective
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### Lecture Description

Protective geometry is more fundamental than euclidean geometry, with applications in physics, biology and perspective art. We shall introduce it visually, without equations.

This video begins by considering a hexagonal tiled hive from a bee's perspective. We then formally define projective geometry, discuss previous approach the study, like perspective art and geometry with only a straight-edge.

We discuss the famous hexagon theorem of Pappus. A foundation result of projective geometry. We also discuss the work of Descargues, and his brilliant idea about treating projective geometry as the study of constructions which can only be made using a straight-edge.

To contrast projective geometry with Euclidean geometry (which considers constructions using a straight edger `and' a compass), we compare how a hexagon is constructed in each geometry. In the projective case, we form the hexagon starting from three lines which pass through the vanishing line.

In the resulting projective construction of the hexagon we see many interesting qualities which are invariant of how the initial points are chosen. We point out how Pappus's theorem and Desgaurges' theorem can be seen working within this hexagon construction, and show how this construction can be extended to give a beautiful projective tiling of the plane with hexagons.

The Geogebra program can be found here:
www.geogebra.org/

I would like to thank Olive Whicher for his illuminating book on projective geometry, and Norman Wildberger for his insightful teachings on the subject.

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