An Electronic Journal of
Geography and Mathematics.
(Major articles are refereed;
full electronic archives available)

Sandra Lach Arlinghaus

     Often the introduction of added perspective from another dimension sheds light on pattern.  So it is with the following theorem from Projective Geometry which becomes easy to visualize in three dimensions.  Google Earth will assist in creating graphics for such visualization.

DESARGUES'S THEOREM [Coxeter, 1961]:  If two triangles, A, B, C, and A', B', C' are perspective from point O (AA', BB', and CC' are concurrent), then the intersections of corresponding triangle sides, L=AB'.A'B, M=AC'.A'C, and N=BC'.B'C, are collinear.

Figure 1.  Desargues's Two-Triangle Theorem visualized in the plane.

Figure 2.  Desargues's Two-Triangle "Tower" centered on the Sidney Smith Building, home of the Mathematics Department of the University of Toronto and the academic home of Professor H. S. M. Coxeter for many years.

To visualize the theorem in the plane while flying through the .kmz file, tilt the configuration so that it appears to flatten out in the plane.  When triangle sides are "parallel" in the Euclidean sense the theorem still holds.  The lines forming the sides intersect at points at infinity, which, in the projective plane, are no different from any others.  The proof of this theorem is often presented in three, rather than in two, dimensions.


Congratulations to all Solstice contributors.


Solstice:  An Electronic Journal of Geography and Mathematics
Volume XVIII, Number 2
Institute of Mathematical Geography (IMaGe).
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