SOLSTICE: An Electronic Journal of Geography and Mathematics. (Major articles are refereed; full electronic archives available) THE ANIMATED PASCAL
Sandra Lach Arlinghaus

There are a number of elegant theorems about conics in the projective plane.  The animations below illustrate a few associated constructions.

• Figure 1 shows an animated illustration of Pascal's Theorem:
PASCAL'S THEOREM [Coxeter, 1961]:  If A, B, C, A', B', C' are six vertices of a hexagon inscribed in a conic, then the points of intersection of opposite sides of the hexagon, L, M, and N, are collinear (lying along line z). Figure 1.  Pascal's Theorem.

• Figure 2 shows an animated illustration of the dual of Pascal's Theorem:
BRIANCHON'S THEOREM [Coxeter, 1961]:  If a, b, c, a', b', c' are six sides of a hexagon circumscribed about a conic, then the lines joining opposite vertices (the diagonals) of the hexagon, l, m, and n, are concurrent (at the point Z). Figure 2.  Brianchon's Theorem:  dual of Pascal's Theorem.

• Figure 3 shows an animated illustration of the converse of Pascal's Theorem used to create a specific, important construction:
BRAIKENRIDGE-MACLAURIN CONSTRUCTION [Coxeter, 1961]:  Construction of a conic through 5 given points based on: Fiugre 3a.  This construction is the converse of Pascal's Theorem.  Choose line z through L, leading to point C' on the conic through the 5 given points, A, B, C, A', B'. Figure 3b.  This construction is independent of the choice of z.  Choose a different line line z through N.  It, too, will produce a point C' in a location different from that of Figure 3a but the new C' will also lie on the conic through the 5 given points!

 Part of the motivation for creating these animations lies in offering helpful graphics to illuminate notation.  Another part of it is to build a foundation on which to continue ongoing research linking non-Euclidean geometries and the  compression of (geo)graphics [Arlinghaus and Batty, 2005] -- within the digital world but perhaps not beyond the "limits" of the Escher series! [Escher, Circle Limit series; kmz file link). References:

• Solstice was a Pirelli INTERNETional Award Semi-Finalist, 2001 (top 80 out of over 1000 entries worldwide)
• One article in Solstice was a Pirelli INTERNETional Award Semi-Finalist, 2003 (Spatial Synthesis Sampler).
• Solstice is listed in the Directory of Open Access Journals maintained by the University of Lund where it is maintained as a "searchable" journal.
• Solstice is listed on the journals section of the website of the American Mathematical Society, http://www.ams.org/
• Solstice is listed in Geoscience e-Journals
• Solstice is listed in the EBSCO database.
• IMaGe is listed on the website of the Numerical Cartography Lab of The Ohio State University:  http://ncl.sbs.ohio-state.edu/4_homes.html
Congratulations to all Solstice contributors. Solstice:  An Electronic Journal of Geography and Mathematics,  Volume XVIII, Number 2 Institute of Mathematical Geography (IMaGe). All rights reserved worldwide, by IMaGe and by the authors. Please contact an appropriate party concerning citation of this article: sarhaus@umich.edu http://www.imagenet.org
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