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More on the Desargues Configuration

One of the remarkable properties of the Desargues configuration is its symmetry. Recall that Desargues Theorem states that if two triangles are in perspective from a point j, the center of perspectivity, then the three points of intersection of the corresponding sides of the triangles are collinear. This is illustrated in the configuration on the left below. What is surprising is that any point of the configuration can be chosen as the center of perspectivity, and there will be triangles in the configuration in perspective from that point illustrating Desargues Theorem! For example, if the original point d is chosen as the center j, the relabelling at the right shows another instance of Desargues Theorem.

One way to see why this is so uses yet another interesting property of the Desargues configuration. In the figure to the left below, the points are labeled with pairs of digits from 1 to 5 such that the points on each line are the three pairs in a triplet of digits from 1 to 5. For example, the points labeled with pairs from the triplet 123, namely 12, 13, 23, are on a line. Since this is true for any of the ten triplets, permuting the digits produces another labeling with the same properties.

The triangles in perspective from 12 are 13, 14, 15 and 23, 24, 25, and the collinear points of intersection of corresponding sides are 34, 35, 45. Swapping the digits 1 and 3 produces the labeling on the right, the same configuration illustrating Desargues Theorem but with a different center of perspectivity. With a suitable permutation any of the points can be labeled 12, the center of perspectivity.

Coronavirus Update

The latest estimate I have for the Basic Reproduction Number R0 and a projection for the number of cases are:

Based on this estimate, we are very close the inflection point where the R0 falls below 1, and less than a month away from where the total cases curve flattens.

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