Sunday, April 29, 2007

Moriarty's Treatise on the Binomial Theorem

Of the few writings that can be definitively attributed to Professor Moriarty (for he often published under a pseudonym) is his brilliant work, "A Treatise on the Binomial Theorem." While "The Dynamics of an Asteroid" is better known, Moriarty's work on the applications of Newton's formula is certainly greater proof of his genius.
In this treatise he extends the concept of dimensionality to negative numbers. A line segment can be measured in feet; a planar region in square feet; a volume in cubic feet. A negative-one-dimensional region is measured in per-feet, a negative-two-dimensional region in per-square-feet, and so forth. A negative-one-dimensional region is the intersection of an arbitrary point and a line which are both confined to a plane. While this is usually understood to be the empty set, it is distinguishable from (and somehow less empty than) the intersection of two arbitrary points in the plane. It is a measure of the degree of emptiness, the extent of hopelessness of ever finding something even resembling an intersection.
The binomial theorem, supplied with positive integers, gives rise to Pascal's triangle. These coefficients can be taken as the number of components of an n-dimensional simplex. For example, in three dimensions, the tetrahedron has
1 volume, 4 faces, 6 edges, and 4 vertices.
The missing 1 from the end of this list corresponds to a negative-one-dimensional object in Moriarty's scheme.
There is a thread throughout Moriarty's work of seeking out whatever is negative, queer, or monstrous in science. It is clear that this is what attracted him to this unusual mathematical subject...

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