By Simon Donaldson
If we have four objects—say A, B, C, D—there are just three ways of dividing them into pairs:
(AB)(CD) (AC)(BD) (AD)(BC).
The salient property is that 3 is less than 4. This simple fact expresses something special about the number 4. For example if we take 6 objects there are 10 ways to divide them into triples; there are 35 ways to divide 8 objects into quadruples, 126 ways to divide 10 objects into quintuples and so on. We will discuss two famous applications of this special property of 4: one going back five centuries and one underlying important concepts in contemporary differential geometry and physics.
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Intersection of conics. Image Simon Donaldson
![Grassmann](https://scgp.stonybrook.edu/wp-content/uploads/2016/10/Grassmann-241x300.jpg)
Jim Simons speaks at the Dedication of the Iconic Wall, standing before Maxwell’s Equations. May 8, 2015. Photo Stony Brook University