Constructing the Nolan Chart so that each of the five divisions, "Libertarian," "Liberal," "Centrist," "Conservative," and "Authoritarian," contains an equal area poses a nice problem in geometry. The challenge really consists in how to place the diagonals for the "Centrist" division, which consists of a square, with one fifth the area of the whole Chart, rotated 45 degrees with respect to the larger square.
If the whole square of the Chart is 10 units on a side, then its area is 100 square units. Each of the five divisions will then be exactly 20 square units. A nice round number. This means that each side of the "Centrist" square must be the square root of 20. That makes the diagonal of the "Centrist" square that number times the square root of 2, or the square root of 40. But the square root of 40 is 2 times the square root of 10. So half that diagonal, or the distance from the center of the whole Chart to a vertex of the "Centrist" square, is just the square root of 10 (= 3.162). The remaining distance from the center of the whole square to the edge of the whole square is then 5 minus the square root of 10 (= 1.838).
Sometimes Nolan Charts are constructed without attention to this kind of precision, perhaps with the "Centrist" division smaller in size than the other four. However, there is no reason for that-why should there be fewer "Centrists" than other types?-and presenting an equal area chart is the most honest test: It is also a nice geometry exercise, like Plato's classic geometry construction, demonstrating Socratic Method, in the Meno, to which this problem bears some similarities.
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