## Four-dimensional graph

18 Jan 98

The picture (application explained as "Mixed-ethnic stain") shows a function of three variables:

r = r(c,b,h)

so four dimensions are involved.

How can a 4-dimensional graph be shown in such a way that people will understand it?

When there are only three variables:

z=z(x, y)

the x and y axes are depicted by lines at 90 degree angles to one another in a plane, and z occupies a third dimension perpendicular to the x-y plane. Finally the 3-dimensional picture can be projected to two dimensions and still be comprehended.

What is special about the present situation is that the "independent" variables c, b, and h are not completely independent of one another. Rather, since they represent ethnic proportions of a person, they are related by

c+b+h=1.

Therefore among them they represent only two degrees of freedom. Can they be represented in a plane? There is of course not room in the plane for 3 mutually perpendicular lines, so what about putting them at 60 degrees to one another?

Remarkably, this works out beautifully. As shown at the right, every point of the triangle uniquely represents a mixture of the three constituents c, b, h. By Viviani's theorem of plane geometry (not so well-known, but not hard to prove) the sum of the distances from any point to the three edges of an equilateral triangle is a constant. (I don't need to say "interior point" providing you are willing to interpret a distance as negative when the point is on the wrong side of the edge line.)

Application: "Mixed-ethnic stain"