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In three dimensions, its a little more complicated, because there are different kinds of anisotropy. If the material isn't isotropic (on the left), the ink blot could take on this cigar shape (center top) or a pancake shape (center bottom), depending on whether you have, as its called, "linear" or "planar" anisotropy, respectively. These three ellipsoids span the variety of diffusion patterns possible in three dimensions. Mathematically, you can represent the tensor with a 3x3 symmetric semi-positive definite matrix. The connection between the matrix and the ellipsoid is that the ellipsoid is the image of the unit sphere under the linear transform of the matrix. Also, the eigenvectors of the ellipsoid are the axes of symmetry of the ellipsoid, and the eigenvalues are the scalings along those axes.