discussion focuses on the ellipsoid because of its physical relevance to diffusion and because of its historical significance in the development of diffusion anisotropy imaging in MRI. The following matrix displays the components of the diffusion tensor: The same matrix of numbers can have a simultaneous second use to describe the shape and orientation of an ellipse and the same matrix of numbers can be used simultaneously in a third way for matrix mathematics to sort out eigenvectors and eigenvalues as explained below. Physical tensors The idea of a tensor in physical science evolved from attempts to describe the quantity of physical properties. The first properties they were applied to were those that can be described by a single number, such as temperature. Properties that can be described this way are called scalars; these can be considered tensors of rank 0, or 0th-order tensors. Tensors can also be used to describe quantities that have directionality, such as mechanical force. These quantities require specification of both magnitude and direction, and are often represented with a vector. A three-dimensional vector can be described with three components: its projection on the x, y, and z axes. Vectors of this sort can be considered tensors of rank 1, or 1st-order tensors. A tensor is often a physical or biophysical property that determines the relationship between two vectors. When a force is applied to an object, movement can result. If the movement is in a single direction, the transformation can be described using a vector—a tensor of rank 1. However, in a tissue, diffusion leads to movement of water molecules along trajectories that proceed along multiple directions over time, leading to a complex projection onto the Cartesian axes. This pattern is reproducible if the same conditions and forces are applied to the same tissue in the same way. If there is an internal anisotropic organization of the tissue that constrains diffusion, then this fact will be reflected in the pattern of diffusion. The relationship between the properties of driving force that generate diffusion of the water molecules and the resulting pattern of their movement in the tissue can be described by a tensor. The collection of molecular displacements of this physical property can be described with nine components—each one associated with a pair of axes xx, yy, zz, xy, yx, xz, zx, yz, zy. These can be written as a matrix similar to the one