(A) The Ι-beam is a structural element that is commonly used in civil engineering. If the beam is clamped in a horizontal position at the end furthest from view (grey rectangle), and a transverse force is applied at the other end (in this case upwards), then the beam's resistance is defined by the scalar transverse stiffness, EIxx, which is the product of the Young's modulus, E, and the area moment of inertia, Ixx. The area moment of inertia can be computed by dividing the cross-section into area elements, dA, and multiplying each area by the square of their distance, y, from the neutral axis (dotted line). A sum or integration of this quantity can then be used to compute the area moment of inertia, Ixx. The neutral axis is perpendicular to the applied force and passes through the centre-of-mass of the beam's cross-section. Intuitively, an increased stiffness can be achieved by placing the beam's material as far from the neutral axis as possible. This property is the rationale behind the design of the I-beam, which typically bears loads in the direction given by Ixx. (B) A similar calculation to A can be used to calculate the beam's stiffness in an orthogonal direction. It is usually the case that Ixx > Iyy for Ι-beam designs. (C) The generalised response of a beam to forces in an arbitrary direction, denoted by the unit vector n, can be represented by the area moment of inertia tensor, J, with entries Ixx, Iyy, Ixy. The tensor, J, is a matrix quantity that relates the direction of the applied force to the beam's deflection (Landau et al., 1986). For a general tensor matrix, there exist two orthogonal vectors (known as eigenvectors) that represent the directions of the beam's maximal and minimal stiffness. (D) The eigenvectors or principal axes of the Ι-beam point along the x and y-axes. (E) Under purely compressive forces a prismatic beam with length, L, will buckle in the direction of the most compliant principal axis, which defines the critical force, Fc, for beams of this type. (F) Beams with a mechanically isotropic transverse organisation, such as microtubules, have a scalar stiffness tensor (Imin = Imax) and degenerate principal axes. The ratio Imax/Imin ≥ 1 can be used to quantify the beam's degree of anisotropy. The ratio is one for mechanically isotropic structures such as a single microtubule or a bundle of 4 microtubules arranged in a 2 × 2 square motif.