Next, based on equation of state for the stress on any microelement in an object under external load and (14) and (15), the shear stress, [[tau].sub.p], and the normal stress, [[sigma].sub.p], at any point on the logarithmic spiral are

To simplify the stress analysis, the shear fracture surface of the rock during cutting is treated as a plane whose starting and end points are where the profile of the impact fracture pit (a logarithmic spiral) intersects the cut and uncut surfaces of the lunar rock simulant.

Assume uniformly distributed force, q, in logarithmic spiral AB satisfies the following condition:

The device used four spring-loaded cams, each shaped like a logarithmic spiral. When placed into a vertical crack in the rock, the cams rest unless the hiker misses a step or slips, causing his or her weight to pull the cams down the crack.

The logarithmic spiral (far left) maintains a constant angle, the camming angle, between its tangent and its radial line.

Phi is also expressed in the logarithmic spiral and is the source of the Golden Angle, as well as many other mathematical forms too numerous to mention.

(26) Interestingly, the shell of the Murex trunculus, the snail from which tekhelet dye is made, appears to follow the phi-based logarithmic spiral.

The general equation of a logarithmic spiral, attributed to Descartes, is given in polar form by

In order to find c, use is made of a diagram in which part of a logarithmic spiral is enclosed by a rectangle of golden proportion; the sides of which are tangential to the spiral (see Figure 2).

181), one can construct a

logarithmic spiral (a spiral in which the logarithm of the radial distance from the center increases in proportion to the total angle traversed along the spiral).

The logarithmic spiral can be defined as a curve that exhibits a constant angle between the radius vector (a line from the centre to a point on the curve) and the tangent vector (a line oriented along the path of travel).

The logarithmic spiral has many special properties that make it very useful in both nature and engineering: it is self-similar, in that its shape remains unaltered by scaling and angular growth; the distance between arms increases in a geometric progression; (15) any straight line passing through the origin makes a constant angle with the curve (Figure 1); a degenerate logarithmic spiral is a straight line at one extreme and a circle at the other; it can be produced using incredibly simple rules, such as 'move forward a bit, turn left 30 degrees'.