For a linear load, these tracks constitute two logarithmic spirals, as shown in
where [r.sub.p0] is the distance between the intersection point of the two logarithmic spirals and pole O and [[lambda].sub.f] is the angle between the tangent at any point along the shear crack's track and the maximum principle stress at that point.
Handzic started with Archimedean spirals and logarithmic spirals. The Archimedean spirals didn't work well because they didn't generate a constant backward force.
In a logarithmic spiral, this angle, the spiral's "camming angle," never varies.
Almon highlights the beauty of the natural forms and processes Thompson observed: "the growth / of the nautilus shell with logarithmic spirals
" and "why a moth flies to the lamp in a decaying orbit / rather than a straight line." Using a similar circling technique, Almon begins and ends the poem with a mother who "worked for the Ideal Laundry" and"feared those instruments of order: / the wringer, the mangle, the trouser-press".
Phyllotaxis also shows that not only do leaves grow around a stem using this angle, and not only do the florets on the head of a sunflower grow accordingly, but they also grow in interleaving logarithmic spirals which number according to Fibonacci pairs.
Furthermore, if we were to draw quarter circles from the diagonal corners of each of these squares and connect them, as shown in Figure 2, we would achieve a "logarithmic spiral." (21) Due to its wonderful property of never changing its shape when it increases in size this spiral was dubbed the spira mirabilis (wonderful spiral) by the seventeenth-century Swiss mathematician Jacques Bernoulli.
There has been a lot of material written about logarithmic spirals of golden proportion but I have never come across an article which states the exact equation of the spiral which ultimately spirals tangentially to the sides of the rectangles as in Figure 1.
In articles associated with such diagrams, it is often stated, that the curve thus obtained approximates a logarithmic spiral of golden proportion.
I started out with logarithmic spirals
that vary in intervals as they move outward.
Thus, the processes that lead to the formation of logarithmic spirals
in seashells and animal horns seem to operate through the intricate relationships between the Fibonacci sequence and the golden section.
The logarithmic spiral
is also known as the growth spiral or the equiangular spiral.