Paul CalterRR1, Box 425Randolph Center, VT 05061 USA
This spiral was discovered by the French mathematician René
Descartes in 1638. It is seen in nature in the chambered Nautilus,
animal horns, and certain plants. In the 19th century the mathematician
Bernoulli discovered that the angle f between the tangent It is helpful to compare the logarithmic spiral to the circle. The circle intersects its own radii everywhere at the same angle of 90°, while the logarithmic spiral intersects its own radii everywhere at the same angle, which may differ from 90°.
Figure 2 shows a rosette from the Baptistery of S. Giovanni in Florence; Figure 3 shows another rosette from Pompeii. In the S. Giovanni rosette, notice that the shapes of the curvilinear triangles change as we move from the center outwards, but that the shapes in the Pompeii rosette do not. That's a sure sign that the spirals in the Pompeii rosette are logarithmic.
We have to choose the values of three dimensions: the radius of the inner circle; the number of its subdivisions; the height of the first row of triangles. After that all other dimensions follow automatically.
aq
= ln rwhere r = e^{a}qFor equal increments of q e^{a}, e^{2a}, e^{3a}
. . .
e^{a}, (e^{a})^{2},
(e^{a})^{3} . . .We recognize this as a geometric progression with a common
ratio of r, for equal increments of the polar
angle q,
form a geometric progression. Conversely, if the radius vectors
to a curve form a geometric progression for equal increments
of the polar angle, the curve is a logarithmic spiral.Returning to our construction in Figure 5, let .OA = r, _{1}OB = r,
_{2}OC = r, etc_{3}The radii r are
two sides of triangle _{2}OAB. Let us denote the ratio of
r to _{2}r by the letter k._{1}r= _{2} / r_{1} kor, r_{2} = k r_{1}Here r_{3} / r_{2} = kr_{3} = kr_{2} = k(kr_{1})
= k^{2}r_{1}Also, triangle r_{4} = kr_{3} = k(k^{2}r_{1})
= k^{3}r_{1}and so forth. Our radii, for equal increments of angle, thus form the geometric progression r_{1} , kr_{1} , k^{2}r_{1},
k^{3}r_{1}, ...proving that the points The straight-line segments connecting these points do not,
of course, lie on the spiral, but give its approximate location.
If we want a logarithmic rosette with
2. H. E. Huntley,
3. Paul and Michael
Calter,
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his M.S. from Columbia University, both in engineering, and his Masters of Fine Arts Degree at Vermont College of Norwich University. Calter has taught mathematics for over twenty-five years and is the author of ten mathematics textbooks and a mystery novel. He has been an active painter and sculptor since 1968, has participated in dozens of art shows, and has permanent outdoor sculptures at a number of locations in Vermont. Calter developed the MATC course "Geometry in Art & Architecture" and has taught it at Dartmouth and Vermont Technical College, as well as giving workshops and lectures on the subject. He is presenting a paper on the survey of a doorway by Michelangelo in the Laurentian Library in Florence at the Nexus 2000 conference on architecture and mathematics.
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