What Is a Logarithmic Rosette?
Rosettes (Figure 1) are common features of pavements from Roman times through the Renaissance, and have been well documented by Kim Williams.[1] They usually feature a number of concentric circles and other curves whose distance from the center of the rosette continuously increases, their intersections forming a number of curvilinear triangles whose size increases with their distance from the center of the rosette. Those other curves are usually circular arcs that are not concentric with the center of the rosette. But occasionally we see a rosette in which such a curve is a logarithmic spiral, and we call it a logarithmic rosette.

What Is a Logarithmic Spiral?
Figure 4 shows a polar plot of a portion NPQ of a logarithmic spiral. A logarithmic spiral is a plane curve whose vectorial angle q is proportional to the logarithm of the length r of the radius vector OP.

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 T to the spiral at any point P and the radius vector OP to that point does not change as the spiral grows (Figure 2). Hence the logarithmic spiral is also called the equiangular spiral. It may have been this property of keeping its shape that prompted Bernoulli to call it the spira mirabilis or miraculous spiral.[2]

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°.

Identifying a Logarithmic Rosette
Since the logarithmic spiral maintains a constant angle with the radius vector anywhere on the curve, it follows that the geometric figures formed by the intersection of logarithmic spirals in a rosette will preserve the same shape anywhere in the rosette. They will be similar figures, in the geometric sense, that the angles are preserved under magnification.

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.

Construction of a Logarithmic Rosette
Our object is to construct a rosette in which all the white tiles are similar isosceles triangles, and all the black tiles are similar isosceles triangles, but not similar to the white tiles. Later we will show that we indeed get a logarithmic rosette by this procedure, even if we had never heard of the logarithmic spiral.

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.

First Band
1. Choose the radius r₁ of the inner circle (Figure 4).

2. Subdivide the circle into n equal intervals. Let w be the width of each interval.

3. Choose the height h for the triangular black tiles in the first band.

4. Cut n black tiles, all isosceles triangles of base w and height h.

5. Install the black tiles, with base vertices on the inner circle and heights in the radial direction.

6. Connect the vertices of the black tiles. This defines the shape of the white tiles in the first band.

7. Cut n white tiles and install these in the spaces between the black tiles.

Second Band

8. Cut and install n black tiles with base equal to the base of the first-band white tiles, and height chosen geometrically similar to the first-band black tiles. Similarity requires that the angles be the same as for those in the first band. A simple template can be cut to transfer the angles.

9. Connect the vertices of the second-band black tiles to give the shape of the second-band white tiles. You should find these to be geometrically similar to the white tiles in the first band. Install between the spaces between the black tiles.

10. Repeat the procedure for the subsequent bands.

Proof That this Construction Gives a Logarithmic Spiral
If r = OP (Figure 4), the equation of the spiral in polar form is

where a is a constant. Rewriting this equation in exponential [3] form gives

For equal increments of q = 1, 2, 3, . . ., our values of r are:

or,

We recognize this as a geometric progression with a common ratio of e^a. Thus, in a logarithmic spiral, the radius vectors 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

The radii r₁ and r₂ are two sides of triangle OAB. Let us denote the ratio of r₂ to r₁ by the letter k.

or,

Here k is a constant whose value depends upon our initial choices of the radii of the inner circle and the height of the first band. Now triangle OBC is similar to triangle OAB, and since corresponding dimensions of similar figures are in proportion we can write,

Also, triangle OCD is similar to triangle OAB, so,

and so forth. Our radii, for equal increments of angle, thus form the geometric progression

proving that the points A, B, C, D, . . . lie on a logarithmic spiral.

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 curvilinear triangles, we would lay out the rosette on thin metal or wood, as shown above, and connect the vertices of the triangles with smooth curves. The bases of the curvilinear triangles would be circular arcs concentric with the center of the rosette. Such a layout could then be cut apart and the pieces used as templates for the individual tiles.

NOTES
1.Kim Williams, "Spirals and the Rosette in Architectural Ornament", Nexus Network Journal, vol. 1 (1999), pp. 129-138. To order this book from Amazon.com, click here. return to text

2. H. E. Huntley, The Divine Proportion (New York: Dover, 1970), p. 168. return to text

3. Paul and Michael Calter, Technical Mathematics with Calculus (New York: Wiley 2000) p. 571. return to text

FOR FURTHER READING
Boles, Martha and Newman, Rochelle. The Golden Relationship: Art, Math & Nature. 4 Vols. Bradford MA. Pythagorean Press. To order this book from Amazon.com, click here.

Bord, Janet, Mazes and Labyrinths of the World. NY. Dutton, 1975.

Calter, Paul. Technical Mathematics. NY: Wiley, 2000. To order this book from Amazon.com, click here.

Cook, Theodore, The Curves of Life. NY. Dover, 1979. First published 1914. To order this book from Amazon.com, click here.

Cowen, Painton. Rose Windows. London: Thames and Hudson, 1979.

Demus, Otto. The Mosaic Decoration of San Marco, Venice. Chicago: U. Chicago, 1988. To order this book from Amazon.com, click here.

Edgerton, Samuel. The Heritage of Giotto's Geometry. (Cornell University Press, 1994). To order this book from Amazon.com, click here.

Edwards, Edward. Pattern and Design with Dynamic Symmetry. NY: Dover 1967. Reprint of Dynamarhythmic Design, 1932. To order this book from Amazon.com, click here.

Emmer, Michele, Ed. The Visual Mind: Art and Mathematics. Cambridge: MIT Press, 1993. To order this book from Amazon.com, click here.

Fisher, Sally. The Square Halo. NY: Abrams, 1995.

Gombrich, E. H. Art and Illusion. NY: Pantheon, 1960. To be re-released in April 2000. To pre-order this book from Amazon.com, click here.

Ghyka, Matila. The Geometry of Art and Life. NY: Dover, 1977. To order this book from Amazon.com, click here.

Hambidge, Jay. The Elements of Dynamic Symmetry. NY: Dover, 1967. To order this book from Amazon.com, click here.

Hargittai, István, and Pickover, C.A. eds. Spiral Symmetry. NY: World Scientific, 1991. To order this book from Amazon.com, click here.

Harris, Cyril. Illustrated Dictionary of Historic Architecture. NY: Dover, 1977. To order this book from Amazon.com, click here.

Hersey, George. The Lost Meaning of Classical Architecture. Cambridge: MIT 1988. To order this book from Amazon.com, click here.

Hofstadter, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid. NY: Vintage, 1979. To order this book from Amazon.com, click here.

Huntley, H.E. The Divine Proportion. NY: Dover, 1970. To order this book from Amazon.com, click here.

Janson, H.W. History of Art. Fifth Edition. NY: Abrams, 1995. To order this book from Amazon.com, click here.

Kappraff, Jay. Connections: The Geometric Bridge between Art and Science. NY: McGraw, 1990.

Kemp, Martin, The Science of Art. New Haven: Yale U. Press, 1990. To order this book from Amazon.com, click here.

Kepes, Gyorgy, ed. Structure in Art and Science. NY: Braziller, 1965.

Lawlor, Robert. Sacred Geometry. NY: Thames & Hudson, 1982. To order this book from Amazon.com, click here.

Linn, Charles. The Golden Mean. NY: Doubleday, 1974.

Pedoe, Dan. Geometry and the Visual Arts. NY: Dover, 1976. To order this book from Amazon.com, click here.

Richter, Irma. Rhythmic Forms in Art. London: John Lane, 1932.

Shlain, Leonard. Art and Physics: Parallel Visions in Space, Time, and Light. Morrow, 1991. To order this book from Amazon.com, click here.

Thompson, Darcy. On Growth and Form. NY: Dover, 1992. First published 1942. To order this book from Amazon.com, click here.

Williams, Kim, Ed. Nexus: Architecture and Mathematics. Fucecchio: Edizioni dell=Erba, 1996, To order this book from Amazon.com, click here.

Williams, Kim, Ed. Nexus II: Architecture and Mathematics. Fucecchio: Edizioni dell=Erba, 1998, To order this book from Amazon.com, click here.

Williams, Kim. Italian Pavements, Patterns in Space. Houson: Anchorage, 1997 To order this book from Amazon.com, click here.

RELATED SITES ON THE WWW

LOGARITHM IC SPIRALS
Mattew Brand's Logarithmic spirals
The Geometry Junkyard: Spirals
Yah Lee's Equiangular Spiral
Wolfram Research's Equiangular Spiral
The MacTutor History of Mathematics archive: Equiangular Spiral

APPLICATIONS OF SPIRALS
Smith College's Phyllotaxis Tutorial
Spiral Images
A Middle-School Spiral Project
Mathematics and the Liberal Arts: Spirals
Mathematics and Art: Spirals and Hypercones

ABOUT THE AUTHOR
Paul A. Calter is a Visiting Scholar at Dartmouth and Professor Emeritus of Mathematics at Vermont Technical College. He has interests in both the fields of mathematics and art. He received his B.S. from Cooper Union and
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.

Copyright ©2000 Kim Williams