Fermat's Natural Spirals

A typical daisy is a star-like flower. It features a fringe of white or colored petals and a central disk of tubular florets. Each floret is itself a tiny flower.

This daisy has 21 white petals and a yellow central disk of tubular florets. Photo by Kenneth Peterson.

The tightly packed florets at a daisy's center have an intriguing arrangement. The florets get larger at greater distances from the center. And there are hints of clockwise and anticlockwise spirals in the pattern.

A close-up view of a daisy's disk of florets reveals intriguing spiral patterns. Photo by Kenneth Peterson.

One way to model such a pattern is to start with a curve called Fermat's spiral. This curve is also known as a parabolic spiral. It's given by the polar equation

r = k a^1/2

where r is the distance from the origin, k is a constant that determines how tightly wound the spiral is, and a is the polar angle.

This type of spiral has the property of enclosing equal areas with every turn.

Fermat's spiral.

By placing points (disks or polygons) centered at regular angular intervals along such a spiral, you can create a variety of intriguing patterns—depending on the angle you choose to use. Using the angle 222.49 degrees (a value related to the golden ratio, 1.618034. . . ), you get a pattern with an even packing of polygons (or disks). It closely resembles a daisy's florets.

By placing points at regular angular intervals along Fermat's spiral, you get a pattern resembling that of a daisy's central florets. Courtesy of Robert Krawczyk.

By choosing other angles, you get intriguing variants. Each choice gives a different pattern of secondary spirals, some winding clockwise and others anticlockwise, which form an interlocking system. Robert Dixon explores some of these possibilities in his book Mathographics.

Using larger numbers of points and smaller angles produces patterns with a variety of secondary spirals and, often, with radial lines that become evident toward the edges. Michael Naylor has investigated a variety of such patterns (see "Golden Blossoms, Pi Flowers").

Robert J. Krawczyk has taken the generation of these patterns another step further, creating striking images of eerie ripple patterns. He calls these circular designs "Fermat's spiral mandalas."

Krawczyk starts by combining several spirals to create one complex pattern.

Four different individual spirals (top) can be combined in various ways to create new, complex patterns (bottom). Courtesy of Robert Krawczyk.

By placing points at fixed angular intervals along these curves, he gets very elaborate patterns that show a variety of features.

An example of a Fermat's spiral mandala. Courtesy of Robert Krawczyk.

To finish his images, Krawczyk enhances the texture and gives them a coppery glow. Examples of such mandalas can be seen at his website.

Courtesy of Robert Krawczyk.

A mandala's complex circular design, with its symmetrical and radial balance, is intended to draw the eye to its center, Krawczyk says. Fermat's spiral, in particular, "is a natural basis for this inward draw."

For those who prefer a more natural look, all it takes is a close view of a daisy's central disk.

Photo by Kenneth Peterson.

Originally posted September 5, 2005.

References:

Dixon, R. 1991. Mathographics. Dover.

Krawczyk, R.J. 2005. Fermat's spiral mandalas.

Naylor, M. 2002. Golden, √2, and π flowers: A spiral story. Mathematics Magazine 75(June):163-172.

Lighting Up a Polyhedron with a Hole

For most people, the word "polyhedron" conjures up an image of a cube, a tetrahedron, or something similar—a solid figure with flat faces. If the polyhedron is regular, each face has the same size and shape.

But polyhedra come in all sorts of guises. The faces of a polyhedron can have different sizes and shapes, just as long as each one is a polygon. The polyhedron itself can have a hole (or two or more).

One particularly intriguing polyhedron was discovered in 1977 by Hungarian mathematician Lajos Szilassi. This polyhedron has seven faces, 14 vertexes, 21 edges, and a hole. Topologically, if it were smoothed out, it would be equivalent to a doughnut (or torus). You could describe as a toroidal heptahedron. Each face is a six-sided polygon.

The Szilassi polyhedron forms the basis for this lamp, created by Hans Schepker.

Like the tetrahedron, the Szilassi polyhedron has the remarkable property that each of its faces touches all the other faces. Templates are available for anyone who would like to make a model of the Szilassi polyhedron.

The Szilassi polyhedron also provides insight into the problem of coloring maps. On a flat surface (or the surface of a sphere), a map must be colored with four colors so that no two adjacent regions are the same color. For a map on the surface of a torus, the number is seven. So, each face of the Szilassi polyhedron's seven faces must be a different color to ensure that no two adjacent faces have the same color.

Another view of the Szilassi polyhedron.

By replacing each face of the Szilassi polyhedron with a vertex and each vertex with a triangular face, you end up with another unusual polyhedron known as the Császár polyhedron. It's the only known polyhedron, aside from the tetrahedron, that has no diagonals. It also has a hole, making it topologically equivalent to a torus. This polyhedron was discovered in 1949 by Ákos Császár.

Hans Schepker of Harrisville, N.H., has been crafting "mathematically correct" sculpture out of paper, glass, and other materials for many years. So, when he learned about the Szilassi polyhedron, he had to make one. In March 2006, he presented his first glass model to Szilassi himself, who was attending the 7th Gathering for Gardner in Atlanta.

Schepker also installs lights inside his glass creations. Another version of the Szilassi polyhedron, featuring four lights, was the centerpiece of Schepker's display at the 2007 Joint Mathematics Meetings, held in New Orleans.

Hans Schepker can turn all sorts of polyhedra into colorful lamps.

Originally posted Jan. 22, 2007.

References:

Gardner, M. 1992. Minimal sculpture. In Fractal Music, Hypercards and More . . .: Mathematical Recreations from Scientific American Magazine. W.H. Freeman.

______. 1988. The Császár polyhedron. In Time Travel and Other Mathematical Bewilderments. W.H. Freeman.

Photos by I. Peterson

Mathematical Knitting Network

More than 5,000 mathematicians come annually to the Joint Mathematics Meetings (JMM), held in January. In recent years, amidst the usual lectures, poster sessions, job interviews, and much else, these gatherings have also featured an evening event for those particularly interested in mathematical crafting.

Devoted to knitting, crocheting, beading, needlework, paper folding, and more, this informal session is organized by sarah-marie belcastro (Smith College) and Carolyn A. Yackel (Mercer University).

belcastro and Yackel edited and contributed to the book Making Mathematics with Needlework: Ten Papers and Ten Projects (A K Peters, 2007), which contains not only instructions for creating mathematical objects but also insights into the underlying mathematics.

The JMM knitting network event brings together a wide variety of people, both experts and beginners. Participants and projects can vary widely from year to year.

In the realm of counted cross stitch, Mary Day Shepherd (Northwest Missouri State University) creates painstakingly woven symmetry patterns. For the type of cloth and technique that she uses, the fabric is a grid of squares, and one cross stitch covers one square of the fabric. The only possible subdivision of this square is with a stitch that "covers" half a square on the diagonal, Shepherd says.

These features constrain the number of symmetry patterns that you can weave. Of the 17 possible wallpaper patterns, for example, only 12 can be done in counted cross stitch.

The 12 wallpaper patterns that can be done in counted cross stitch needlework.

Courtesy of Mary D. Shepherd.

Shepherd has also worked on frieze and rosette symmetry patterns. Rosette patterns, for example, give a nice visualization of the symmetries of a square (technically, the group D4 and all its subgroups), she says. See Shepherd's article "Groups, Symmetry and Other Explorations with Cross Stitch" (Word document) and her chapter "Symmetry Patterns in Cross Stitch" in the book Making Mathematics with Needlework.

Rosette patterns for visualizing the symmetries of a square (the dihedral group of the square).

Courtesy of Mary D. Shepherd.

David Jacob "Jake" Wildstrom (University of Louisville) has a passion for crocheting in relief.

One of the few fractals that's amenable to crochet is the Sierpinski triangle. Wildstrom has turned this remarkable geometric figure into blankets, wispy shawls, and even a hat. His instructions for crocheting such figures can be found in the chapter "The Sierpinski Variations: Self-Similar Crochet" in Making Mathematics with Needlework. More information is available at Wildstrom's "crochetgeek" website.

Jake Wildstrom's relief crocheting has turned a fractal known as the Sierpinksi triangle into a shawl.

Photo by I. Peterson.

Tom Hull (Western New England University) specializes in mathematical origami design.

One of Tom Hull's modular origami creations.

Photo by I. Peterson.

Laura M. Shea strings tiny crystal beads to form polyhedra or geometric tilings.

This beadwork bracelet, created by Laura Shea, is based on a triangular tiling.

Photo by I. Peterson.

Creating polyhedra with beads is an interesting way to learn the properties of regular and semi-regular solids, Shea says. In a bead polyhedron, each face becomes open space, each edge becomes one bead, and each vertex becomes a thread void. The resulting structure is light and open.

The "Plato Bead," created by Laura Shea, is a dodecahedron. A bead stands in for each of this polyhedron's 30 edges. Each of the 20 vertices becomes a void surrounded by three beads and thread. The 12 faces of the form become open spaces.

Courtesy of Laura Shea.

The Bridges website shows additional examples of Shea's work.

No mathematical crafts session of the knitting network at JMM would be complete without a Möbius strip—that mind-bending, one-sided, one-edged mathematical object. 

Josh Holden (Rose-Hulman Institute of Technology) displays a Möbius band that he crocheted.

Photo by I. Peterson.

Carolyn Yackel works on her knitting during a JMM knitting network session.

Photo by I. Peterson.

Originally posted Jan. 27, 2007

References:

belcastro, s.-m., and C. Yackel. 2006. About knitting . . . . Math Horizons 14(November):24-27.

Klarreich, E. 2006. Crafty geometry. Science News 170(Dec. 23&30):411-413.