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.

Designer Decimals

Calculate 100/89. You get the decimal expansion 1.1235955056 . . .

Look closely, and you'll see that this fraction generates the first five Fibonacci numbers (1, 1, 2, 3, and 5) before blurring into other digits. Recall that, starting with 1 and 1, each successive Fibonacci number is the sum of the two previous Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on.

Calculate 10000/9899. This time, you get 1.0102030508132134559046368 . . .

This fraction generates the first 10 Fibonacci numbers (using two digits per number). Going further, the fraction 1000000/998999 generates the first 15 Fibonacci numbers (using three digits per number).

Note that, in successive fractions, two 0s are appended to the numerator and a 9 to the beginning and end of the denominator.

Will the next fraction, 100000000/99989999, generate the first 20 Fibonacci numbers? Does the pattern continue forever? The answer appears to be yes.

James Smoak discovered this curious phenomenon, and he and Thomas J. Osler went on to prove that this class of fractions always produces decimal expansions containing terms of the Fibonacci sequence. They described it as "a magic trick from Fibonacci."

A little later, Marjorie Bicknell-Johnson found a formula, or "generalized mathematical magician," that identifies fractions whose decimal representations include successive values belonging to a variety of other sequences. She called them designer decimals.

Smoak (with O-Yeat Chan) then continued his adventures in the realm of designer decimals, as reported in the November 2006 College Mathematics Journal.

Consider, for example, the fraction 10000/9801. It has the decimal expansion 1.0203040506 . . . , suggesting the existence of a new class of fractions with curious properties.

Smoak and Chan ask: Do all the integers from 1 to 99 occur in the sequence? Given that the decimal expansion must repeat, what is the length and nature of the repeating part?

The key, Smoak and Chan say, is to note that 9801 = 99². So 10000/9801 = (100/99)² = (1.0101010101...)².

Then, it's possible to show that the repeating part is 0203 . . . 97990001.

In general, fractions of the form [10ⁿ/(10ⁿ – 1)]^k yield the sequence of integers in their decimal expansions.

It's amazing what can lie hidden in simple fractions!

Originally posted November 6, 2006

References:

Bicknell-Johnson, M. 2004. A generalized magic trick from Fibonacci: Designer decimals. College Mathematics Journal 35(March):125-126.

Chan, O-Y., and J. Smoak. 2006. More designer decimals: The integers and their geometric extensions. College Mathematics Journal 37(November):355-363.

Smoak, J., and T.J. Osler. 2003. A magic trick from Fibonacci. College Mathematics Journal 34(January):58-60.

Cool Rationals

One of my more distinct recollections of math class in grade school involves the decimal representation of rational numbers and the discovery of wonderful patterns among those digits.

Consider the fraction 1/7. Expressed as a decimal, it has the form 0.142857142857. . . , where the digits 142857 are repeated, ad infinitum. The surprise to me as a child was learning that the fraction 2/7 has the same decimal digits but in a different order: 0.285714285714. . . . This is also true for 3/7 (0.4285714285714. . . ), 4/7 (0.571428571428. . . ), 5/7 (0.714285714285. . . ), and 6/7 (0.857142857142. . . ).

To my young mind, that was an amazing, inexplicable pattern—a glimpse into the mysteries of numbers. What made the thrill of discovery even stronger for me was how the digits emerged one by one as I laboriously performed the long division operations needed to get the answers. There were no calculators in those days, and it's possible that using a calculator would have eliminated much of the suspense and surprise.

I was reminded of this scene from my distant past when I happened to come across an article by Francesco Calogero of the University of Rome "La Sapienza" in the Fall 2003 issue of the Mathematical Intelligencer.

In his report on "cool" irrational numbers and their rational approximations, Calogero starts off with the example of 10/81. Expressed in decimals, this fraction has the value 0.123456790, with these digits endlessly repeated in the same order. Only the digit 8 is missing from the sequence.

According to Calogero, that defect can be corrected by subtracting from 10/81 a number of order 10^–9 so as to change the last two of the first nine decimal digits from 90 to 89. He comes up with the following expression:

10/81 – 10^–9 (3340/3267).

In decimal form, it has the value (to 101 decimal places) 0.12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455. . . .

That's remarkable!

Try the fraction 1000/998001, or (2³ x 5²)/(3⁶ x 37²).

In decimal form, it has the value (to 100 decimal places) 0.001002003004005006007008009010011012013014015016017018019020021022023024025026027028029030031032033034. . . .

In his article, Calogero goes on to provide an explanation for such "numerology" and offers several additional examples of numbers that display remarkable patterns when written out in decimal form.

Ah, sweet mystery of rational number!

Originally posted Nov. 17, 2003

Reference:

Calogero, F. 2003. Cool irrational numbers and their rather cool rational approximations. Mathematical Intelligencer 25(No. 4):72-76.