Patterns in the Santa Fe Botanical Garden

When we look at the world around us, we don't usually think about mathematics, or even notice math that may be right in front of our eyes. Yet an eye for math can greatly enrich our appreciation and understanding of what we see.

The Santa Fe Botanical Garden is a wonderful place to explore with mathematics in mind, from the bilateral symmetry of leaves to branching fractal forms and Fibonacci numbers embedded in spiral patterns.

Counting and Measuring

Most people associate the term mathematics with numbers and, indeed, numbers do play a role in mathematics. At the same time, we encounter numbers in all sorts of ways in everyday life.

Let’s start by characterizing the Santa Fe Botanical Garden, noting how we use numbers as key parts of these descriptions.

The Garden sits about 7,200 feet above sea level, near the southern end of the Rocky Mountains, which were formed 80 million to 55 million years ago.

The Sangre de Cristo Mountains near Santa Fe represent the southernmost subrange of the Rocky Mountains.

The Garden gets about 9 to 13 inches of precipitation (rain and snow) annually, putting the area in the climate category of semi-arid steppe.

This botanical garden is relatively new; its oldest section opened to the public in 2013.

The developed part of the Garden covers about 8 acres and has three sections: the Orchard Gardens (2.5 acres) and, on the other side of the Arroyo de los Pinos, the Ojos y Manos ethnobotanical garden (2.5 acres) and the Piñon-Juniper Woodland (3.25 acres). The focus of the Garden is on plants selected for their beauty and adaptation to the Santa Fe environment.

Note how numbers help us describe, measure, and understand what we experience or encounter.

Another number: The Garden’s address is 715 Camino Lejo (though you won’t see that number anywhere on the site). We generally take for granted the use of numbers as parts of addresses, but there are places around the world where a location is more often defined by its position relative to some landmark than by a number.

You might also notice that the number 715 itself is divisible by 5. Indeed, it is a composite number, the product of the three prime numbers 5, 11, and 13.

For centuries, only mathematicians and number enthusiasts cared about and studied prime numbers and their multiplicative offspring. That changed about 40 years ago when the distinction between prime and composite numbers became a key part of a digital cryptosystem widely used for protecting data.

The so-called RSA public-key cryptosystem relies on the observation that a computer can multiply large numbers remarkably quickly, but typically takes much, much longer to determine the prime factors of a given large number.

But there’s much more to mathematics than just numbers and counting (and arithmetic). More broadly, we can think of mathematics as the study (or science) of patterns, though those patterns may themselves sometimes involve numbers.

Four Edges

The Garden's Rose and Lavender Walk features a wide variety of roses and several types of lavender (Lavandula).

Feel the stem of a lavender plant. You'll notice that the stem is not rounded but has edges. Indeed, the stem has (roughly) a square cross section.

Lavender stems have a square cross section.

The square stem is a characteristic of plants in the mint family (Lamiaceae). This family includes not only mint and lavender but also basil, rosemary, sage, thyme, salvia, and others.

In the Garden, you'll see that the stems of a variety of plants, all belonging to the Lamiaceae family, have square cross sections: Mintleaf bergamot (Monarda fistulosa), hummingbird mint (Agastache cana), English thyme (Thymus vulgaris), garden sage (Salvia officinalis), and Mojave sage (Salvia pachyphylla).

Garden sage (Salvia officinalis) is a member of the mint family.

Five Petals

The number 5 comes up repeatedly when you examine members of the rose family of plants (Rosaceae). The flowers of these plants typically have five sepals and five petals.

Flowers of the rose family typically have five sepals.

Wild roses have just five petals, as do a few varieties of cultivated roses such as 'Golden Wings.' The sweetbriar rose (Rosa eglanteria) is another example of a rose with five petals found in the Garden. However, most cultivated roses, which are bred for their appearance, have many more petals (though they still have just five sepals).

The 'Golden Wings' rose has five petals.

The fruit trees in the Orchard Garden are all members of the Rosaceae family. In springtime, the apple, apricot, cherry, plum, peach, and pear trees all produce blossoms with five petals.

The number 5 can also come up in surprising ways. Cut across an apple to reveal its core, and you'll find a five-pointed star shape in the center.

The Garden has a number of other plants, beyond roses and fruit trees, that belong to the Rosaceae family. They include crabapple, Apache plume, mountain mahogany, serviceberry, and fernbush.

Apache plume (Fallugia paradoxa) blossoms have five petals.

Cactus Spirals

The Dry (Xeric) Garden includes plants that thrive despite a dry climate and humus-poor mineral soils. Partially enclosed by a stone wall and featuring a stone walkway, the "Hot Box" portion relies on natural precipitation for moisture and serves as a home for cold-hardy but heat-loving plants, including various kinds of cactus, agave, yucca, and Mojave sage.

The "Hot Box" of the Xeric (Dry) Garden includes various types of cactus and plants such as Mojave sage.

If you look closely at a cactus, you can often detect distinctive patterns (though the spines may sometimes hide the underlying pattern), particularly spirals and helixes. Note, for example, the way in which the spines and ridges on a cane cholla (Cylindropuntia spinosior) create a helical (spiral) pattern.

Cane cholla (Cylindropuntia spinosior) helix.

The helical pattern is even more evident in the woody skeleton that serves as the framework for a cholla cactus.

The woody skeleton of a cholla cactus shows a helical pattern, as seen in the offset slits of the limb.

Similarly, observe how the leaves of an agave appear to grow in a spiral fashion. The leaves are not lined up like the spokes of wheel.

Spiral growth pattern of Havard's agave (Agave havardiana).

An agave's spiral growth pattern is also evident when a stalk forms at the end of the plant's life.

An early stage in the growth of an agave stalk reveals a spiral pattern.

You might also notice a resemblance between an agave stalk and the young shoot of an asparagus plant. It turns out asparagus, agave, and yucca are genetically related and all belong to the Asparagaceae family.

Yucca plants also produce flowering stalks with a spiral pattern, but they do so annually, unlike an agave.

Yucca stalk.

Spheres and Hexagons

The leaves of a beaked yucca (Yucca rostrata) form a distinctive spherical shape. In effect, the plant looks the same from any direction, displaying spherical symmetry. For a sphere, the distance from its center to any point on the surface is the same.

Beaked yucca (Yucca rostrata) has a roughly spherical shape.

Here’s an interesting botanical question: How does this species of yucca achieve its spherical shape? What “rules” do its cells follow so that each leaf ends up roughly the same length?

Take a look at the sculpture Gift by Elodie Holmes and Caleb Smith adjacent to the "Hot Box." It displays a symmetric pattern of hexagons, representing a honeycomb pattern. If you were to look inside a beehive, you would see such an array of cells constructed from wax and used for storing honey.

Gift by Elodie Holmes and Caleb Smith displays a honeycomb grid of regular hexagons.

The basic unit is a regular hexagon, with six equal sides and angles. The only other such shapes (regular polygons) that fit together to cover a surface without gaps are the square (four sides of equal length) and the equilateral triangle (three sides of equal length). These are examples of tilings (tessellations).

Bees have been making such hexagonal structures for millennia. It was only in recent times (1999) that mathematicians were able to prove that this particular pattern is the most efficient way to divide an area into equal units while using the least wax (smallest perimeter). That’s something that bees “knew” all along.

The number 6 also arises in another context. Note the six ridges characteristic of a  claret cup cactus (Echinocereus coccineus).

Sixfold rotational symmetry of a claret cup cactus (Echinocereus coccineus).

This cactus has the same sixfold rotational symmetry as the regular hexagon of the honeycomb.

Triangles, Squares, and Symmetry

Kearny's Gap Bridge is a recycled structure, originally built in 1913 for a highway near Las Vegas, New Mexico, and installed at the Garden in 2011 to connect the two sides of the Arroyo de los Pinos.

Like many human-made structures, the bridge features several types of symmetry.

In general, an object has some form of symmetry when, after a flip, slide, or turn, the object looks the same as it did originally.

Reflection is arguably the simplest type of symmetry. Notice, for example, that the two sides of the bridge mirror each other. What other forms of symmetry do you see at the bridge?

The most important geometric element is the use of equilateral triangles, characteristic of what is called a Warren truss, named for British engineer James Warren, who patented the weight-saving design in 1846.

A truss is a framework supporting a structure. A Warren truss consists of a pair of longitudinal (horizontal) girders joined only by angled cross-members (struts), forming alternately inverted equilateral triangle-shaped spaces along its length.

It’s a particularly efficient design in which the individual pieces are subject only to tension or compression forces. There is no bending or twisting. This configuration combines strength with economy of materials and can therefore be relatively light.

Look at the pattern of struts along the “railing.” This is an example of translational symmetry. Shifting the pattern to the left or right leaves the pattern the same.

Behind the railing is another geometric feature: a protective fence in the form of a square grid.

As seen from the bridge, the sides of the arroyo are partially lined with gabions—wires cages filled with rocks to help control erosion. These gabions were constructed in the 1930s by the Civilian Conservation Corps.

Many of the wire cages of gabions in the Arroyo de los Pinos have a square grid pattern.

In some locations, the wire cages have a hexagonal grid.

In general, the repeated patterns of a symmetrical design make it easier for engineers to calculate and predict how a structure will behave under various conditions. They are characteristic of a wide range of human-built structures.

Horno Circles

The adobe structures found near the north end of the bridge are outdoor ovens, called hornos.

The design originated many centuries ago in North Africa, and it was brought to Europe when the Moors occupied Spain for several centuries starting in the year 711. The Spanish ended up adopting the design and brought these ovens to their colonies around the world, mainly for baking bread. In New Mexico, the indigenous people of the Pueblos also found the technology useful, and hornos became a commonplace sight in their villages.

How would you characterize an horno’s geometry? Many people describe the basic shape as a beehive. That means its horizontal cross-section is a circle, and the circles get smaller as the height increases.

Each example found in the Garden is essentially half of a sphere. What advantages would such a shape have? How would you go about constructing one, making sure that the structure is spherical? Recall that the distance from the center to any point on its surface is the same.

Counting Petals

During seasons when flowers are in bloom, it can be rewarding to examine the blossoms of individual plants, paying close attention to the number of petals characteristic of a given type of blossom.

A chocolate daisy (Berlandiera lyrata) blossom appears to have eight "petals."

Certain numbers come up over and over again: 3, 5, 8, 13, 21, 34. We don’t often find flowers with four, seven, or nine petals, though they do exist. For example, sundrop (Oenothera hartwegii) blossoms have four petals.

The larger numbers are generally characteristic of daisies, asters, and sunflowers, all belonging to the Asteraceae family. However, in this case, each "petal" is actually a flower, known as a ray floret.

The 'Arizona Sun' blanket flower (Gaillardia x grandiflora 'Arizona Sun') belongs to the Asteraceae family of plants. This particular example has 34 ray florets.

The numbers 3, 5, 8, 13, 21, and 34 all belong to a sequence named for the 13th-century Italian mathematician Leonardo of Pisa (also known as Fibonacci). Each consecutive number is the sum of the two numbers that precede it. Thus, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34, and so on.

Is it just a coincidence that the number of flower petals is more often than not a Fibonacci number, or does it point to something deeper—a pattern—about the way plants grow? That’s a question that’s been pondered for centuries.

Perhaps the statistics are skewed. For example, the number of flower petals can be characteristic of large families of plants. The flowers of plants in the rose family (Rosaceae), which includes many fruit trees such as apple, peach, and cherry and shrubs such as fernbush, serviceberry, and mountain mahogany, typically have five petals. So, we are likely to find the number 5 come up again and again when counting petals in the Garden.

Fibonacci numbers also come up in other ways. Take a look at the bottom of a pine cone. Pine cones have rows of diamond-shaped markings, or scales, which spiral around both clockwise and counterclockwise. If you count the number of these spirals, you are likely to find 5, 8, 13, or 21.

The overlapping scales of a pine cone produce intriguing spiral patterns.

You find similar spirals among the seeds at the center of sunflowers and in the helical patterns that many cacti and succulents such as agave feature.

The number of ray florets (above) displayed by a sunflower is often a Fibonacci number, as is the number of clockwise and counterclockwise spirals of seeds at a sunflower's center (below).

The patterns are intriguing (though sometimes difficult to discern and count), and mathematicians, physicists, and other scientists have, over the years, proposed various sets of “rules” that might govern how plants grow and produce the patterns observed in nature. One set, for example, posits (or puts forward as an argument) rules that lead to efficient three-dimensional packing of “cells.” It's a growth pattern that results in the optimal spacing of scales or seeds to reduce crowding.

Branches and Patches

Examine the leaf of a bigtooth maple (Acer saccharum).

Bigtooth maple (Acer saccharum) leaf in autumn.

You’ll notice that the left side of the leaf is just about identical to the right side. These maple leaves have mirror (or bilateral) symmetry: one side is a reflection of the other. The leaves of many plants, large and small, display the same left-right symmetry.

But there’s another pattern on display. If you look closely, you will also see a network of veins: a main vein that branches into smaller veins, and these veins in turn branch into smaller veins, and so on.

The leaves of a bur oak (Quercus macrocarpa) have a distinctive pattern of veins, particularly visible in the fall.

Such branching structures are characteristic of many natural forms. Cypress and juniper trees, for example, have fronds that show this type of pattern.

The fronds of an Arizona cypress (Cupressus arizonica) have a distinctive branching structure.

In many cases, the branches look (at least roughly) like miniature versions of the overall structure. Such patterns are said to be self-similar.

Mathematicians can create self-similar forms simply by repeating the same geometric structure on smaller and smaller scales to create an object known as a fractal. Each part is made up of scaled-down versions of the whole shape.

This example illustrates the first few steps in creating a simple geometric branching structure that has a self-similar, or fractal, pattern. 

The notion of self-similarity can also apply in other ways to natural forms. Just as a tree's limbs and twigs often have the same branching pattern seen near its trunk, clouds keep their distinctive wispiness whether viewed distantly from the ground or close up from an airplane window.

The edge of a cloud may have many indentations, and those indentations when examined closely reveal smaller indentations, and so on. 

Take a look at a raw stone surface. Do you see any straight lines, circles, triangles?

Instead, you might see some large hollows and ridges, and when you look closely, you see smaller hollows and ridges within these features, and so on. So there is a kind of pattern, even if the features are irregular.

The patchiness of lichen growth on a stone surface has a fractal quality.

In general, in nature, you often see patterns in which shapes repeat themselves on different scales within the same object. So clouds, mountains (rocks), and trees wear their irregularity in an unexpectedly orderly fashion. In all these examples, zooming in for a closer view doesn’t smooth out the irregularities. Objects tend to show the same degree of roughness at different levels of magnification or scale.

The characteristic furrows and ridges of Ponderosa pine (Pinus ponderosa) bark have a self-similar, or fractal, quality.

Where else might you find fractal patterns? Try a grocery-store produce department, where you’ll find striking fractal patterns in such vegetables as cauliflower and Romanesco broccoli.

This image looks like a fern, but the self-similar, or fractal, form on display was actually generated point by point by a computer following a simple set of rules.

Although the Garden doesn't have any ferns, it does have fernbush (Chamaebatiaria millefollum). Its leaves have roughly the same branching pattern displayed by fern fronds.

Fernbush (Chamaebatiaria millefollum) leaves display a branching structure similar to that of a fern.

Several artworks along the Garden's Art Trail highlight the contrast between the curves and lines of traditional Euclidean geometry and the fractal geometry characteristic of many natural forms.

Blaze by Greg Reiche. This sculpture contrasts the straight lines and shapes of traditional geometry with the branching structure of tree limbs. 

Sentinel by Greg Reiche. Note the contrast between the straight lines and curves of one part of the sculpture with the rough (fractal) surface of a stone slab.

There are many other patterns to observe in the Garden. For example, you could study and catalog the arrangements of leaves on plant stems (phyllotaxis).

Possible arrangements of leaves on plant stems.

Studying pattern is an opportunity to observe, hypothesize, experiment, discover, and create. By understanding regularities based on the data we gather, we can predict what comes next, estimate if the same pattern will occur when variables are altered, and begin to extend the pattern.

In the broadest sense, mathematics is the study of patterns—numerical, geometric, abstract. We see patterns all around us, in a botanical garden and just about anywhere else, and math is a wonderful tool for helping us to describe, understand, and appreciate what we are seeing.

See also "DC Math Trek" and "Where's the Math?"

Murder and the Economist

Henry Spearman is a prominent economics professor at Harvard University. He is also an amateur detective, applying reason, logic, and economic theory to solve puzzling murders at a Virgin Islands resort and among members of Harvard's promotion and tenure committee.

Spearman is a fictitious character, the hero of a series of murder mysteries written by Marshall Jevons. Short and balding, the economist-detective does, however, share several characteristics with the eminent, real-life economist Milton Friedman.

Clearly a good sport, Friedman himself contributed the following comment for the jacket of The Fatal Equilibrium, the second Spearman mystery, which was published in 1985: "It is hard to conceive of a more pleasant and painless way of imbibing sound economic principles than reading this fascinating, absorbing, and well-written mystery story."

MIT Press

The prominent role of economic principles is certainly evident in all the mysteries. It doesn't take long for Spearman to begin invoking economic theory in Murder at the Margin, which started the series off in 1978. On his way to the island of St. John in the Caribbean, he ponders the apparent paradox of finding less time to enjoy vacations and leisure activities as his income increased substantially.

"But the paradox was not puzzling to an economist who understood the doctrine of 'opportunity cost,'" Jevons writes. "For each evening spent enjoying his stamp collection, Spearman gave up the opportunity to work on a lecture, article, or book that would bring him a large monetary return. On balance he decided to choose work over leisure. As his book sales and fees rose, the cost to him of that leisure time went up accordingly. Consequently vacations were rare, his stamp collection generally went unattended, and many extracurricular books remained neglected."

Like a professor determined to instruct, the author can't resist the temptation to halt the narrative to present minitutorials on various subjects, from the basics of constructing the drums used by a Caribbean steel band to the pricing strategy of Filene's Basement in Boston and the intricacies of operating an ocean liner.

Marshall Jevons is actually a pseudonym for two economics professors: William L. Breit and Kenneth G. Elzinga. The pseudonym was formed by combining the last names of Alfred Marshall (1842-1924) and William Stanley Jevons (1835-1882), two distinguished British economists. Both are explicitly mentioned in the second story, The Fatal Equilibrium.

In one aside, we learn that Spearman ranks Marshall over John Maynard Keynes as the preeminent economic thinker of the twentieth century. Marshall is quoted as saying that economics is "the study of mankind in the ordinary business of life."

Jevons co-developed marginal utility theory, which explains the value of goods and services in terms of the subjective valuation of consumers. That principle plays a crucial role in unmasking the perpetrator in The Fatal Equilibrium.

As a fan of detective stories in offbeat settings, I enjoyed reading the first two books in the Spearman series. Though entertaining, Murder at the Margin, however, seemed to me somewhat contrived and the writing rather stilted. I found The Fatal Equilibrium much more engrossing and lively, almost as if it were written by a different team of authors. I haven't yet read the third book in the Spearman series, A Deadly Indifference.

In recent years, college students appear to have been the primary audience for the series, which sometimes shows up on supplementary reading lists for introductory economics courses. Online student reviews of the books have not been kind.

The Spearman books haven't convinced me that economic theory, which generally posits a world of total economic rationality, represents an all-purpose, widely applicable explanation of human behavior. Spearman solves mysteries by believing that if someone does something that is apparently irrational, there must be hidden rationality behind it, and he seeks to discover what it is.

"So there are two puzzles in Murder at the Margin," economist Herbert Stein wrote in the foreword to the 1993 Princeton University Press edition of the book. "One is who killed A and B. The other is how much resemblance the world of rational economics in which the story takes place bears to the real world. The second puzzle adds to, rather than subtracts from, the fascination of the first."

Originally posted January 12, 1998

Ronald L. Graham (1935-2020)

Mathematician Ronald L. Graham died on July 6, 2020, at the age of 84. He was one of the principal architects of the field of discrete mathematics and made important contributions to computational geometry, Ramsey theory, notions of randomness, and other topics. He also served as president of the American Mathematical Society (1993-1994) and the Mathematical Association of America (2003-2004).

Ron Graham in August 1990 at the 75th anniversary of the Mathematical Association of America, held in Columbus, Ohio.

For more than 20 years while I was a reporter and writer at Science News, I relied on Ron Graham for advice, comments, and news tips. He was especially helpful in pinpointing significant developments in mathematics and patiently explaining their relevance and importance.

Graham's remarks on the difficulties of establishing mathematical certainty inspired the title of my second book: Islands of Truth: A Mathematical Mystery Cruise (W.H. Freeman, 1990).

I wrote about Graham's own research on a number of occasions.

Inside Averages

What can you deduce when you know certain averages but the original data from which the averages were computed are missing?

Considering that question in the 1980s, Ron Graham wondered, for example, to what extent secret data contained in confidential files can be uncovered if the right questions are asked. Such considerations led him to collaborate with Persi Diaconis to explore the discrete Radon transform and its inverse, an important mathematical tool in tomography.

Cracking a confidential database can be likened to the old parlor game of twenty questions. A player receiving only yes-or-no answers yet asking the right sequence of pertinent questions can often deduce the identity of some hidden object or person.

In the same way, the answers to a series of general questions addressed to a particular database could add up to a revealing portrait of something that is supposed to be secret.

A simple example shows how such a scheme might work. Suppose someone wants to find out Alice's salary. The inquisitor has access to information revealing that the average of Alice's and Bob's salaries is $30,000; the average of Alice's and Charlie's salaries is $32,000; and the average of Bob's and Charlie's salaries is $22,000. This provides enough information to deduce that Alice's salary is $40,000.

Researchers often face a situation in which certain averages are known but the original data are missing. If eight data points happen to be identified with the eight vertices of a cube and each of the eight numbers is the average of its three nearest neighbors, then it's possible to deduce the actual but currently hidden value associated with each vertex.

The actual value of vertex A is the sum of the averages shown at B, D, and E minus twice the average at G: (3 + 5 + 5) − 2(5) = 3. The values of the other vertices are 6, 3, 6, 9, 3, 12, and 9.

In this situation, the actual value at each vertex is equal to the sum of the nearest-neighbor averages minus double the average at the corner farthest from the point of interest. Curiously, the point that makes the biggest contribution to the answer is the one that's farthest away.

Diaconis and Graham developed a mathematical theory, based on the idea of discrete Radon transforms, that helps to decide how many and which averages are needed to crack a database or to analyze statistical data. Their results appear in the article "The Radon Transform on Z₂^k," published in the Pacific Journal of Mathematics.

At the root of their exercise is the mathematical concept of how completely a bunch of averages captures the mathematical relationship underlying a data set.

See also "Pennies in a Tray."

A Mathematical Space Odyssey

Preface

Mathematics can take you places where you have never been before. When you explore a mathematical universe you observe amazing patterns, solve intriguing puzzles, and stretch your imagination into the far reaches of the mind.

Samples of the phenomenal shapes and patterns you encounter in mathematics also show up in daily life right here on Earth! Look at a daisy, kick a soccer ball, or untangle a string, and you are in touch with the exotic wonders of mathematics.

You don't have to be a professional to enjoy mathematical patterns and puzzles. Solving mathematical puzzles can be a passion, just as playing baseball, making music, or riding horses can be a passion for those who enjoy it,

There is fun and challenge at every level, so set your mind on the launching pad and be ready for a wild trek into the mathematical universe.

1. A Consequential Countdown

2. Planet of the Shapes

3. The Buckyball Asteroid

5. The Alien Baseball Field

8. The Bumpy Bike Path

13. Pi in the Sky

21. Galactic Gridlock

34. Hyperspace Hangout

55. Triangle Tribulations

89. Back to Home Base

Adapted from Math Trek 2: A Mathematical Space Odyssey by Ivars Peterson and Nancy Henderson

Bibliography

The Cow in the Classroom

“Miss Zarves drew a triangle on the blackboard. ‘A triangle has three sides,’ she said, then pointed to each side. ‘One, two, three.’ She drew a square. ‘A square has four sides. One, two, three, four.’

“She walked around the cow to the other side of the board. She drew a pentagon, a hexagon, and a perfect heptagon.”

There it was: a little mathematics lesson in the middle of a storybook, Wayside School Gets a Little Stranger by Louis Sachar. As I was reading the book to our children, Kenneth, who was 6 years old at the time, interrupted immediately. “What’s a heptagon?” he wondered. He didn’t ask what a cow was doing in the classroom. That was part of the wackiness you could expect at Wayside School. The book promptly answered his question: A heptagon has seven sides.

However, it was the next paragraph that really stuck in my mind because it nonchalantly captured one of those little truths of classroom life. Sachar wrote: “Miss Zarves was very good at drawing shapes. When most people try to draw heptagons, there is always one side that sticks out funny. But Miss Zarves’s heptagon was perfect. Every side was the same length, and every angle the same degree.” Of course, she had drawn a regular heptagon. But that refinement merely enhanced the wonder of her singular talent.

A regular pentagon (left), hexagon (middle), and heptagon (right).

A few months earlier, we had enjoyed another book that featured mathematics even more prominently. Math Curse by Jon Scieszka and Lane Smith neatly spoofs the types of word problems that educators and textbook writers invent to dress up arithmetic exercises and, supposedly, to demonstrate the relevance of mathematics to everyday life.

Eric, who was eight and in the third grade, laughed loudly through much of the book. He particularly enjoyed the following example:

“The Mississippi River is about 4,000 kilometers long. An M&M is about 1 centimeter long. There are 100 centimeters in a meter, and 1,000 meters in a kilometer.

“Estimate how many M&Ms it would take to measure the length of the Mississippi River. Estimate how many M&Ms you would eat if you had to measure the Mississippi River with M&Ms.”

Scieszka and Smith weren't the first ones to tackle Mississippi math. More than 100 years earlier, in Life on the Mississippi, Mark Twain discussed the river’s penchant for shortening its length from time to time when a straight, new channel cut off a deep bend.

Twain noted that in 176 years, the Lower Mississippi had shortened itself by 242 miles, a rate of just over 1.3 miles per year. Thus, he reasoned, a million years ago, the river was at least 1.3 million miles long and “stuck out over the Gulf of Mexico like a fishing rod.” By the same token, in 742 years, “the Lower Mississippi will be only a mile and three-quarters long."

Twain concluded, “There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.”

Canadian political economist and humorist Stephen Leacock also had fun with some of the peculiarities of “concrete problems” often used in classroom mathematics. In a piece titled “A, B, and C—The Human Element in Mathematics,” which was one of a collection of comic stories originally published in 1910 under the title Literary Lapses, Leacock wrote:

“The student of arithmetic who has mastered the first four rules of his art and successfully striven with money sums and fractions finds himself confronted by an unbroken expanse of questions known as problems. These are short stories of adventure and industry with the end omitted and, though betraying a strong family resemblance, are not without a certain element of romance.

“The characters in the plot of the problem are three people called A, B, and C; the form of the question is generally of this sort: ‘A, B, and C do a certain piece of work. A can do as much work in one hour as B in two or C in four. Find how long they work at it.’”

Drawing on clues tucked away in the myriad problems featuring A, B, and C, Leacock reconstructed the tragic, misspent lives of these characters. “Now to one who has followed the history of these men through countless pages of problems, watched them in their leisure hours dallying with cordwood, and seen their panting sides heavy in the full frenzy of filling a cistern with a leak in it, they become something more than mere symbols,” Leacock mused. “They appear as creatures of flesh and blood, living men with their own passions, ambitions, and aspirations like the rest of us.”

Books and articles that base their humor on mathematical quirks and quibbles don't come along very often. One handy, though difficult-to-find collection of mathematical poems, cartoons, jokes, anecdotes, and puzzles is the booklet called Mathematics and Humor, published in 1978 by the National Council of Teachers of Mathematics.

Additional examples can be found in the more recent book Twenty Years Before the Blackboard: The Lessons and Humor of a Mathematics Teacher, in which mathematics teacher Michael Stueben describes his use of mathematical humor and curiosities as a way to enliven his classes.

Fortunately, several wonderful collections of stories, essays, and poems featuring mathematical themes were reprinted a while ago. You'll find it well worthwhile taking a look at Fantasia Mathematica and The Mathematical Magpie, both edited by Clifton Fadiman, and the fourth volume of The World of Mathematics, edited by James R. Newman.

Stephen Leacock's story about A, B, and C appears in The Mathematical Magpie. Two other Leacock stories related to mathematics, "Mathematics for Golfers" and "Common Sense and the Universe," appear in The World of Mathematics. Interestingly, Leacock starts off his commentary on cosmology and modern physics by quoting Mark Twain's speculations on the Mississippi River.

There is also some humor among the stories featured in another anthology, Imaginary Numbers: An Anthology of Marvelous Mathematical Stories, Diversions, Poems, and Musings, edited by William Frucht.

Louis Sachar, author of Sideways Stories from Wayside School and its sequels, has written two books that feature arithmetic problems and logic puzzles in the guise of wacky stories: Sideways Arithmetic from Wayside School and More Sideways Arithmetic from Wayside School.

Hans Magnus Enzensberger weaves intriguing sequences of numbers (and snippets of mathematics involving infinite series, binary numbers, primes, probability, and other topics) into an engaging tale in his book The Number Devil: A Mathematical Adventure.

For a more up-to-date take on mathematics, check out episodes of the TV series The Simpsons for surprisingly frequent references to math topics and vocabulary. But you have to watch or listen closely to catch most of them. The pace can be so quick that they easily zip by unnoticed. See The Simpsons and Their Mathematical Secrets by Simon Singh. Or Sarah Greenwald's online guide to mathematics on The Simpsons.

For old-movie or classic-TV buffs, the routines of the comic duo Bud Abbott and Lou Costello sometimes contain wacky mathematical exercises. See, for example, Abbott's "proof" that 7 times 13 is 28.

My favorite mathematical spoof is in The Phantom Tollbooth by Norton Juster. The book is about a boy named Milo, who is traveling in a strange land, accompanied by the Humbug and Tock (the dog who ticks). In one episode on the road to Digitopolis, Milo encounters not only the Dodecahedron (a mathematical shape with 12 faces) but also the mysterious, somewhat forbidding Mathemagician. Prompted by the Mathemagician, Milo plunges into the intricacies of arithmetic, the paradoxes of infinity, the absurdity of averages, and other mathematical surprises.

At one point, Milo meets the fraction of a child that completes the average family, which consists of a mother, a father, and 2.58 children. After discussing the meaning and relevance of averages, the child remarks, "… one of the nicest things about mathematics, or anything else you might care to learn, is that many things which can never be often are… . it's very much like your trying to reach Infinity. You know that it's there, but you just don't know where—but just because you can never reach it doesn't mean that it's not worth looking for."

Humor has a place in the mathematics classroom. The smiles or laughter that accompany a mathematical joke, cartoon, or anecdote signal understanding gained and lessons learned.

The examples I have cited, from drawing geometric figures to deciphering word problems, also serve as gentle reminders that there is a difference between mathematical exercises disguised as episodes of everyday life and real mathematics applied in the real world. It’s a distinction that’s not always apparent in the mathematics classroom.

Originally posted December 1, 2003