November 15, 2022

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 PiƱon-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 has 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.

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

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 petals (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?"