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.
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.
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 helical pattern is even more evident in the woody skeleton that serves as the framework for a cholla cactus.
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.
An agave's spiral growth pattern is also evident when a stalk forms at the end of the plant's life.
Yucca plants also produce flowering stalks with a spiral pattern, but they do so annually, unlike an agave.
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?
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 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 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
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.
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.
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?"
