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 are seeing.
I have been taking pictures of mathematical patterns for a long time. Some of my earliest photos date back to 1967. I had just graduated from high school, and I had a chance to visit the world's fair in Montreal that summer. The fair was full of fascinating architecture.
Here's one example.
You're standing inside the building, looking up into its spire. How is that spire put together?
The basic unit is a hexagon, with six equal sides. So the spire is a stack of hexagons, with the corners of an upper hexagon resting on the sides of a lower hexagon. A simple recipe for an elegant structure. [Hexagon Spire]
Here's another example.
This is the U.S. pavilion: a gigantic geodesic dome. When you get close to or step inside the dome, you see hexagons. Lots of hexagons.
But you know that the dome can't consist of just hexagons. Why?
Hexagons fit together to cover a flat surface, as seen in the floor tiles of a subway station in Washington, D. C. [From Hexagon to Square]
So how would you make such a flat surface curve? One way is to introduce some units with fewer than six sides. You can see the effect in this playground climber, which has pentagons scattered among the hexagons.
If the climber were a complete sphere, you would find exactly 12 pentagons.
Back to the U.S. pavilion. Where are the pentagons? [Hexagons, Pentagons, and Geodesic Domes]
You have to look closely, but they are there.
And, if the dome were a complete sphere, you would find 12 pentagons.
Here's another intriguing structure, this time a sculpture on the National Mall in Washington, D.C.
Made from rods and cables, this sculpture rises about 50 feet into the air. Notice that no rod is directly connected to any other rod. So how can this structure possibly stand up?
Here you see that the tower stands on three legs.
It is designed so that the cables push on the ends of the rods, compressing them, while the rods in turn pull the cables taut, stretching them. It is this interplay between tension and compression that allows the structure to stand erect.
If you step underneath and look up toward the tower's top, you'll see a clue about how this interplay is achieved, in a beautifully symmetric pattern. [Needle Tower]
Across the mall, you'll find the East Wing of the National Gallery of Art.
Its design is based on a trapezoid sliced into two triangles.
One result of this design is that there are very few right angles, and that's unusual for buildings. [Splitting a Trapezoid]
Here's a view of the gallery's western face. On the extreme right, you see an especially sharp edge, 19 degrees.
Looking more closely at that edge, you can see a dark smudge in the marble. What do you think that might be?
So many people have felt the urge to touch this sharp edge that they have left their mark on the building.
The dark area represents all the people who have touched the edge. Most probably reached out at about shoulder level, so the smudge is, in effect, a population distribution by height.
Notice a second, smaller dark area higher up. This might represent the show-offs who tried to reach as high as possible. [An Irresistible Edge]
People leave their marks all over the place, and we can learn from those wear patterns.
This is the door to a men's restroom at East Tennessee State University. Notice the patch where the paint has worn away.
This is a swing door, so the patch represents the many men who have pushed the door open. The up-down direction gives the height distribution. The left-right direction says something about how hard you have to push on the door. Pushing nearer to the edge is easier.
The metal plate, which is lower down, is opposite a handle used to pull the door open. People tend to pull from waist level but push at shoulder level, so the wear pattern is higher than the metal plate.
What do you suppose the wear pattern on the women's restroom door looks like?
The pattern is lower down, which suggests that women tend to be shorter than men. And it seems to be a bit closer to the door's edge. [Statistical Wear]
Here's a set of elevator buttons.
From the wear pattern, what can you say about the elevator's location and how it was used? [Elevator Buttons and Stone Steps]
Here's the wear pattern on stone steps in Wells Cathedral in England. What can you say about the foot traffic over the centuries?
Here's another sort of distribution: icicles hanging from a gutter. What can you say about the water flow or the shape of the gutter? [Icicle Distributions]
And here's still another kind of distribution: a set of organ pipes in a church. [Organ Pipes]
The pipes on the left seem to be a mirror image of the pipes on the right. But it can't be an exact mirror image. Why?
The sound that an organ pipe makes depends on its length, so each note is represented by a pipe of a different length. You could arrange these pipes in an array from shortest to longest. So, the mirror symmetry of the organ pipes in the photo is an illusion.
Math can crop up in surprising places. This is a sign for an organization on the campus of Westminster College in Salt Lake City. [Moebius Inclusion]
The twisty loop at the top is actually a mathematical object called a Moebius strip. You can make one from a strip of paper simply by twisting one end through 180 degrees, then taping the ends together.
The resulting loop has a number of remarkable properties. For example, if you start anywhere on the loop and draw a line down the middle, you will go around twice and end up right where you started, as if the strip has only one side. This display at the Boston Museum of Science demonstrates that property with an arrow train running along a track down the loop's middle. [Endless Train Track]
The Moebius strip has fascinated all sorts of people, including engineers and artists. Here's a Moebius strip carved from granite, on display at the Baltimore Museum of Art. [Endless Ribbon]
This object came out of mathematical research more than 150 years ago, when mathematicians were trying to identify and classify all possible surfaces.
At the same time, we all encounter the Moebius strip somewhere just about every day, in its guise as the symbol for recycling.
If you join the three bent arrows into a loop, you end up with a Moebius strip. And that was deliberate. The designer of the recycling symbol in 1970 wanted to represent "continuity in a finite entity" and turned to the Moebius strip as his model. [Recycling Arrows]
Look closely at the three bent arrows.
You'll notice that two arrows are identical, and the third arrow bends the opposite way.
Now look at this example. What's different about it?
The three arrows are identical. What happened?
I suspect that someone didn't pay close enough attention to the original design and simply used three bent arrows, all bent the same way. I call this the mutant form of the recycling symbol. And you'll now find it all over the place.
What is this mutant form? It turns out to be one-sided, just like the standard Moebius strip. But if you join the arrows into a loop, you'll find that it forms a knot, as seen in the recycling symbol for Santa Clara, California.
Another everyday object: a fire hydrant. What's there to say about a fire hydrant?
There's something odd about the valve used to turn on the water. It has five sides: a pentagon. Why?
A standard wrench has parallel jaws, so it works best on nuts with an even number of sides: four, six, eight. It doesn't work so well on a pentagonal nut.
Instead, you need a special tool, designed especially for a pentagonal knob.
This helps discourage just anyone from turning on the water. [Fire Hydrant Pentagons]
For my final example, I turn to breakfast. How many of you eat Cheerios or some other ring-shaped cereal? Have you noticed that the floating rings tend to clump together and stick to the sides of the bowl?
This is a surface-tension effect. The liquid rides up the sides of the rings and the bowl. When two rings get close enough together, they attract each other, so clumping occurs. You can look it up online as the "Cheerios effect." [Clumping Cheerios]
When you have enough rings still floating in the liquid, you can also see them organize themselves into an orderly pattern.
The rings pack into orderly offset rows. But there's another pattern here, too. Towards the middle, each ring is surrounded by six other rings, so you also have a hexagonal array.
And the same hexagonal pattern appears when you stack oranges or apples into a pyramid. [Market Packing]
This is just a small sampling of the many ways in which math is embedded in the world around us. At the same time, whenever we look at anything closely, questions are sure to come to mind, and math is a wonderful tool for helping us understand what we are seeing.
So, where's the math? Everywhere, once we start looking.
Photos by I. Peterson