National Museum of Mathematics

Dallas Arts District: One Arts Plaza

At this location, you can ask your participants to imagine that they were working on construction for this building, and that the panes of glass in the main grid of windows on this face of the building needed to be ordered. How many panes would you need to order? In the discussion, bring out the fact that there are terraces on the upper levels, so they don’t need that same pane of glass; restrict your count to just the panes that are right on the surface of the building, amidst (the two sections of) the concrete grid.

Some problem-solving strategies that may come up: divide the count into the two sections of the concrete grid. In the right-hand section, the glass panes form a perfect rectangle, so you can count them by multiplying 17 windows across horizontally by 11 windows high to get 187 panes of glass. We will have to add this to the result from the other section of the building. That one is not a rectangle, so we could either divide it into a 4-by-7 rectangle (on the bottom) plus a 3-by-4 rectangle (on the top), or visualize it as a 4-by-11 rectangle with the four top-left panes missing. Either way you work, you will get 40 panes in this section, for a total of 227 panes in all. This exercise provides a good opportunity to illustrate how there can be more than one way to attack a problem, and each different way can give further insight into the nature of the problem.

You can also discuss, time permitting, whether having made the count, you would order exactly 227 panes. Presumably not, because it would be too risky; absolutely nothing could go wrong with any of the panes. Since glass can be fragile, you would likely order some extra. Depending on budgets, you might order as much as 10% extra. How many would that be? Well, 10% of 227 rounds to 23, and adding that to 227 gives a nice round 250 panes of glass as a conservative order. (Note that you will likely be spending money on glass that you will never use, but that’s the price of avoiding the risk of ending up short a pane of glass in case of breakage.)

There’s one other interesting story about this building. In a first draft math tour, the tour leader Glen Whitney asked what was the shape the window panels counted in the challenge above. “Well, they look like squares.” And then he asked what the shape of the large structure projecting out from the building near the top was. “That looks like a square, too.” Glen then stated that they couldn’t both be squares. Why? Well, if you count window units across, the big structure is clearly six units across. And counting window units up the side of the big structure, it appears to be five units high. If one rectangle is five-by-six in terms of another rectangle, they clearly can’t both be squares; one or the other might be, or neither, but they can’t both be. However, one of the tour participants quietly pointed out to Glen later in the tour, “If you look at the building from far away, I think you will see that the big structure extends past the top of the building and both it and the windows are squares.” Judging from the picture below, I believe that’s absolutely correct. The structure does extend past the top of the building, so that it appears to be exactly six units high, so that it and the individual windows are squares.

I didn’t include this story here with any intention that it would be mentioned on a typical tour. Instead, I think it holds several important lessons for the tour process as a whole. (1) You will make mistakes. Acknowledge that. Try your absolute best to minimize those mistakes as close to zero as possible, but realize that no matter how hard you try, you will make a mistake now and then. So don’t present yourself as an infallible authority. (2) Math tours, like all mathematical experiences, are works in progress; they can always be refined, extended, polished, improved, corrected, or expanded. (3) The folks that come along on the tours are a valuable resource. Engage them. Listen to them. And most important of all, be open to learning from them. If you are, you certainly will, and these interactions will enrich your experience as a tour leader and make leading math tours more rewarding.

Dallas Arts District: Booker T. Washington School Pegasus

Obviously for a walk in the Dallas Arts District, it’s important to highlight ways in which math and art interact with and enrich each other. One opportunity to do that is the sculpture of Pegasus by artist Stuart Kraft at the Booker T. Washington High School for the Visual and Performing Arts. There are at least two points of contact between math and the arts that you might highlight here. One is that the choice of Pegasus as subject for this artwork is almost certainly not coincidental, as Pegasus is at least to some extent a symbol of Dallas, if for no other reason than the long history of the red Pegasus sign on the Magnolia Oil building. And like art, mathematics depends on the disciplined use of symbols. While the meaning and uses of mathematical symbols may be more narrow and regimented that of artistic symbols, it’s important to remember that in both cases, the use of symbols stems from the central, human drive to communicate. Both art and math communicate ideas, concepts, and relationships, and the use of symbols enhances the power of those communications.

More concretely, if you ask visitors to look closely at the pieces of steel that make up Pegasus and describe their shapes, you will see that although some individual pieces, as in the mane, are free-form and defy easy description, the bulk of the pieces generally conform to simple, familiar geometric shapes. You will see triangles, for example in the neck of the horse; pieces of cylinders, for example in the wings; circles, particularly in the rear hip of the horse. This echoes a familiar technique that underlies much of mathematics, which is to take a complicated problem, situation, or object, and analyze it by breaking it down into many small constituent parts. Moreover, the details of those parts can help to understand the function of the whole. So for example, in Pegasus, the circle at the hip reflects how that joint can freely rotate over a large range of motion, just as a circle can be rotated any amount and yet still match up with itself. To produce the most striking rendition of his subject, the artist needs to understand and appreciate how these simple forms, in a very mathematical way, make up something as flowing and dynamic and subtly contoured as Pegasus about to spring into flight.

Dallas Arts District: Winspear Opera House louvers

This post is the first in a series detailing a large number of math tour stops designed for the walkSTEM tours of the Dallas Arts District. The tours begin at the Winspear Opera House, so let’s start by examining several mathematical ideas that you can bring out by examining the louvers above the plaza surrounding the opera house.

Length

A first, simple challenge that you can give is to ask tour participants how we can determine the length of the louvers above. Even if we have a tape measure here on the ground, we can’t fly up to the level of the louvers and measure them directly. There are many possible suggestions, such as guessing, using the shadows of the louvers, using ladders to get to the level of the louvers, etc. — brainstorming should be encouraged. But one very productive idea is to find something we can measure on the ground that will be the same length as the louvers. To do that, we use the mathematical properties of a basic shape: the rectangle. (Actually we are using a property that’s true of any parallelogram at all, regardless of whether it is right-angled, but rectangles are the easiest parallelograms to spot an most architectural structures.) Namely, the opposite sides of a rectangle are equal in length, so if we use the rectangle shown in this annotated version of the photo at right, all we have to do is measure along the ground at the base of the building to find that the louvers are 28 feet long.

Number Patterns

The first observation here is that the numbers of louvers in the various rectangular sections or cells are not the same, as one might expect from the way most buildings are designed. That leads to a natural question — is there any pattern to the different numbers of louvers? Or any relationship between the number of louvers in adjacent cells? Well, is there any physical relationship between the louvers in one cell and the adjacent cell? A bit of observation and brainstorming will result in noticing that whenever there are more louvers in one cell than an adjacent cell, all of the “beams” from the cell with fewer louvers extend into the adjacent cell. In other words, the cell with more louvers has louvers in all of the same positions as the cell with fewer, plus some extra louvers. And where do these extra louvers go? Well, on the north side of the building, it is easy to see that one additional louver is added in every “space” created between louvers from the adjacent cell, as shown in the diagram on the left. What does this operation do to the number of louvers? It might seem as though it doubles the number of louvers, since we are adding one louver in every space, but actually it goes from three to seven louvers. Why is that? It’s because dividing a rectangle into sections with three lines creates four sections (one more section than line), and if we put one new louver in each section, we get 3 + 1×4 = 7 louvers. On the other side of the building , you can see that the architects (Foster & Partners) added two new louvers in every space between louvers of the adjacent cell, so we go from three louvers to 3 + 2×4 = 11 louvers. Following that line of cells, we see on the other side of the building that same count of eleven louvers comes from inserting one in each space among five louvers. So finding these two different ways of breaking down eleven using arithmetic operations, 3 + 2×4 = 11 = 5 + 1×6, exemplifies the kind of playing with numbers that the architects used to create this building.

Width

Measuring the width of the louvers poses similar challenges to measuring their length, but is slightly more intricate. As a first step, we can use the same technique we used to measure the length of the louvers to measure the length of the rectangular cells, but employing a different rectangle to project the length to something measurable on the ground (again shown in the annotated photo to the right). You should find that the cells are 48 feet long.

However, in this case it’s not enough just to measure the length of a cell, because we’re after the width of a single louver. So the number of louvers that fit in one cell is relevant. Should we look at the cells that have the most louvers? No, because the louvers are at an angle, more louvers fit than would if they were all perfectly horizontal. So we want to estimate which cell looks as though the louvers will just fit if they were all level. We can’t be certain, but it appears as though the louvers would just touch each other when horizontal in the cells with eleven louvers. That means that the width of the louvers is the same as the spacing between the beams in the cells that have eleven louvers. How do we get that spacing? Do we divide 48 feet (the length of the cell) by 11 (the number of louvers)? No; we have to remember that the eleven beams divide the 48-foot-long cell into twelve equal spaces, so the beams are four feet apart. Hence, we estimate that the louvers are four feet wide, which seems pretty reasonable as a nice, round number.

Angle

We noticed in the previous activity that the the louvers are not horizontal. Why might that be? There’s lots of possible ideas, but if we think about what the purpose of the louvers is, that may help. Namely, the louvers are there to block the sun, and so we can ask what angle is best for blocking the sun. This brings us to the idea of different problems that are mathematically equivalent; can we think of a problem that people have worked on in great detail, that maybe sounds different from blocking the most sun, but is really equivalent to that problem? Yes — the problem of mounting solar panels that soak up the most sun. It stands to reason that if a panel is absorbing the greatest amount of sunlight possible, then it must be blocking the greatest amount of sunlight possible to viewers behind the panel. So we can use the considerable amount of mathematical analysis that’s out there on optimal solar panel positioning to understand at what angle the louvers should be tilted.

Starting from the basics, why shouldn’t the louvers be completely horizontal? The answer lies in Dallas’s latitude. Because Dallas is sufficiently north of the equator, the sun is never directly overhead, so we can block more sun by angling the louvers toward where the sun actually is. How much should we angle them? A louver will block the greatest amount of sunlight when it is perpendicular to the sun. Dallas’s latitude determines the highest angle the sun can reach in the sky, as shown in the accompanying diagram (which is simplified in that it ignores the tilt of the Earth’s axis, so it is only exactly valid on the spring and fall equinoxes). From the diagram, you can see that the tilt angle of the panels to directly face the sun when it’s at its highest is equal to Dallas’s latitude, or 32.8°. (In the context of this diagram, in which we are clearly using geometry to understand the relationship of the Earth and the Sun, it may be interesting to point out that the origin of the word geometry comes from the Greek roots geo- for “Earth” and -metry for “measure,” i.e., geometry is literally the science of measuring the Earth, from which it has expanded to its present-day meaning.)

So should we angle the louvers at 32.8°? Well, that will block the most sun possible when the sun it at its highest point on an equinox. But will that block the most sun on average? Does the sun spend most of its time at its highest in the sky? No; it rises at the horizon and sets at the horizon each day. And because of the tilt of the earth, the sun reaches higher in the sky in the summer, but not as high in the winter. Based on all of that, it is possible to calculate a better angle to block more sunlight on average. According to the site linked above, the best year-round tilt for Dallas is 28°. But there are always deeper layers of analysis — if you read that other site carefully, you will see that the 28° really only applies when the panels are tilted due south. Since the Winspear opera house does not happen to be oriented along lines of the compass, finding the true optimal tilt for the louvers would require further computations that we’re not going to go into here.

Everyday Regular Pentagons

You might be doing a shape scavenger hunt and want to add the otherwise elusive regular pentagon to your list of shapes, or you may just want to ask what interesting geometric shape do your participants see on this everyday item, but either way, take advantage of the fact that fire hydrants in many municipalities employ bolts with regular pentagonal heads. You can dig a little deeper and ask, “Why?” Here are some reasons that have come up:

Since the regular pentagon has an odd number of sides, no two sides are parallel. Hence ordinary wrenches, which are based on gripping two opposite parallel sides, cannot grip on the pentagonal shape, reducing the chance that unauthorized people can tamper with the hydrant. Instead, you have to use a special tool:
that only firefighters have.
The special hydrant wrench actually grips on three sides of the pentagon, instead of the two for usual bolt/wrench combinations, meaning that you can still operate the bolt even if it is worn/damaged.
The unusual shape serves as a flag/reminder that hydrant bolts turn righty-loosey instead of lefty-loosey.

This last item can serve as a jumping-off point for a discussion of how in three dimensions we can have two structures that are mirror images of each other, and yet cannot be superimposed on each other. Such objects are called chiral. Thus, there is such a thing as a left-handed or right-handed screw, and the direction of turning to tighten or loosen is opposite depending on which type of screw it is. Another illustration of the usefulness of the ability to have two different handednesses is that the early New York subways were lit using light bulbs with left-handed threads. Riders therefore could not unscrew the subway bulbs and use them at home, because they simply would not screw into ordinary right-hand threaded lamps. Thus, the basic mathematical phenomenon of chirality enabled a theft-prevention measure.

Hyperbolas from Light Fixtures

When you see light from a fixture shining against a wall, you often observe some very clear, sharp curves as the boundaries between light and dark areas. Those curves are generally hyperbolas. Why? The light from the source is generally blocked off to produce a cone of light, and then that cone is intersected with the wall, creating a conic section. For typical arrangements of light fixtures, that section is generally a hyperbola.

Tours are Modular

Most math tours will consist of a series of “stops” or vignettes, in which something observed in the world around us spurs discussion of a particular mathematical topic, idea, or result. Often these different vignettes can be mixed and matched and regrouped to create new tours in new places or even new tours in places you’ve toured before, but with a different theme. So in constructing tours, it’s extremely useful to have a large “bag of tricks” — a big collection of different things that can come up in the natural world or the built environment, and bits of interesting math that you can hang off of them. So, the idea for this blog in posts going forward is to share such a collection. We’ll post different topics that have come up in different MoMath tours, starting with a rapidly-designed tour that took place at the ASTC 2015 convention. And please feel free to comment, including describing similar activities you’ve done in math tours, or totally new ideas. If you have an interesting one in a comment, we’ll invite you to do a guest post in this blog.

So: go out there and create some great tours and let us know about them.

Dream a Theme

✹ You want the tour to gel into an experience for the participants.
✹ Distills and reinforces a “take-home message”.
✹ Helps to filter the myriad of ideas that you will encounter once you’re looking at the world through mathematical lenses.
✹ Conversely, also helps you to generate fresh ideas, as you flesh out your theme.

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Use the Built Environment

✷ Shows math is useful and that people use it to make our world a better place.
✷ Mathematical ideas are used so often for and in decoration and adornment. Shows math off as a creative and aesthetic endeavor.
✷ These items will be the mainstay of your tour construction.
✷ They provide a source of reliable “modules” that can be inserted into virtually any tour.