mathwalks.org

Category: Symmetry

  • Calatrava “Wave”

    Another three quickies, this time about the iconic sculpture “Wave” by Santiago Calatrava. (1) Is this sculpture curved or straight? (2) Is it symmetric? (3) How did Calatrava get each one of the copper beams to stay at the particular angle it sits at?

    Again, it’s best to give participants time to chew on these questions and generate their thoughts. Then you can bring everyone together to see if there’s consensus. Here are my thoughts. (1) This sculpture is both curved and straight! It’s a remarkable fact that there are truly curved surfaces that can be broken up completely into a collection of individual straight lines. Such surfaces are called ruled surfaces, and they have been the subject of much mathematical study. This “wave” surface is one, and others include the elliptical hyperboloid and hyperbolic paraboloid shown below.

         

    It is also very interesting that when a curved surface like this can be constructed from straight lines, it also has other straight lines that cross one such set of lines. That can be seen in Wave, as highlighted in the image below.(2) The question of the symmetry of the sculpture is a tantalizing one, becuase there is not a mirror symmetry along the central red line in the in the above picture (since a hump that is high on the left is low on the right and vice versa) nor is there a rotational symmetry around a vertical axis.  But, since each individual line runs through the central red line, there is a 180-degree rotational symmetry about that axis, which brings every one of the beams back to coincide with itself.

    (3) The secret to the different slopes of the different copper beams lies under the sculpture just to the left of the triangular support seen in the image above. If you peer under there, you will see a strut attaching to each beam, which otherwise can pivot at the central red line. Each of those struts is at a different height, and that height sets the slope of the corresponding beam. So as you peer under there, you will see the wave pattern of the entire sculpture replicated in those struts. In fact, there’s a mechanism inside the sculpture which allows each of those struts to move up and down, animating the entire sculpture.

  • Dallas Hall Dome

    The dome of Dallas Hall at Southern Methodist University is one of the most beautiful, striking, and symmetric structures on the entire campus.The entire structure has 16-fold rotational symmetry: if you stand at the center of the rotunda, look up at the dome, and then turn one-sixteenth of a full rotation, you will get exactly the same view of the dome. Not only is it beautiful (demonstrating, I think, an innate reaction to high symmetry), the structure of the dome allows you to count the elements in the decorations easily, because it sets up a linear relationship between the number of items in one section of the design and the number in the entire dome. So for example, to figure out how many brown scrolls there are around  the entire domain, you count that there are seven in a single section, and then multiply by 16 to figure out that there are 112 scrolls in all. Or you can count the number of small carved rosettes near the edge of the design in one section, and multiply that number by 16 to get the total number of rosettes that had to be carved to decorate this dome.

    If you  look more closely, you will see that each one of the sections of the dome is composed of two mirror-symmetric halves. So there are really 32 copies of the same design adorning this dome, half in one orientation, and half as the mirror image. Overall, this is a remarkable illustration of structure and beauty of a highly symmetric design.

  • Hughes-Trigg Student Center Interior

    The study in symmetry continues inside the Hughes-Trigg Student Center, this time providing an excellent illustration of another important type, rotational symmetry. Check out the central gallery of the building. If you stand in the center of the room (right on top of the ubiquitous stallion), pick any direction and memorize what you see, and then turn ninety degrees, you will see an almost identical view. The arches, the moldings, the columns, the well in the center of the floor, the beams in the glass roof — all appear exactly the same when rotated  a quarter turn. Only decorative details such as the sculpture of the boy with birds, or practical considerations like what you can see down each hallway, break the symmetry.

    Given that the overall shape of the room is an octagon, you might think that the room also has an element of eight-fold, rather than just four-fold, rotational symmetry. But if you look closely, you will see that alternating walls are shorter and longer, so that the scene really does change in a fundamental way if you rotate only one eighth of a turn.

    This room is also a good location to compare and contrast rotational versus mirror symmetry, because there are some elements of the room that obey one versus the other. For example, the plants you can see in the photo above are placed in a mirror-symmetric way about an imaginary vertical plane right through the clock and the stallion on the wall, as are the two columns of balloons (which may not generally be there). A good activity is to list different features of the room and classify whether they are compatible with mirror symmetry, rotational symmetry, both, or neither.

  • Hughes-Trigg Student Center Facade

    Many mathematical concepts are important in architecture, and a particularly important one is symmetry. Rarely will you see a building so carefully displaying mirror symmetry as the lovely, neo-Classical Hughes-Trigg Student Center on the campus of Southern Methodist University. If you imagine a vertical line running from the highest point of the building straight down through the center line of the doors and the center of the steps, you will see that virtually every element of the building on the left of that imaginary line is replicated identically, except flipped as if in a mirror, to the right of that vertical line. That replication goes for columns, windows, doors, railings, moldings, drainpipes, and more. It even goes so far as to dictate the placement of lightning rods, and to cause the leftmost and rightmost doors into the building to be hinged in opposite directions.

    Can you find any violations of this near-perfect, rigid symmetry? I was only able to find two. The first is the inscription above the columns; that asymmetry is unavoidable, as English text, like most languages, has an inherent directionality or asymmetry, in the case of English reading left-to-right. And the other exception, as you’ll see below, is an allowance for SMU pride.In this detail, you can see that the revered SMU stallion logo maintains its universal run to the right on both sides of the main entrance doorway, instead of being reversed on the right-hand side to maintain the mirror symmetry. Overall, the Hughes-Trigg facade is an interesting study in the power of symmetry to shape the impression created by an image, and power of breaking that symmetry to emphasize a detail like the stallion.