Symmetry (or not!) in Aircraft


Author: Robin A. Ward, California Polytechnic State University-San Luis Obispo


Audience: 4-6 graders


Mathematical Topics: reflection symmetry


Rationale: According to the NCTM Standards 2000,one of the Focus Areas for students in grades 3 -5 is to recognize the usefulness of transformations and symmetry in analyzing mathematical situations. In particular, students should:

  • predict the results of sliding, flipping, and turning two-dimensional figures;

  • identify and describe line and rotational symmetry in various two-dimensional shapes; and

  • explore symmetry in three-dimensional objects (p170-71).

    • This activity allows students in grades 3 - 5 to begin to expand their notion of symmetry and explore real-life shapes with more than one line of symmetry. In particular, students will identify lines of symmetry in various NASA Dryden air and space craft and discuss the importance of symmetry in the design of air and space craft.


    Materials:

    overhead #1 of the F16-XL aircraft
    overhead #2 of the F16-XL aircraft
    overhead of the AD-1 aircraft


    Background: Symmetry is found everywhere in the world around us. One can find symmetry in nature; for example, in the wings of a butterfly, the petals of a flower, or in a sea shell. (The lines of symmetry in all of these images are shown in red.)

    Symmetry is also found in art. The background of this web page shows the work of Leonardo DaVinci.

    Symmetry can also be found in architecture, such as the sketch of the Capitol building shown below, or in real life objects, such as a pair of scissors.


    Many if not all of the rockets, planes, and space vehicles used at the NASA Dryden Flight Research Center are symmetrical. In this activity, students will be introduced to the definition and concept of symmetry by identifying the lines of symmetry in various aircraft being tested at NASA Dryden. Students will discuss the importance of symmetry in the design of rocketry and learn about two of NASA Dryden's aircraft, namely, the F-16XL, and the AD-1, which are not symmetrical; that is, they are asymmetrical. Finally, students will be given the opportunity to extend their knowledge about symmetry by engaging in a "Symmetry Scavenger Hunt" in which they search the NASA Dryden Flight Research Center's photo gallery and identify various planes and spacecraft that are, or are not, symmetric. Students might also search for other familiar real-life shapes in the classroom that contain symmetry.

    View several photos of the AD-1.
    View a movie clip of the AD-1.

    View several photos of the F-16XL.
    Learn some facts about the F-16XL.


    The Activity:

  • Begin a discussion with students about symmetry. Ask students to first think about what symmetry means and then write down a definition of it. Most students will have a general notion about what symmetry means, but may have difficulty trying to express it in their own words. Instead of defining symmetry verbally, students might instead provide examples of objects that are symmetric. Encourage them to provide a verbal description, as opposed to giving examples, in order to use mathematical language.

  • After the students have had ample time to jot down their own definition of symmetry, ask students to form groups of fours and let them each individually share, within their groups, their definition. After everyone has had an opportunity to present his or her definition within their group, instruct the group to develop, as a team, a precise, mathematical definition of symmetry. Within each group, students should select someone to write their definition on the board. Once all of the definitions are placed on the board, each team should read each definition and then choose whose group developed the best definition of symmetry.

  • Next, the instructor should share with the students a working definition of symmetry. In mathematics, symmetry is described in terms of motions called transformations. When an object undergoes a transformation, it does not change in size or shape; instead, it is simply displaced from one location to another. Transformations are also known as "rigid motions," because the object undergoing the transformation does not change in size or shape; that is, it remains rigid. There are four types of rigid motions: translations, rotations, reflections, and glide reflections. Rigid motions are sometimes called isometries. The word isometryis derived from the Latin word "iso" (meaning "the same") and "metry" (meaning "measure"). Thus, an isometry has the same measure; that is, it is the same size and shape!

  • What we mean by symmetry is an "isometry that leaves the whole figure and its position unchanged; that is, although the isometry may have moved points within the figure, the figure still looks exactly the same." (Edwards and Lee, 1999)

  • Kaleidomania! is a superb dynamic tool that students can use to create symmetric designs and explore the mathematics of symmetry.

  • If students need more practice with understanding the concept of symmetry, demonstrate to students how some letters of the alphabet are symmetric. For example, consider the name John Glenn, the first American to orbit the Earth. Are any of the letters in his name symmetric? See if you can predict which letters might have a line of symmetry. Do some letters have more than one line of symmetry? Click here to answer these questions and to learn more about reflection symmetry.

  • Next, introduce to students two aircraft being tested at NASA Dryden. Showthe students the photos of the F-16XL and the AD-1. Ask students if they think these aircraft are symmetric. They are not symmetric.

    The reason the F-16XL is not symmetric is because its right wing is designed differently from its left wing. The F-16XL was designed to improve laminar airflow on aircraft flying at sustained supersonic speeds. It was the first program to look at laminar flow on swept wings at speeds representative of those at which a high speed civil transport might fly. Strips of tubing along the leading edge to the trailing edge sense static on the wing and obtain pressure distribution data. The left wing has three types of instrumentation - preston tubes to measure local skin friction, boundary layer rakes to measure boundary layer profiles (the layer where the air interacts with the surfaces of a moving aircraft), and hot films to determine boundary layer transition locations. Given the different functionalities of the two wings, the wings consequently are asymmetric.

    Click here for more information about the F-16XL.

    The AD-1 has a pivoting wing and, depending on the position of the wing, the aircraft may or may not be symmetric. The Ames-Dryden (AD) -1 is a research aircraft designed to investigate the concept of an oblique, or "pivoting" wing. The oblique wing was first conceived by NASA aeronautical engineer Robert T. Jones, who theorized that by using an oblique wing, supersonic transport might achieve twice the fuel economy of an aircraft sporting more conventional wings. The oblique wing pivots about its center point, the fuselage, remaining perpendicular to it during slow flight, and swinging to angles of up to 60 degrees as aircraft speed increases. Thus, the oblique wing can be positioned at the most efficient angle for the speed at which the aircraft is flying.

    Click here for more information about the AD-1.

  • Engage students in a "Symmetry Scavenger Hunt" in which they search the NASA Dryden Flight Research Center's photo gallery and identify various planes and spacecraft that are, or are not, symmetric. Students might begin by linking to the photos of the aircraft listed below that are, or have been, tested at NASA Dryden. Are these aircraft symmetric? Where are the line(s) of symmetry located?

    • Centurion
    Gossamer Albatross
    Perseus
    X-36 Tailless Fighter

    Click here to learn more about Centurion.
    Click here to learn more about the Gossamer Albatross.
    Click here to learn more about Perseus.
    Click here to learn more about the X-36 Tailless Fighter.


    Enrichment Activity: Students might partake in a Symmetry Scavenger Hunt in which they search for objects in the classroom that are symmetric. Ask each student to show the line(s) of symmetry in the object.

    As a homework assignment, ask students to bring in objects from home that are symmetric. Allow each student to show the line(s) of symmetry in the object.


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    • Funded by the NASA Dryden Flight Research Center


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