GEOMETRY/ HISTORY LESSON PLAN

TITLE: Symmetry

LEVEL: Middle School Math or Informal Geometry

OBJECTIVES: Explore translations, rotations and reflections. Explore line symmetry and rotational symmetry.

Student will identify when a figure has been translated, rotated and/or reflected.

Student will identify and state the number of any lines of symmetry of a figure.

Student will identify and state the degree of any rotational symmetry of a figure.

Student will design a pattern using some combination of translation, rotation and reflection.

History motivates students to both appreciate and learn ideas in mathematics. Much is discovered by observing the patterns surrounding us in nature. By trying to solve more difficult problems, many postulates and theorems have been examined and proven. When you study the history surrounding a particular idea, such as symmetry, your student can get a sense of the excitement and a feel of the creativity that surrounded that particular idea when it was a "hot" topic.

I would begin this lesson on symmetry by asking my students a series of questions similar to those posed in Ian Stewart's, Nature's Numbers.¹

Do you think that early man noticed that starfish are equipped with a set of symmetric arms?

Do you ever wonder when man first noticed that if you twirled a daisy in your hand, it looked the same no matter how far you turned it, but if you twirl an iris, it looks different at certain times?

Do you think early man studied how his footprints looked as he walked along the shore?

When do you think he first noticed the pattern of his camel's footsteps as they crossed the desert?

What do tile floors, wallpaper, snowflakes, virus, crystals, and stacks of oranges at the supermarket have in common?

Perhaps the history of the study of symmetry should begin with the Egyptians. Knowing that ancient Egyptians covered their walls and floors with tile makes one wonder if ancient mathematicians, such as Thales, looked at the "concrete geometry of the Egyptians ... and observed that six equilateral triangles could be placed round a common vertex. Although the theorem that 'only three kinds of regular polygons --the equilateral triangle, the square, and the hexagon-- can be placed about a point so as to fill a space,' is attributed by Proclus to Pythagorus or his school (Proclus, ed.Friedlein, p.305), yet it is difficult to conceive that the Egyptians --who erected the pyramids-- had not a practical knowledge of the fact that tiles of the forms above mentioned could be placed so as to form a continuous plane surface."² The ancient Greeks also discovered much about symmetry while trying to construct the five regular polyhedra, namely, the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. Much was also learned in trying to ascertain whether these were the only regular solids that could be constructed.

Thinking about these regular solids and the Archimedean (semi-regular) solids, perhaps, then led to the study of the patterns involved in packing objects in an efficient manner. If you look at a pile of oranges at a supermarket and wonder if this is the most efficient method of packing them, you will be investigating a subject that interested mathematicians "at least as far as work by Kepler in the seventeenth century".³ "Kepler's interest in sphere packing, it has to be said, was not driven by an overpowering interest in piles of fruit, but it was, nonetheless, motivated by an equally real phenomenon, that of the shape of a snowflake."⁴ A snowflake is a regular six sided figure, a hexagon, and Kepler noticed that the symmetry of a snowflake was such that, if you rotate a snowflake through a 60^o angle, its appearance remains unchanged. "His idea was that natural forces impose a regular geometric structure on the growth of such seemingly diverse objects as snowflakes, honeycombs, and pomegranates."⁵

So too, man, in noticing the geometric shape of crystals, may have speculated on their internal structure and their degree of symmetry. "The precise geometrical form --the mirrorlike perfection of the flat facets, and the overall shape which shows in different species varying degrees of symmetry-- was to the early naturalists the most enigmatic, the most significant feature of crystals. Here was a demonstration that geometry is not just an abstraction conjured up by mathematicians, for it appears spontaneously in rocks. Here was a strong hint that in geometry is a clue to the inner structure of solid matter"⁶. Perhaps the repetitive patterns studied by the Greeks formed the basis for analyzing the internal structure of crystals. "The... surmise that the spontaneous appearance of plane surfaces and a symmetrical shape was as indication of a regular and precise pattern of internal structure ...seems to have been published by Robert Hooke in London in a book called Micrographia " about 350 years ago.⁷ (At this point in the story, I would try to borrow from the science teacher her various samples of crystals since our science curriculum includes the study of crystals in 8th grade.) The essential characteristic of crystals is their regular internal structure. They are "...packed in precise ways to form solid internal patterns, repeating over and over again in all directions in space. The plane surfaces are just simple ways of finishing off the stack...and the symmetries shown by crystal shapes are indications of the symmetries of the stacking pattern."⁸

Crystallographers, scientists who study crystals, noticed, like the Greeks, that "regular pentagons cannot pack without leaving spaces. This is why fivefold axes do not occur in crystals."⁹ "Consequently, the mathematics of symmetry plays an important role in crystallography. Indeed, most of the work on classifying the 230 different symmetry groups was carried out by crystallographers, during the nineteenth century."¹⁰ In fact, much of the language of symmetry was the result of such work.

Also, by delving into the history of symmetry, we can see how it developed from two dimensions to three dimensions. Man was able to observe the seven possible strip patterns and the seventeen possible wallpaper patterns and discover the thirty-two types of repetitive patterns of crystals. By studying the various patterns given to us through tilings, wallpaper, and border prints (also known as frieze patterns), mathematicians were eventually able to generalize from two-dimension to three-dimension and, in the later part of the 1900's, to n-dimensional space.¹¹

There is much to be learned in taking a course in the history of geometry. The cavalier attitude I once had regarding the importance of symmetry has been replaced with a new respect for its place in the development of many different fields of study. Perhaps, this is best summed up in the following quotation. "Proclus made a long list of what he called the lower and humbler applications of his science. They included geodesy, mechanics, optics, astronomy, perspective and fortification. There is surveying, mensuration, gauging, geography, cartography and 'stratarithmetic', the art of military fortifications; also, perspective, astronomy, astrology, statics, architecture, navigation and... stage tricks."¹²

Students should have had lessons on angle relations, polygons and equilateral triangles.

1. Worksheets #1 and #2, from Appendix B, photocopied for students

2. Directions for above worksheets in Appendix B

3. Scissors

4. Any examples of symmetry from the "real" world, such as, wallpaper patterns, border prints, pictures of rugs, and crystal samples. If you have access to the internet, I suggest visiting the following sites:

http://forum.swarthmore.edu/geometry/rugs/gallery/04lg.html

http://www/ucs.mun.ca/~mathed/Geometry/Transformations/frieze.html

I have attached pages, in Appendix A, from these sites as excellent examples of symmetry in rug weaving, architecture and wallpaper.

5. Graph paper

Begin this lesson by using some of the information detailed in the HISTORICAL PERSPECTIVE. Any examples from the "real" world should be passed around the room so that students will have a visual representation of your discussion. Discuss the meanings of translation, rotation, reflection and glide reflection. Give students a piece of graph paper and have them practice these transformations with a letter of the alphabet. For an example of this activity, I have included, in Appendix A, a copy of two pages from "Transformations" by John Grant McLoughlin at the internet site: http://www.ucs.mun.ca/ ~mathed/ Geometry/ Transformations/Transformations.html#translation.

Using Worksheets#1 and #2, explore line symmetry and rotational symmetry by rotating and reflecting the equilateral triangle and observing various possible combinations of rotation and reflection. The directions for using the EQUILATERAL TRIANGLE SYMMETRY GROUP is attached in Appendix B.

If computers are available, the following programs offer a wealth of practice with transformations:

1. The Geometer's Sketchpad, c. 1995 Key Curriculum Press

I have used the book, EXPLORING GEOMETRY WITH THE GEOMETER'S SKETCHPAD, by Dan Bennett, c. 1999 Key Curriculum Press to explore translations(p.33), rotations (p.34), reflections (pp.34-35), and glide reflections (pp.48-49). Examples of my use are included in Appendix C.

2. The Factory Deluxe, Sunburst Communications

Students can experiment with transformations in a "game" type atmosphere, where they are in charge of manufacturing geometrical objects. They do this with four types of "machines". The first rotates a figure through an angle determined by the student. The second punches squares of triangular holes through the figure. The third paints stripes of varying widths on the figure. The fourth cuts off and

discards various areas of the figure.

3. If the internet is available to the students, they can visit the site:

scienceu.com/geometry

Students can work on activity sheets from the link, Discovering Frieze Patterns with Kali. I have included examples of these in Appendix C. Students can also design their own frieze and wallpaper patterns at this site. Be forewarned, that this is a commercial site and there were advertisements selling tools to make geometric models or offers to print the patterns on a T-shirt for a cost to the students.

1. Stewart, Ian, Nature's Numbers, The Science Masters Series, c. 1995BasicBooks

2. Allman, George Johnston, Greek Geometry from Thales to Euclid, c. 1976 Arno Press Inc., p.12

3. Devlin, Keith, Mathematics: The Science of Patterns, c. 1994, 1997 Scientific American Library, p. 152

4. Ibid., p.157

5. Ibid., p.158

6. Bunn, Charles, Crystals: Their Role in Nature and in Science, c. 1964 Academic Press Inc., p.4

7. Ibid., p.5

8. Ibid., p.8

9. Ibid., p. 86

10. Devlin, Keith, Mathematics: The Science of Patterns, c.1994, 1997 Scientific American Library, p. 164

11. Joyce, David E., "Wallpaper Groups: History of Crystallographic Groups and Related Topics", c. 1997, Internet site: http://aleph0.clarku.edu/~djoyce/wallpaper/

12. Heilbron, J.L., Geometry Civilized: History, Culture, and Technique, c. 1998 Clarendon Press, Oxford

1. Have students take note of the marking of the vertices of the triangle on both sides of worksheet #1 for their convenience in making observations. The start position is when all vertices are in this original position. When making observation, note how the marked vertices move with respect to the outside markings. For example, A-->B means that the vertex marked A inside the triangle moved to the vertex B position, which is marked on the outside of the triangle.

2. Have students cut out triangle. (An easy way to do this is to fold the worksheet in half and cut along the sides of the triangle without cutting the sides of the worksheet.)

3. With triangle at the start position have students rotate the triangle in a clockwise direction 120^o. Ask the students to tell you what happens to the vertices.

(Answer: A-->C, B-->A, C-->B). Call this rotation R¹.

4. With triangle in the start position have students rotate the triangle in a clockwise direction 240^o. Ask the students to tell you what happens to the vertices.

(Answer: A-->B, B-->C, C-->A). Call this rotation R².

5. With triangle in the start position have students rotate the triangle in a clockwise direction 360^o. Ask the students to tell you what happens to the vertices.

(Answer: A-->A, B-->B, C-->C). Ask if this reminds them of any of the properties they have studied? (Answer: Identity Property). Call this rotation R³.

6. With triangle in the start position have students reflect the triangle with X as the line of symmetry or mirror line. To do this the student flips the triangle over, keeping vertex A at the same place. What happens to the vertices? (Answer: A-->A, B-->C, C-->B). Call this reflection X.

7. With triangle in the start position have students reflect the triangle with Y as the line of symmetry or mirror line. To do this the student flips the triangle over, keeping vertex B at the same place. What happens to the vertices? (Answer: A-->C, B-->B, C-->A). Call this reflection Y.

8. With triangle in the start position have students reflect the triangle with Z as the line of symmetry or mirror line. To do this the student flips the triangle over, keeping vertex C at the same place. What happens to the vertices? (Answer: A-->B, B-->A, C-->C). Call this reflection Z.

9. Now have the students try a combined rotation and reflection. They should notice, no matter what combination they try, the vertices will move in one of the six ways in the answers from steps #3 - #8.

10. Using Worksheet #2, have students work in groups to complete the chart of the possible combinations using any two of the operations from #3- #8.

NOTE: To read the table proceed as follows:

Rows Columns

denotes that the first move was a reflection of the triangle over Y, and the second move was a reflection of the triangle over Z and the result was equivalent to rotating the triangle 120^o or an R¹ .

NOTE: For uniformity of notation, please write any combination of moves (reflection or rotation) as the following example shows:

Rows Columns

denotes that the first move was the rotation, R², and the second move was a reflection of the triangle over X and the result was equivalent to reflecting the triangle over Y. (In other words, after doing these two moves the vertices have moved as follows: A-->C,

B-->B, C-->A). So the letter Y should be placed in the column labeled R², and in the row labeled X..