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Colors, geometric forms, art, and mathematicsAuthor(s): THOMAS P. HILLMANSource: The Arithmetic Teacher, Vol. 14, No. 6 (OCTOBER 1967), pp. 448-452Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187405 .

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Colors, geometric forms, art, and mathematics

THOMAS P . HILLM AN

Njala University College, Freetown, Sierra Leone

'' ith the rising popularity of "pop" art, "op" art, and stunning geometric de- signs in feminine fashions, the author is reminded how easily art and mathematics can be closely related, especially in the upper elementary school or junior high school classrooms. For students who have been convinced that mathematics is al- ways associated with calculations or mani- pulations (or even hallucinations!), the following suggestions can arouse new interest in mathematics - and perhaps re- establish vigorous efforts when regular mathematical topics are pursued again. These lessons are introductory art lessons that have mathematical overtones, and it is important that the student not become too burdened with the mathematical ideas at the expense of losing interest in art. The concepts in both subject areas are simple and basic, and each student is given an opportunity to discover relationships for him- self - in art as well as in mathematics.

Lesson 1 : An introduction to color

Color plays an important part in the world of the artist, and a good artist has learned the basic relationships between colors. He understands the physical reac- tions caused by certain colors and the resulting psychological effects. The object of this lesson is to identify the set of primary colors and evolve the set of secondary colors.

Each student is given three containers of paint - red, blue, and yellow - and brushes and white paper. The teacher identifies the colors of the paint as the primary colors and writes more formally on the chalkboard:

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P = {Red, Blue, Yellow} (where P is the set of primary colors). With a casual remark such as "I wonder what will happen if we mix these different colors?" or "I wonder how we could make a can of orange paint?" the teacher places on the chalkboard a mathematical table (shown in Fig. 1) that defines an operation on a set of elements. (The teacher's role in this lesson is to refrain from audi- bly emphasizing the mathematical notation of the situation, but to be very precise via the notation on the board.) Recalling other tables used in the study of mathematics, the teacher explains the notation, if necessary, and remarks that this is one way of recording the results of mixing different colors of paints.

* R Y B

R

Y

В

Figure 1

The operation * can be explained as a mixing operation. R * В means "Take a drop of red paint and place it on the clean white paper; then take a drop of blue paint and mix it with (or add it to) the drop of red paint." В * R means to do the same thing but place the drops on the paper in reverse order. (Drops of paint that are mixed together should be as nearly equal as possible.) Each student is left to complete the chart on his own.

The Arithmetic Teacher



The completed table is given in Figure 2.

* R Y B

R R О V

Y O Y G

В V G В

Figure 2

Students are asked to observe what has happened. The teacher can discuss with the class the new colors and define them to be a set S, where S = {Violet, Orange, Green), the set of secondary colors.

After discussing the results of mixing the different paints, the teacher should ask students if they can observe what is true about the mathematical system defined by the table in Figure 2. (A mathematical system has three characteristics: a set of elements, an operation, and some proper- ties.)

It is hoped students will observe that the mathematical system is not closed under the operation * (since the resulting colors are not members of the original primary colors) but that the operation * is a commutative operation. There is no identity element for the operation *.

A very natural question to ask at this time is, "Can we tell from the table whether * is an associative operation?" It should be discerned by the students that since *

is not a closed operation, it is impossible to find from the table an answer to that question. For example, the problem "Does (R * B) * Y = R * (B * Y)?" cannot be solved, since there is no defined result for V * Y (as well as R * G).

Since the students have been presented with a problem that requires additional information not obtainable in the table, suggestions for solving the problem with a different mathematical system are in order. (Hopefully, the students will want

October 1967

to mix the six colors together to see what happens - regardless of whether they de- sire an answer to the mathematical question! If the teacher has been executing his role correctly, showing interest in mixing the colors, filling in the table, and expressing amazement at the results, there should be no problem in motivating the students to solve the new problem.) The teacher can suggest constructing a new table (shown in Fig. 3) and encourage students to begin the task set before them.

* R О Y G В V

R

О

Y

G

В

V

Figure 3

Some teachers may want to challenge good students to fill in the table before mixing the paints and to guess at the an- swers to questions such as these:

Do you think the operation is closed? Do you think the operation is commutative? Will there be an identity element? Will the operation be associative?

Teachers may complete the chart on their own and make their own observa- tions. (Teachers should have and enjoy the same experiences that will be ex- pected of their students!) Among the set of new colors formed is the set of tertiary colors, Set T, which is customarily defined as T = {Red-Violet, Red-Orange, Yellow-Orange, Yellow-Green, Blue- Violet, Blue-Green). Students who wish to solve the problem without mixing the paints will have to be careful with ratios, since each resulting color is a mixture of blue and red, red and yellow, or yellow and blue.

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(It can also be a combination problem: RRRR, RRRY, RRYY, RYYY, YYYY, YYYB, etc.) Students can name the new colors, and students should check with neighbors to see if they have similar results - the same colors appearing in certain positions in the table.

The entire lesson should be a fun ex- perience for the students, a wonderful laboratory opoortunity to examine relationships between colors, and, in addition, a chance to see an application of some basic mathematical ideas. Students who wish to continue the study of the relationship between colors should be encouraged to do so. For example, the teacher might suggest to students to define W, the set of warm colors, or C, the set of cool colors.

Lesson 2: An introduction to form

An artist is interested in geometric form as well as color. When an artist represents an object on canvas, he has to reproduce a three-dimensional object on a two-dimensional surface. The height, width, and length of an object must be captured (except when an artist specializes in non- representational painting), and the artist begins this task by considering the basic geometric form of the object. Artists know that most common objects can be reduced to one of four basic geometric forms, or a modification or combination of these forms. No matter what the object may seem to be, it can be represented by a sphere, a cone, a cube, a cylinder, or a combination of these.

Potential artists achieve realism in their painting more easily if they learn to draw the basic forms on paper. First they identify real-world objects in terms of their pure geometrical forms and represent them on paper in terms of their pure forms. Finally, they modify the form to represent more closely the real-world object, and in this way, they will improve their drawing skills.

Each student is shown the four basic forms. Wood or plastic models are used, and students are given an opportunity to handle them, study them, and observe

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their characteristics. Finally each student is asked to represent each form on paper. No suggestions of any kind are given, but the teacher mentions trying to achieve a three-dimensional look on a two-dimensional surface. (Sketches can be drawn from several vantage points: eye level with the object, looking down on the object, etc.) When all students have been given an opportunity to draw the basic forms, the teacher can select certain students to show their work and explain their technique. A discussion should bring out the idea that sometimes the artist must see through the shape and draw more than meets the eye. For example, a cube may be drawn by one student as shown in Figure 4. Another student may see

Figure 4

through the object and represent the cube in the manner illustrated in Figure 5. Sim-

Figure 5

ilarly, the cone, cylinder, and sphere may be given depth by supplementing the figure with additional descriptive lines, as shown in Figure 6. Every student should be given

Figure 6




opportunities to practice drawing basic forms until reasonable likenesses are made.

Before the student is asked to draw real-world objects in terms of their pure geometric forms, he must learn to identify these forms correctly. A classroom exercise can quickly help students identify the pure geometric forms of objects. Ob- jects that can be used are a globe, desk, chair, light, doorknob, drinking glass, book, and so on.

At this time students are given opportunities to sketch objects, first representing them in their pure geometric forms and then filling in the details. The teacher can illustrate with a drinking glass. A cylinder is drawn first and then the simple modifications are made. (See Fig. 7.)

Figure 7

It should be emphasized that seeing the object first in terms of its pure geometric form and representing it on paper in terms of this form, then modifying the form, is the best way to become a good artist.

The following exercise can give each student additional practice and furnish opportunities to use his imagination. The four geometric models are placed upon a plane (table) in some arrangement. Each

Figure 8

October 1967

Figure 9

student is asked to represent the geometric forms on paper and translate the arrangement into a real-world setting. The final result may be a still life or a country scene, like the one in Figure 8. Rearrange- ments of the objects can require the student to extend his imagination and create new and different pictures.

This additional practice prepares the student to represent real objects that are combinations of the four basic forms. For example, a lamp on a table is a combination of the four basic forms. This is pictured in Figure 9, above.

Summary

The above lessons clearly illustrate how mathematics and art can be interrelated. A good teacher will not hesitate to connect these subjects and will use art to increase an interest in mathematics and vice versa. The language of the mathe- matician can be associated with the world of the artist, and the books listed in the bibliography will help the teacher expand his ability to connect these subjects. Both

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subjects will delight young students when presented correctly, imaginatively, and en- thusiastically.

Bibliography Arnheim, Rudolf. Art and Visual Perception.

Berkeley, Calif.: University of California Press, 1954.

Gardner, Helen. Art Through the Ages. New York: Harcourt, Brace & World, 1964.

Goldstein, Harriet. Art in Everyday Life. New York: The Macmillan Co., 1940.

Kline, Morris. Mathematics - a Cultural Ap- proach. Reading, Mass.: Addison-Wesley Pub- lishing Co., 1962.

Lamancusa, Katherine С. Source Book for Art Teachers. Scranton, Pa.: International Textbook Co., 1965.

Weyl, Herman. Symmetry. Princeton, N. J.: Princeton University Press, 1952.

Life-membership plan instituted The NCTM Life Membership Plan suggested by many members has now been adopted. This plan enables one to become a life member of the NCTM by making a single payment, which is determined on a sliding scale accord- ing to the applicant's age. Details will be sent to most members during the coming year with their membership renewal notices but will be furnished at any time, upon request, by the NCTM Washington office.

Foreign texts and journals collected The Committee on International Mathematics Education has established a collection of foreign mathematics texts and journals. This collection may be examined at the Washington office of the NCTM. Parts of the collection are displayed at each Annual Meeting and many of the Name-of-Site Meetings of the Council.

A bibliography of the collection may be obtained free of charge from the display at NCTM meetings or by writing to the National Council of Teachers of Mathematics, 1201 Sixteenth Street, N.W., Washington, D. C. 20036.

Science and mathematics teaching conference to be held in October

Dates for the Fourteenth Annual Conference for the Advancement of Science and Mathe- matics Teaching (cosponsored by the Texas Council of Teachers of Mathematics) have been announced by the general chairman, Dr. David L. Miller, professor of philosophy at The Uni- versity of Texas at Austin.

The science session will be held October 12-14 and the mathematics session October 19-21. Both will be conducted in the convention hall of Terrace Motor Hotel at Austin.

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Other sponsors of the conference, which at- tracts teachers from all over the state, are the Science Teachers Association of Texas, Texas Council of Teachers of Mathematics, The University of Texas, Texas Science Super- visors Association, Texas Association of Super- visors of Mathematics, and the Texas Academy of Science, in cooperation with the Texas Education Agency, National Science Teachers Association, and National Council of Teachers of Mathematics.


