Mathematical Patterns in African American Hairstyles
MATHEMATICAL<br>PATTERNS
IN AFRICAN AMERICAN HAIRSTYLES
by GLORIA<br>GILMER
MATH-TECH, MILWAUKEE
BACKGROUND.
The discipline of mathematics includes<br>the study of patterns. Patterns can be found everywhere in nature.<br>See Figure 1 with two bees in a beehive. Often these patterns<br>are copied and adapted by humans to enhance their world. See the<br>pineapple in Figures 2a and the adapted hairstyle in Figure 2b.<br>Ethnomathematics is the study of such mathematical ideas involved<br>in the cultural practices of a people. Its richness is in exploring<br>both the mathematical and educational potential of these same<br>practices. The idea is to provide quicker and better access to<br>the scientific knowledge of humanity as a whole by using related<br>knowledge inherent in the culture of pupils and teachers.
fig1 TWO BEES IN A BEEHIVE
FIGURE 2a PINEAPPLE
TESSELATING HEXAGONS
FIGURE 2b GIRL
Going into a community, examining its languages<br>and values, as well as its experience with mathematical ideas<br>is a first and necessary step in understanding ethnomathematics.<br>In some cases, these ideas are embedded in products developed<br>in the community. Examples of this phenomena are geometrical designs<br>and patterns commonly used in hair braiding and weaving in African-American<br>communities. For me, the excitement is in the endless range of<br>scalp designs formed by parting the hair lengthwise, crosswise,<br>or into curves.
INTRODUCTION.
The main objective of my work with black<br>hair is to uncover the ethnomathematics of some hair braiders<br>and at the same time answer the complex research question: "What<br>can the hair braiding enterprise contribute to mathematics education<br>and conversely what can mathematics education contribute to the<br>hair braiding enterprise?" It is clear to me that this single<br>practical activity, can by its nature, generate more mathematics<br>than the application of a theory to a particular case.
My collaborators include Stephanie Desgrottes,<br>a fourteen year old student of Haitian descent, at Half Hollow<br>Hills East School in Dix Hills, New York and Mary Porter, a teacher<br>in the Milwaukee Public Schools. We have each observed and interviewed<br>hair stylists at work in their salons along with their customers.<br>Today's workshop for middle school teachers will focus on the<br>mathematical concept of tesselations which is widely used and<br>understood by hair braiders and weavers but not thought of by<br>them as being related to mathematics.
TESSELATIONS.
A tesselation is a filling up of a two-dimensional<br>space by congruent copies of a figure that do not overlap. The<br>figure is called the fundamental shape for the tesselation. In<br>Figure 1, the fundamental shape is a regular hexagon. Recall that<br>a regular polygon is a convex polygon whose sides all have the<br>same length and whose angles all have the same measure. A regular<br>hexagon is a regular polygon with six sides. Only two other regular<br>polygons tesselate. They are the square and the equilaterial triangle.<br>See Figures 3a and 3b for parts of tesselations using squares<br>and triangles. In each figure, the fundamental shape is shaded.<br>To our surprise two types of braids found to be very common in<br>the salons we visited were triangular braids and box braids which<br>describe these tesselations on the scalp!
FIGURE 3a TESSELATING SQUARES
FIGURE 3b TESSELATING TRIANGLES;
FIGURE 4 TESSELATING A NON-STANDARD FIGURE.
Box Braids.
In the tesselations we saw, the boxes were<br>shaped like rectangles and the pattern resembled a brick wall<br>starting with two boxes at the nape of the neck and increasing<br>by one box at each successive level away from the neck. The hair<br>inside the box was drawn to the point of intersection of the diagonals<br>of the box. Braids were then placed at this point. You may notice<br>in Figure 3a that braids so placed will hide the scalp<br>at the previous level in the tesselation. In this style, the scalp<br>is completely hidden. In addition, we were told that braids so<br>placed are unlikely to move much when the head is tossed.
Triangular Braids.
In the tesselations we saw, the triangles<br>were shaped like equilateral triangles and the pattern resembled<br>the one shown in Figure 3b . The hair inside the triangle<br>was drawn to the point of intersection of the bisectors of the<br>angles of the triangle. Again, this style allowed hair to move<br>less liberally than hair drawn to a vertex and then braided.
Tesselations can be formed by combining translation,<br>rotation, and reflection images of the fundamental shape. Variations<br>of these regular polygons can also tesselate. This can be done<br>by modifying one side of a regular fundamental shape and then<br>modifying the opposite side in the same way. See Figure 4 .
CLASSROOM ACTIVITIES.
1. Draw tesselations using different fundamental<br>shapes of squares and rectangles.
2. Draw a tessalation using an octagon and<br>square connected along a side as the fundamental shape.
3. Draw tesselations with modified squares<br>or...