An Example of Using the
History of Iranian Mathematics for the Math Classroom
Dividing a right angle into five equal angles with only a straightedge and a compass is used to construct particular tiling patterns in Islamic arts, and one of these patterns is presented in figure 1 (10-petal rose construction). The methods and ideas that I explain in this paper were obtained from Iranian math history (Jazbi, S. A. (translator), Applied Geometry, appendix2. Soroush Press, ISBN 964 435 201 7, Tehran 1997). All figures have been created by the author using the Geometer’s SketchPad (GSP) software program. These samples have been used at the Isfahan Math House (IMH) in workshops for teaching math history to secondary students and mathematics teachers.
Figure 1. 10-petal rose construction (girih construction)
1) Task 1: Dividing a right angle into five congruent angles with only a straightedge and a compass Construct arbitrary arc (figure 2).
2) Construct D the midpoint of OA then find D‘ as OD = OD‘ (figure 3).
3) Then construct a circle with center D‘ and radius D‘B. This circle cuts OA at point E (figure 4).
4) Now construct segment BE (figure 5).
5) Finally, construct a circle with center B and radius BE, and label the intersection point of the green circle and new circle (in magenta), F (figure 6).
6) Construct segment OF, and then (figure 7). (Prove it!)
7) Now divide into four equal angles (explain your work!). Now you have five angles.
- a) Task 2: Dividing a right angle into six congruent angles with only a straightedge and a compass
1) Construct a circle with center O and radius OB (figure 8).
2) Construct a circle with center B and radius BO. Label the intersection point of the two circles, D (figure 9).
3) Construct segment OD, then . Why? (figure 10)
4) Construct the bisector, and repeat again for created angles. Construct bisector, then you have six angles (figure 11).
Figure 11. Dividing a right angle into six angles.
Dividing a right angle into six equal angles can be used to construct Islamic art patterns. One of them is named a 12-petal rose pattern like the one shown in figure 12.
Figure 12. 12-petal rose pattern
Narges Assarzadegan, Math teacher, math history researcher, Isfahan Mathematics House (IMH) (Iran)