id: 06455410
dt: j
an: 2015e.00544
au: O’Dell, Robin S.
ti: Not so complex. Iteration in the complex plane.
so: Math. Teach. (Reston) 107, No. 8, 592-599 (2014).
py: 2014
pu: National Council of Teachers of Mathematics (NCTM), Reston, VA
la: EN
cc: F50 M80 G60 G90
ut: number concepts; complex numbers; complex plane; iteration; orbits;
mathematics and art; complex number multiplication; geometric
interpretation of orbits; rotating polygons
ci:
li: http://www.nctm.org/publications/article.aspx?id=41400
ab: From the text: The simple process of iteration can produce complex and
beautiful figures. In this article, I present a set of tasks requiring
students to use the geometric interpretation of complex number
multiplication to construct linear iteration rules. When the outputs
are plotted in the complex plane, the graphs trace pleasing designs
reminiscent of hypotrochoids, pursuit curves, and star polygons.
Students are challenged to duplicate designs, and in doing so they
explore and generalize patterns, apply basic trigonometry and number
theory ideas, and integrate their findings with the tools of iteration
‒ iteration rules, seeds, orbits, and fates. The first part of this
article covers the basics of complex linear iteration and the geometric
interpretation of complex number multiplication. Included are a number
of relatively simple starter tasks for students to master before
creating the beautiful designs presented in the second part of the
article. The second part presents four sets of designs for students to
create using their knowledge from part 1.
rv: