Butterfly spirals

Abstract

In this article special sequences involving the Butterfly theorem are defined. The Butterfly theorem states that if M is the midpoint of a chord PQ of a circle, then following some definite instructions, it is possible to get two other points X and Y on PQ, such that M is also the midpoint of the segment XY. The convergence investigation of those sequences reveals bounded piecewise spirals, and is done by using the free dynamic mathematics software GeoGebra. It is also supported with rigorous reasoning concerning the convergence of a special sequence. Moreover, it raises related problems that encourage further investigation. This article suggests a content that can be useful for students at secondary as well as undergraduate levels since it incorporates principles taught to students during their first courses in Algebra, Calculus and Geometry especially those related to convergent sequences, the algebra of complex numbers, plane isometries and basic Euclidean geometry.

Keywords:

• Butterfly theorem
• Butterfly spirals
• bounded piecewise spirals
• sequence convergence
• GeoGebra geometry applet
• Butterfly iterates

2010 Mathematics Subject Classifications:

• Convergence and divergence of infinite limiting processes
• Geometry
• Technological tools

Acknowledgments

The author is thankful to the referees and the editor who read the manuscript carefully, and whose valuable comments improved it substantially. Many thanks to everyone who read, commented and helped me improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author.