Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-14T07:53:21.790Z Has data issue: false hasContentIssue false
  • English
  • Français

Iterating circum-medial triangles

Published online by Cambridge University Press:  21 October 2019

Hans Humenberger  and
Franz Embacher
Hans Humenberger
Affiliation:
University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, A – 1090 Vienna, Austria e-mail: hans.humenberger@univie.ac.at; franz.embacher@univie.ac.at
Franz Embacher
Affiliation:
University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, A – 1090 Vienna, Austria e-mail: hans.humenberger@univie.ac.at; franz.embacher@univie.ac.at
Get access Rights & Permissions [Opens in a new window]

Extract

When considering ‘convergence’ many people think of number sequences or contexts arising from calculus. But there are also interesting phenomena of convergence – easy to visualize – arising in elementary geometry. Some of them are so elementary that they can be dealt with at school, for instance an example of iteration that is described in [1, p. 171f] (see also [2, p. 59], [3, p. 222f] and [4, p. 42ff]).

Type
Articles
Information
The Mathematical Gazette , Volume 103 , Issue 558 , November 2019 , pp. 480 - 487
Copyright
© Mathematical Association 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ismailescu, D., Jacobs, J., On sequences of nested triangles, Periodica Mathematica Hungarica, 53 (1–2) (2006) pp. 169–184.10.1007/s10998-006-0030-3CrossRefGoogle Scholar
Jones, Stephen, Two iteration examples, Math. Gaz. 74 (March 1990) pp. 58–62.10.2307/3618858CrossRefGoogle Scholar
Abbott, S., Averaging sequences and triangles, Math. Gaz. 80 (March 1996) pp. 222–224.CrossRefGoogle Scholar
De Villiers, M., Over and over again: two geometric iterations with triangles, Learning & Teaching Mathematics 16 (July 2014) pp. 40–45, also available at http://dynamicmathematicslearning.com/geometriciteration-examples.pdfGoogle Scholar
Humenberger, H., Embacher, F., Convergence with respect to triangle shapes – elementary geometric iterations, Learning & Teaching Mathematics 25 (December 2018) pp. 15–19.Google Scholar
Trimble, S. Y., The limiting case of triangles formed by angle bisectors, Math. Gaz. 80 (November 1996) pp. 554–556.CrossRefGoogle Scholar
Leversha, G., The geometry of the triangle, United Kingdom Mathematics Trust (2013).Google Scholar
Bourgeois, G., Lechêne, J.-P., Etude d’une itération en géométrie du triangle, Bulletin de l’APMEP 1(409) (1997) pp. 147–154 also available at https://www.apmep.fr/IMG/pdf/AAA97013.pdfGoogle Scholar
Humenberger, H., Embacher, F., Konvergenz bei Dreiecken – interessante geometrische Iterationen, Mathematische Semesterberichte 65 (2) (2018) pp. 253–275 also available at http://link.springer.com/article/10.1007/s00591-017-0206-3Google Scholar
Bourgeois, G., Orange, S., Dynamical systems of simplices in dimension two or three. In: T. Sturm, Ch. Zengler, (Ed., 2011): Automated deduction in geometry. 7th International Workshop, Shanghai, China, September 22–24, 2008. Springer Berlin-Heidelberg, 1–21 also available at https://arxiv.org/pdf/0906.0352.pdfCrossRefGoogle Scholar
Honsberger, R., Episodes in nineteenth and twentieth century Euclidean geometry, The Mathematical Association of America (1995).Google Scholar
Gomez, M. F., Taslakian, P., Toussaint, G. T., Convergence of the shadow sequence of inscribed polygons, 18th Fall Workshop on Computational Geometry, New York, EEUU (2008). http://ac.aua.am/ptaslakian/web/publications/shadowConvergence.pdf Google Scholar