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Complex Continued Fractions
Author

Doug Hensley
Revision date

2005-07-21
Description

Schmidt Circles The Asmus Schmidt algorithm effectively partitions the plane into regions, then subregions, and so on, and in this way, each point is encoded by the sequence of regions to which it belongs. Since it cuts it infinitely fine in the end, visualizing the partition requires some compromises.
It is by this method that Complex Continued Fractions are calculated. Seven matrices are used.
  • v1={{1,I},{0,1}}
  • v2={{1,0},{-I,1}}
  • v3={{1-I,I},{-I,1+I}}
  • e1={{1,0},{1-I,I}}
  • e2={{1,-1+I},{0,I}}
  • e3={{I,0},{0,1}}
  • c={{1,-1+I},{1-I,I}}
Each use of these matrices divides the plane into seven parts. Mathematica is well-suited for studying successive generations. A PDF and EPS of many generations is shown.
Subjects

*Mathematica Technology > Programming > 3D Graphics
*Mathematics > Calculus and Analysis > Complex Analysis
URL

http://www.maa.org/editorial/mathgames/mathgames_03_15_04.html
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schmidtcircles.pdf (403.9 KB) - PDF Document
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schmidtcircles.eps (199 KB) - Enhanced postscript file
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schmidtcircles.nb (261.5 KB) - Mathematica Notebook