Number Systems Using a Complex Base

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The points are the set , which is the approximation of the fractional parts of all the numbers in the base system with digits, , . (Integer parts () are suppressed because the result would be almost the same.)

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Use the locator to change .

The controls on the left let you control manually, set its absolute value or angle, or choose the number of digits and levels .

You can see how the convex hull is constructed by selecting the number of supporting lines, which also displays its area and the length of the circumference. The next slider lets you choose one of the periodic cases and see the equation for and the Hausdorff dimension of the boundary.

Finally, you can choose to draw lines to the origin or to the center of symmetry, choose the size of points, specify the number of most important digits to be distinguished by colors, manipulate the visible range, or turn the axes on and off.

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Contributed by: Jarek Duda (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

For more information on the theory behind this Demonstration, see:

http://xxx.arxiv.org/pdf/0712.1309

The width function and the equation for the center of symmetry are restricted to a finite number of levels for better appearance, but the length of the circumference and the area of the convex hull are calculated for the full convex hull.

The method of calculating analytically the Hausdorff dimension of the boundary using recurrence is explained in:

http://demonstrations.wolfram.com/TheBoundaryOfPeriodicIteratedFunctionSystems



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