Number Systems Using a Complex Base

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.)
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.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+