Advances in Applied Clifford Algebras

, Volume 21, Issue 3, pp 647–659 | Cite as

Generating Fractals Using Geometric Algebra

  • R. J. Wareham
  • J. Lasenby


In this paper we investigate how, using the language of Geometric Algebra [7, 4], the common escape-time Julia and Mandelbrot set fractals can be extended to arbitrary dimension and, uniquely, non-Eulidean geometries. We develop a geometric analog of complex numbers and show how existing ray-tracing techniques [2] can be extended. In addition, via the use of the Conformal Model for Geometric Algebra, we develop an analog of complex arithmetic for the Poincaré disc and show that, in non-Euclidean geometries, there are two related but distinct variants of the Julia and Mandelbrot sets.

Mathematics Subject Classification (2010)

28A80 37F45 37F50 15A66 


Geometric algebra Clifford algebra Mandelbrot set Julia set fractals hyperbolic geometry 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. A. Brannan, M. F. Esplen and J. J. Gray, Geometry. Cambridge University Press, 1999, Ch. 6.Google Scholar
  2. 2.
    Yumei Dang, Louis H. Kauffman and Daniel J. Sandin. Hypercomplex interations: distance estimation and higher dimensional fractals. World Scientific, 2002.Google Scholar
  3. 3.
    D. Dunham, Transformation of Hyperbolic Escher Patterns. Visual Mathematics, 1 (1) 1999.Google Scholar
  4. 4.
    D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A unified language for mathematics and physics. Reidel, 1984.Google Scholar
  5. 5.
    A. Lasenby, Recent applications of conformal geometric algebra. In International Workshop on Geometric Invariance and Applications in Engineering Xi’an, China, May 2004.Google Scholar
  6. 6.
    A. Lasenby, J. Lasenby and R. Wareham, A Covariant Approach to Geometry using Geometric Algebra. Technical repot, Cambridge University Engineering Department, 2004. CUED/F-INFENG/TR-483.Google Scholar
  7. 7.
    A. Rockwood, D. Hestenes, C. Doran, J. Lasenby, L. Dorst and S. Mann, Geometric Algebra. Course Notes. Course 31, Siggraph 2001, Los Angeles.Google Scholar
  8. 8.
    Richard Wareham, Computer Graphics using Conformal Geometric Algebra. PhD thesis, University of Cambridge, 2006.Google Scholar
  9. 9.
    Richard Wareham, Jonathan Cameron and Joan Lasenby, Applications of Conformal Geometric Algebra in Computer Vision and Graphics. In Hongbo Li, Peter J. Olver and Gerald Sommer, editors, Computer Algebra and Geometric Algebra with Applications: 6th International Workshop, IWMM 2004, Secaucus, NJ, USA, 2005. Springer-Verlag New York, Inc.Google Scholar
  10. 10.
    Richard Wareham and Joan Lasenby. Mesh vertex pose and position interpolation using geometric algebra. In Proceedings of AMDO 2008, Mallorca, Spain, Secaucus, NJ, USA, 2008. Springer-Verlag New York, Inc.Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of EngineeringUniversity of CambridgeCambridgeUK

Personalised recommendations