Symmetry groups of fractals

Alexander, Carol, Giblin, Ian and Newton, David (1992) Symmetry groups of fractals. Mathematical Intelligencer, 14 (2). pp. 32-38. ISSN 0343-6993

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This paper contains computer pictures of generalised
Mandelbrot and Mandelbar sets, and their associated
Julia sets, from which it is evident that their symmetry
groups possess an elegant and simple structure. We
show that (i) the Mandelbrot set M(p) generated by the
iteration zt+ ~ = ztp + c remains invariant under the
symmetry transforms of the dihedral group Dp_ 1 (i.e.,
these are isomet-ries of M(p)); (ii) the Mandelbar set
M(p) is invariant under the isometries inDp + 1; and (iii)
the Julia sets of points inside M(p) (or M(p)) are invariant
under the isometries in either Dp or just the cyclic
group Cp, depending on whether the seed point is on
or off a symmetry axis of the parent Mandelbrot (or
Mandelbar) set. The proofs are relatively easy, but
showing that there are no other isometries of these sets
is not so straightforward. As is often the case in the
theory of chaos, what is obvious geometrically is difficult
to prove analytically. For the generalised Mandelbrot
and Mandelbar sets with even p we have in fact
proved that the dihedral symmetry transforms are the
only isometries of these sets, but the method does not
appear to be applicable to odd p, or to the Julia sets.

Item Type: Article
Schools and Departments: School of Business, Management and Economics > Business and Management
Subjects: Q Science > QA Mathematics > QA0150 Algebra. Including machine theory, game theory
Depositing User: Carol Alexander
Date Deposited: 26 Sep 2012 10:53
Last Modified: 26 Sep 2012 10:53
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