Cameron Browne. Impossible fractals

1. Introduction

In the field of psychology there is a long tradition of optical illusions that exploit quirks in visual perception to create misleading figures with ambiguous or contradictory perceptual interpretations. One such type of illusion is the impossible object, which is a shape that cannot physically exist despite having an apparently valid visual description. Martin Gardner describes such objects as undecidable figures [1].

A defining characteristic of impossible objects is that each part makes sense but the whole does not; local geometry is satisfied but the figure's global geometry is ambiguous or contradictory, and the viewer must constantly revise their understanding of the figure as their eye travels over it. As Penrose and Penrose put it, each individual part is acceptable but the connections between parts are false [2]. Many examples of impossible objects can be found in Bruno Ernst's The Eye Beguiled: Optical Illusions [3] which provided the inspiration for most of the constructions in this paper.

In the field of mathematics there is a long tradition of objects that display fractal geometry, even though the precise definition of self-similarity that underpins them and their classification as a related group is relatively new. Classical fractals typically involve simple transformations recursively applied to simple shapes to produce more complex shapes. Chaos and Fractals: New Frontiers of Science by Pietgen et al [4] provides a comprehensive overview of fractals, their construction and basic properties.

When drawing impossible objects, artists tend to choose shapes that are as simple as possible in order to emphasise the illusion. This paper investigates whether fractal techniques can be applied to impossible objects to produce new, more complex designs which retain the perceptual effect. The following sections examine some of the more common types of impossible objects, and their development by standard fractal techniques.