Impossible Fractals

ARTICLE IN PRESS

Computers & Graphics 31 (2007) 659–667 www.elsevier.com/locate/cag Chaos and Graphics Impossible fractals

Cameron BrowneÃ

SWiSHzone.com Pty Ltd., The Basement, 33 Ewell St., Balmain 2041, Australia

Abstract

Impossible objects are a type of optical illusion involving ambiguous visual descriptions of ﬁgures that cannot physically exist. It is shown by way of example that such objects can be further developed using standard fractal techniques to create new, more complex designs that retain the perceptual illusion, sometimes allowing additional illusions to emerge from the process. The balanced Pythagorean tree is used to efﬁciently render impossible fractals that display the perceptual effect across decreasing levels of scale. r 2007 Elsevier Ltd. All rights reserved.

Keywords: Impossible object; Optical illusion; Fractal; Art

1. Introduction new. Classical fractals typically involve simple transforma- tions recursively applied to simple shapes to produce In the field of psychology, there is a long tradition of more complex shapes. Chaos and Fractals: New Frontiers optical illusions that exploit quirks in visual perception to of Science by Pietgen et al. [4] provides a compre- create misleading figures with ambiguous or contradictory hensive overview of fractals, their construction and basic perceptual interpretations. One such type of illusion is the properties. impossible object, which is a shape that cannot physically When drawing impossible objects, artists tend to choose exist despite having an apparently valid visual description. shapes that are as simple as possible in order to emphasise Martin Gardner [1] describes such objects as undecidable the illusion. This paper investigates whether fractal figures. techniques can be applied to impossible objects to produce A defining characteristic of impossible objects is that new, more complex designs which retain the perceptual each part makes sense but the whole does not; local effect. The following sections examine some of the more geometry is satisfied but the figure’s global geometry is common types of impossible objects, and their develop- ambiguous or contradictory, and the viewer must con- ment by standard fractal techniques. stantly revise their understanding of the figure as their eye travels over it. As Penrose and Penrose [2] put it, each 2. The tri-bar individual part is acceptable but the connections between parts are false. Many examples of impossible objects can be The tri-bar (Fig. 1, left) is described by Ernst [3] as the found in Bruno Ernst’s The Eye Beguiled: Optical Illusions simplest yet most fascinating of all impossible objects, and [3] which provided the inspiration for most of the is one of the most widely recognised. The illusion is created constructions in this paper. by the ambiguous use of parallel lines drawn in different In the field of mathematics, there is a long tradition of perspectives, so that the figure appears to perpetually turn objects that display fractal geometry, even though the out of the page when traversed in a clockwise direction. precise definition of self-similarity that underpins them The two corners at the end of each bar are interpreted as and their classification as a related group is relatively lying perpendicular to each other, which Ernst points out would give a total internal angle of 3601 and hence defy a fundamental property of triangles; this figure cannot be ÃTel.: +61 7 3371 1576. physically constructed as a closed shape with perpendicular E-mail address: [email protected]. corners and straight arms.

Fig. 1. The tri bar, the Koch snowﬂake and the Sierpinski gasket.

Fig. 2. Two iterations of an impossible snowﬂake (with acute and obtuse generators shown).

Fig. 3. An alternative snowﬂake design that emphasises the perceptual effect (with generator shown).

The tri-bar was invented in 1934 by Oscar Reutersva¨rd, a 3. Triangular fractals Swedish graphic artist who went on to become the world’s greatest exponent of impossible figures, producing several Fig. 1(middle and right) shows two well-known fractal thousand until his death in 2002. The tri-bar is often called developments of the triangle, the Koch snowflake and the the Penrose Triangle after mathematician Roger Penrose, Sierpinski gasket. The snowflake modifies the triangle’s who independently rediscovered it and popularised it in the perimeter shape while the gasket recursively subdivides its 1958 article ‘‘Impossible Objects: A Special Type of Visual interior. Illusion’’ co-written with his father [2]. It is also known as Fig. 2 illustrates a development of the tri-bar as an the Escher Triangle as Dutch graphic artist Escher [5] impossible snowflake. The first iteration can be constructed embraced the principles it represented and included its entirely from a single subshape, the acute generator (top design in many of his works, most famously the perpetual right), repeated six times in a cycle with appropriate stream of his 1961 lithograph ‘‘Waterfall’’. colouring. Further iterations require a combination of We call this figure the tri-bar in keeping with Ernst’s acute and obtuse generators. terminology [3], which may be extended to multibar figures Fig. 3 shows an alternative snowflake development that with more than three sides. Multibars are generally drawn retains parent triangles from previous generations and uses on an isometric grid with the following design rules in them as a framework upon which subsequent triangular mind: struts are added. Although this is not a traditional snowflake and the final design is busier than the previous (1) local geometry and shading should be consistent; figure, this approach only requires a single generator (right) (2) adjacent regions should not share the same colour; and and the struts enhance the ambiguity of perspective to give (3) he least number of colours should be used (three a stronger effect. colours will generally suffice, although four are In both cases, the thickness of all bars in the figure are required in some cases). uniformly reduced with each iteration to retain the shape’s ARTICLE IN PRESS C. Browne / Computers & Graphics 31 (2007) 659–667 661

Fig. 4. Impossible gaskets are more troublesome.

Fig. 5. The Devil’s fork and the Cantor set. definition. Unfortunately, the illusion’s effect diminishes The Cantor set (Fig. 5, right) is the simplest of fractals, with each iteration as the subshapes become harder to involving the recursive removal of the middle third of a distinguish. given line segment and its subsegments. Fig. 6 shows how Turning now to the Sierpinski gasket, Fig. 4 shows how a Cantor-style subdivision may be applied to the Devil’s fork gasket-like impossible object may be developed by successive to increase the number of prongs, allowing some creativity replication of the tri-bar at smaller scales. Note that each of in their configuration. the tri-bar’s three bars has an internal and external face, and Fig. 6(top left) shows the Devil’s fork turned into a that each subtriangle must match colours with the parent’s Devil’s gatling gun after two levels of subdivision and faces to which it joins. This is straightforward for the first replacement of the fork’s alternating spaces with an iteration in which the subtriangle joins the outer faces of the additional layer of prongs to suggest a round configura- parent triangle (left), and for the second iteration in which tion. Fig. 6(bottom left) shows a similar design except that each subtriangle joins the inner faces of its parent (middle), the prongs are realised as square bars of uneven length, but becomes problematic for further iterations which require after a design by Reutersva¨rd [3]. The Devil’s comb (right) subtriangles to join two inner and one outer face (or vice shows the basic fork after three levels of Cantor-style versa) if consistent colouring is to be maintained. This subdivision. problem does not occur with the figure’s central subtriangle (right) which remains consistently coloured for any number 5. Squares of iterations; however, this is not a standard gasket recursion. Fig. 7(left) shows two impossible square multibar designs known as four-bars or Penrose Squares, as they 4. Forks are obvious extensions of the tri-bar or Penrose Triangle. The four-bar on the left can be constructed using The Devil’s fork, also known as the impossible fork, alternating sharp and truncated generators (shown under- Devil’s pitchfork, Devil’s tuning fork or the blivet [1],is neath) while the four-bar on the right requires only the another famous impossible object (Fig. 5, left). Similar truncated generator for its construction. figures had been devised by Reutersva¨rd since 1958 [3] but Fig. 7(middle) shows the development of the left it was not until Schuster’s description in a 1964 article [7] four-bar as one generation of the square Sierpinski and its appearance on the cover of the March 1965 issue of curve, and Fig. 7(right) shows the development of the Mad magazine that the Devil’s fork became widely known. right four-bar as two generations of the square Sierpinski This illusion involves confusion between the figure and its curve. Just as the four-bar is less striking an illusion than background. Its local geometry is valid at its open and the tri-bar, so the impossible square Sierpinski figures are closed ends on the left and right, but a perceptual shift less striking than the impossible snowflake and gasket occurs as the eye travels between the two. figures. ARTICLE IN PRESS 662 C. Browne / Computers & Graphics 31 (2007) 659–667

Fig. 6. The Devil’s gatling gun (two levels), timber offcuts (two levels) and comb (three levels).

Fig. 7. Four-bar designs (with sharp and truncated generators) applied to the square Sierpinski curve.

Fig. 8. An impossible multibar cube and the Menger sponge after one and three iterations.

6. Cubes colours should ideally be mutually exclusive; this figure really requires four colours. Fig. 8(left) shows an impossible multibar cube such as Fig. 8 also shows the Menger sponge, which is the three- the one prominently featured in Escher’s famous litho- dimensional equivalent of the Cantor set, after one graph ‘‘Belvedere’’ [5]. Note that this figure violates the iteration (middle) and after three iterations (right). While second multibar design rule as it contains adjacent same- the impossible multibar cube is topologically similar to the coloured faces where the arms cross. This will be a problem first iteration of the Menger sponge (except for the with the multibar description of any non-planar shape, as impossible crossing) it is difficult to apply the multibar the two overpassing colours and the two underpassing technique to further iterations. However, the idea of cubic ARTICLE IN PRESS C. Browne / Computers & Graphics 31 (2007) 659–667 663

Fig. 9. Subcube designs and Buch’s ‘‘Cube in blue’’.

Fig. 10. Ernst’s ‘‘Nest of impossible cubes’’ and two variations. subdivision does yield another method for describing are recursively subdivided into further subcubes. This impossible objects used extensively by Reutersva¨rd [3]. approach is not amenable to fractal development. This method is epitomised in his ‘‘Opus 2B’’ (Fig. 9, top Fig. 9(right) shows the key elements of Monika Buch’s left) in which three overlapping cubes form a visual 1976 painting ‘‘Cube in blue’’ [3]. This is a clever realisation contradiction. of the impossible multibar cube as a subcube design, and Fig. 9(bottom left) shows the tri-bar realised as a exploits the fact that perpendicular arms share a common subcube design. This design is of particular interest as subcube of the same colour where they cross; it is none of the overlapping subcubes form local contradictions ambiguous where the impossible multibar cube is contra- (as they do in ‘‘Opus 2B’’) yet the overall figure is just as dictory. This design is also topologically similar to the first effective as the tri-bar. This design came before the tri-bar iteration of the Menger sponge and looks promising for and was actually the first of Reutersva¨rd’s many impossible recursive subdivision as a sponge-style fractal. Unfortu- figures, and is hence probably the first example of an nately, further subdivision again becomes overly confusing impossible object designed specifically as an illusion. There as the implied edges of the cubic frames at each level are exist examples of impossible figures in artwork dating back lost—as is the perceptual effect—in the busy detail. 1000 years, before the advent of classical perspective [3]. The impossible is proving difficult to achieve in this case. However, these generally appear to be ‘‘fixes’’ introduced However, Fig. 10 shows a different style of perspective by the artist to address compositional shortcomings (such illusion, based on multiple planes, which may be recur- as a pillar which would occlude an important foreground sively applied to a cube without affecting the illusion. figure if drawn in correct perspective) rather than as Fig. 10 (left) shows the basic design of Bruno Ernst’s 1984 deliberately ambiguous objects. work ‘‘Nest of impossible cubes’’—please note that Ernst’s Fig. 9(middle, top) shows a square made of subcubes and original [3] is more artfully executed both in its perspective Fig. 9(middle, bottom) shows a cube made of subcubes and its texture. The first variation (middle) shows how the with local contradictions added. It can be seen that these effect may be enhanced with curved arches, which allow a figures contain a perceptual illusion but are not very wider view of the interior portion of the wall. The second elegant. The cubic subcube design in particular is confusing variation (right) shows that the effect can be achieved to to the eye and becomes even more so if the subcubes some extent even without the arch. ARTICLE IN PRESS 664 C. Browne / Computers & Graphics 31 (2007) 659–667

Fig. 11. An impossible multibar Peano–Gosper curve.

Fig. 12. Reutersva¨rd’s ‘‘Meander’’ and a Hilbert meander.

7. Area-filling curves Hilbert meander. This design is derived from the Devils’s fork illusion and involves figure/background ambiguity. It is generally possible to create a multibar illusion for However, another illusion not found in the original Meander most curves that can be drawn on the isometric grid. For figure emerges, to yield two types of effect at once: example, Fig. 11 shows the first three iterations of the Peano–Gosper curve or flowsnake [6] realised as an (1) Figure/background ambiguity (as per the Devil’s fork). impossible multibar. Note that the original Meander figure is penetrated by Similarly, Fig. 12(top left) shows Reutersva¨rd’s ‘‘Mean- the background from underneath, whereas the Hilbert der’’ illusion [3] and its application to the first three iterations meander is penetrated from all four sides without of the Hilbert curve, mapped to the isometric grid, to yield a diminishing the effect. ARTICLE IN PRESS C. Browne / Computers & Graphics 31 (2007) 659–667 665

Fig. 13. 451 Pythagorean tree, balanced 301 Pythagorean tree and extended tri-bar.

Fig. 14. An impossible mushroom (451 Pythagorean tree).

(2) Perspective ambiguity (as per a multibar). The top part Fig. 13(top left) shows the first four iterations of a 451 of each Hilbert subfigure appears to run out of the page Pythagorean tree, which is a structure composed of squares towards the left, while the bottom part appears to run such that the three touching squares at each branch form a out of the page towards the right. As the eye follows the 451 right triangle. Fig. 13 (bottom left) shows the first four curve around, each feature is interpreted and the iterations of a balanced 301 Pythagorean tree. This tree is mental map reoriented to the new perspective, and described as ‘‘balanced’’ because the left branching the previous perspective forgotten. squares, which are almost twice the size of the right branching squares, grow twice each iteration, forcing The Hilbert meander is best rendered as a black and greater development in the more visible parts and resulting 1 white drawing as the Devil’s fork and multibar colouring in a more homogenous spread of detail. A traditional 30 schemes are not compatible. Pythagorean tree would require almost twice as many iterations to achieve a similar look; hence, the balanced tree allows significant computational savings. The 451 Pytha- 8. Pythagorean trees gorean tree is balanced by default. Fig. 13(right) shows an extended tri-bar from Penrose The multibar fractals considered so far have involved and Penrose’s 1958 article [2] that suggests a method for bars of uniform thickness (if not uniform length) at each handling branches and adapting a multibar motif to the level of recursion. We now consider a multibar design with Pythagorean tree. Fig. 14 shows the extended tri-bar bars of continuously decreasing thickness between levels adapted to a 451 Pythagorean tree after 15 iterations, and that more precisely matches its fractal equivalent. Fig. 15 shows it adapted to a balanced 301 Pythagorean ARTICLE IN PRESS 666 C. Browne / Computers & Graphics 31 (2007) 659–667

Fig. 15. An impossible fern (balanced 301 Pythagorean tree).

Fig. 16. A spiral tri-bar and hexagonally bound isometric spirals. tree after 10 iterations. Note that the perceptual effect shows a spiral version of the tri-bar; this is no longer an remains evident for several iterations, until the bars become impossible figure but has a valid physical interpretation as too fine to distinguish. a conical spiral. However, impossible spirals may be achieved by again 9. Spirals using the isometric grid. The remainder of Fig. 16 shows an isometric cube iteratively layered to produce impossible Although spirals are not fractals, they are briefly spirals of arbitrary resolution. These are similar to multi- mentioned here as area-filling curves that display aspects bar designs, except that concentric layers share bars with of self-similarity and variable resolution. Fig. 16(top left) their inner and outer neighbours (four colours are ARTICLE IN PRESS C. Browne / Computers & Graphics 31 (2007) 659–667 667 required). Each iteration maintains a perfectly hexagonal recursion that the figures in this paper allow; the effect outline, making these isometric spirals suitable for tilings could then be seen at multiple levels, depending on and other artistic applications. the distance at which the picture is observed. Future work might include the creation of such large format 10. Conclusion pictures that include different types illusions at each level of scale. This paper demonstrates ways in which fractal techniques may be applied to impossible object designs, with mixed success. While some designs resist fractal develop- References ment, others prove amenable and can even yield pleasant surprises, such as the emergence of the second type of [1] Gardner M. Optical illusions. Mathematical Circus. New York: Alfred illusion in the Hilbert meander. A. Knopf; 1968. p. 3–15. [2] Penrose L, Penrose R. Impossible objects: a special type of visual The combination of the extended tri-bar with the illusion. British Journal of Psychology 1958;49(1):31–3. Pythagorean tree demonstrates that the perceptual effect [3] Ernst B. The eye beguiled: optical illusions. Berlin: Taschen; 1986. can be successfully maintained at continuously decreasing [4] Peitgen H, Ju¨rgens H, Saupe D. Chaos and fractals: new frontiers of scales in the one design. Tree balancing provides a way to science. New York: Springer; 1992. render such designs efficiently. [5] Escher MC. The graphic work. Berlin: Taschen; 1989. [6] Weisstein E. Peano–Gosper curve, Mathworld. /http://mathworld. Practical applications for the techniques discussed above wolfram.com/Peano-GosperCurve.htmlS. might include the production of large format art works [7] Schuster DH. A new ambiguous figure: a three-stick clevis. American that display the perceptual effect to many more levels of Journal of Psychology 1964;77:673.