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Computers & Graphics 31 (2007) 659–667

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Chaos and Graphics

Impossible fractals

Cameron Browne�

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

In the field of psychology, there is a long tradition ofoptical illusions that exploit quirks in visual perception tocreate misleading figures with ambiguous or contradictoryperceptual interpretations. One such type of illusion is theimpossible object, which is a shape that cannot physicallyexist despite having an apparently valid visual description.Martin Gardner [1] describes such objects as undecidable

figures.A defining characteristic of impossible objects is that

each part makes sense but the whole does not; localgeometry is satisfied but the figure’s global geometry isambiguous or contradictory, and the viewer must con-stantly revise their understanding of the figure as their eyetravels over it. As Penrose and Penrose [2] put it, eachindividual part is acceptable but the connections betweenparts are false. Many examples of impossible objects can befound in Bruno Ernst’s The Eye Beguiled: Optical Illusions

[3] which provided the inspiration for most of theconstructions in this paper.

In the field of mathematics, there is a long tradition ofobjects that display fractal geometry, even though theprecise definition of self-similarity that underpins themand their classification as a related group is relatively

e front matter r 2007 Elsevier Ltd. All rights reserved.

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ess: Cameron.Browne@swishzone.com.

new. Classical fractals typically involve simple transforma-tions recursively applied to simple shapes to producemore complex shapes. Chaos and Fractals: New Frontiers

of Science by Pietgen et al. [4] provides a compre-hensive overview of fractals, their construction and basicproperties.When drawing impossible objects, artists tend to choose

shapes that are as simple as possible in order to emphasisethe illusion. This paper investigates whether fractaltechniques can be applied to impossible objects to producenew, more complex designs which retain the perceptualeffect. The following sections examine some of the morecommon types of impossible objects, and their develop-ment by standard fractal techniques.

2. The tri-bar

The tri-bar (Fig. 1, left) is described by Ernst [3] as thesimplest yet most fascinating of all impossible objects, andis one of the most widely recognised. The illusion is createdby the ambiguous use of parallel lines drawn in differentperspectives, so that the figure appears to perpetually turnout of the page when traversed in a clockwise direction.The two corners at the end of each bar are interpreted aslying perpendicular to each other, which Ernst points outwould give a total internal angle of 3601 and hence defy afundamental property of triangles; this figure cannot bephysically constructed as a closed shape with perpendicularcorners and straight arms.

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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).

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The tri-bar was invented in 1934 by Oscar Reutersvard, aSwedish graphic artist who went on to become the world’sgreatest exponent of impossible figures, producing severalthousand until his death in 2002. The tri-bar is often calledthe Penrose Triangle after mathematician Roger Penrose,who independently rediscovered it and popularised it in the1958 article ‘‘Impossible Objects: A Special Type of VisualIllusion’’ co-written with his father [2]. It is also known asthe Escher Triangle as Dutch graphic artist Escher [5]embraced the principles it represented and included itsdesign in many of his works, most famously the perpetualstream of his 1961 lithograph ‘‘Waterfall’’.

We call this figure the tri-bar in keeping with Ernst’sterminology [3], which may be extended to multibar figureswith more than three sides. Multibars are generally drawnon an isometric grid with the following design rules inmind:

(1)

local geometry and shading should be consistent; (2) adjacent regions should not share the same colour; and (3) he least number of colours should be used (three

colours will generally suffice, although four arerequired in some cases).

3. Triangular fractals

Fig. 1(middle and right) shows two well-known fractaldevelopments of the triangle, the Koch snowflake and theSierpinski gasket. The snowflake modifies the triangle’sperimeter shape while the gasket recursively subdivides itsinterior.Fig. 2 illustrates a development of the tri-bar as an

impossible snowflake. The first iteration can be constructedentirely from a single subshape, the acute generator (topright), repeated six times in a cycle with appropriatecolouring. Further iterations require a combination ofacute and obtuse generators.Fig. 3 shows an alternative snowflake development that

retains parent triangles from previous generations and usesthem as a framework upon which subsequent triangularstruts are added. Although this is not a traditionalsnowflake and the final design is busier than the previousfigure, this approach only requires a single generator (right)and the struts enhance the ambiguity of perspective to givea stronger effect.In both cases, the thickness of all bars in the figure are

uniformly reduced with each iteration to retain the shape’s

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Fig. 4. Impossible gaskets are more troublesome.

Fig. 5. The Devil’s fork and the Cantor set.

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definition. Unfortunately, the illusion’s effect diminisheswith each iteration as the subshapes become harder todistinguish.

Turning now to the Sierpinski gasket, Fig. 4 shows how agasket-like impossible object may be developed by successivereplication of the tri-bar at smaller scales. Note that each ofthe tri-bar’s three bars has an internal and external face, andthat each subtriangle must match colours with the parent’sfaces to which it joins. This is straightforward for the firstiteration in which the subtriangle joins the outer faces of theparent triangle (left), and for the second iteration in whicheach subtriangle joins the inner faces of its parent (middle),but becomes problematic for further iterations which requiresubtriangles to join two inner and one outer face (or viceversa) if consistent colouring is to be maintained. Thisproblem does not occur with the figure’s central subtriangle(right) which remains consistently coloured for any numberof iterations; however, this is not a standard gasketrecursion.

4. Forks

The Devil’s fork, also known as the impossible fork,Devil’s pitchfork, Devil’s tuning fork or the blivet [1], isanother famous impossible object (Fig. 5, left). Similarfigures had been devised by Reutersvard since 1958 [3] butit was not until Schuster’s description in a 1964 article [7]and its appearance on the cover of the March 1965 issue ofMad magazine that the Devil’s fork became widely known.This illusion involves confusion between the figure and itsbackground. Its local geometry is valid at its open andclosed ends on the left and right, but a perceptual shiftoccurs as the eye travels between the two.

The Cantor set (Fig. 5, right) is the simplest of fractals,involving the recursive removal of the middle third of agiven line segment and its subsegments. Fig. 6 shows howCantor-style subdivision may be applied to the Devil’s forkto increase the number of prongs, allowing some creativityin their configuration.Fig. 6(top left) shows the Devil’s fork turned into a

Devil’s gatling gun after two levels of subdivision andreplacement of the fork’s alternating spaces with anadditional layer of prongs to suggest a round configura-tion. Fig. 6(bottom left) shows a similar design except thatthe prongs are realised as square bars of uneven length,after a design by Reutersvard [3]. The Devil’s comb (right)shows the basic fork after three levels of Cantor-stylesubdivision.

5. Squares

Fig. 7(left) shows two impossible square multibardesigns known as four-bars or Penrose Squares, as theyare obvious extensions of the tri-bar or Penrose Triangle.The four-bar on the left can be constructed usingalternating sharp and truncated generators (shown under-neath) while the four-bar on the right requires only thetruncated generator for its construction.Fig. 7(middle) shows the development of the left

four-bar as one generation of the square Sierpinskicurve, and Fig. 7(right) shows the development of theright four-bar as two generations of the square Sierpinskicurve. Just as the four-bar is less striking an illusion thanthe tri-bar, so the impossible square Sierpinski figures areless striking than the impossible snowflake and gasketfigures.

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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.

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6. Cubes

Fig. 8(left) shows an impossible multibar cube such asthe one prominently featured in Escher’s famous litho-graph ‘‘Belvedere’’ [5]. Note that this figure violates thesecond multibar design rule as it contains adjacent same-coloured faces where the arms cross. This will be a problemwith the multibar description of any non-planar shape, asthe two overpassing colours and the two underpassing

colours should ideally be mutually exclusive; this figurereally requires four colours.Fig. 8 also shows the Menger sponge, which is the three-

dimensional equivalent of the Cantor set, after oneiteration (middle) and after three iterations (right). Whilethe impossible multibar cube is topologically similar to thefirst iteration of the Menger sponge (except for theimpossible crossing) it is difficult to apply the multibartechnique to further iterations. However, the idea of cubic

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

Fig. 10. Ernst’s ‘‘Nest of impossible cubes’’ and two variations.

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subdivision does yield another method for describingimpossible objects used extensively by Reutersvard [3].This method is epitomised in his ‘‘Opus 2B’’ (Fig. 9, topleft) in which three overlapping cubes form a visualcontradiction.

Fig. 9(bottom left) shows the tri-bar realised as asubcube design. This design is of particular interest asnone of the overlapping subcubes form local contradictions(as they do in ‘‘Opus 2B’’) yet the overall figure is just aseffective as the tri-bar. This design came before the tri-barand was actually the first of Reutersvard’s many impossiblefigures, and is hence probably the first example of animpossible object designed specifically as an illusion. Thereexist examples of impossible figures in artwork dating back1000 years, before the advent of classical perspective [3].However, these generally appear to be ‘‘fixes’’ introducedby the artist to address compositional shortcomings (suchas a pillar which would occlude an important foregroundfigure if drawn in correct perspective) rather than asdeliberately ambiguous objects.

Fig. 9(middle, top) shows a square made of subcubes andFig. 9(middle, bottom) shows a cube made of subcubeswith local contradictions added. It can be seen that thesefigures contain a perceptual illusion but are not veryelegant. The cubic subcube design in particular is confusingto the eye and becomes even more so if the subcubes

are recursively subdivided into further subcubes. Thisapproach is not amenable to fractal development.Fig. 9(right) shows the key elements of Monika Buch’s

1976 painting ‘‘Cube in blue’’ [3]. This is a clever realisationof the impossible multibar cube as a subcube design, andexploits the fact that perpendicular arms share a commonsubcube of the same colour where they cross; it isambiguous where the impossible multibar cube is contra-dictory. This design is also topologically similar to the firstiteration of the Menger sponge and looks promising forrecursive subdivision as a sponge-style fractal. Unfortu-nately, further subdivision again becomes overly confusingas the implied edges of the cubic frames at each level arelost—as is the perceptual effect—in the busy detail.The impossible is proving difficult to achieve in this case.

However, Fig. 10 shows a different style of perspectiveillusion, based on multiple planes, which may be recur-sively applied to a cube without affecting the illusion.Fig. 10 (left) shows the basic design of Bruno Ernst’s 1984work ‘‘Nest of impossible cubes’’—please note that Ernst’soriginal [3] is more artfully executed both in its perspectiveand its texture. The first variation (middle) shows how theeffect may be enhanced with curved arches, which allow awider view of the interior portion of the wall. The secondvariation (right) shows that the effect can be achieved tosome extent even without the arch.

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

Fig. 12. Reutersvard’s ‘‘Meander’’ and a Hilbert meander.

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7. Area-filling curves

It is generally possible to create a multibar illusion formost curves that can be drawn on the isometric grid. Forexample, Fig. 11 shows the first three iterations of thePeano–Gosper curve or flowsnake [6] realised as animpossible multibar.

Similarly, Fig. 12(top left) shows Reutersvard’s ‘‘Mean-der’’ illusion [3] and its application to the first three iterationsof the Hilbert curve, mapped to the isometric grid, to yield a

Hilbert meander. This design is derived from the Devils’sfork illusion and involves figure/background ambiguity.However, another illusion not found in the original Meanderfigure emerges, to yield two types of effect at once:

(1)

Figure/background ambiguity (as per the Devil’s fork).Note that the original Meander figure is penetrated bythe background from underneath, whereas the Hilbertmeander is penetrated from all four sides withoutdiminishing the effect.

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

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

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(2)

Perspective ambiguity (as per a multibar). The top partof each Hilbert subfigure appears to run out of the pagetowards the left, while the bottom part appears to runout of the page towards the right. As the eye follows thecurve around, each feature is interpreted and themental map reoriented to the new perspective, andthe previous perspective forgotten.

The Hilbert meander is best rendered as a black andwhite drawing as the Devil’s fork and multibar colouringschemes are not compatible.

8. Pythagorean trees

The multibar fractals considered so far have involvedbars of uniform thickness (if not uniform length) at eachlevel of recursion. We now consider a multibar design withbars of continuously decreasing thickness between levelsthat more precisely matches its fractal equivalent.

Fig. 13(top left) shows the first four iterations of a 451Pythagorean tree, which is a structure composed of squaressuch that the three touching squares at each branch form a451 right triangle. Fig. 13 (bottom left) shows the first fouriterations of a balanced 301 Pythagorean tree. This tree isdescribed as ‘‘balanced’’ because the left branchingsquares, which are almost twice the size of the rightbranching squares, grow twice each iteration, forcinggreater development in the more visible parts and resultingin a more homogenous spread of detail. A traditional 301Pythagorean tree would require almost twice as manyiterations to achieve a similar look; hence, the balanced treeallows significant computational savings. The 451 Pytha-gorean tree is balanced by default.Fig. 13(right) shows an extended tri-bar from Penrose

and Penrose’s 1958 article [2] that suggests a method forhandling branches and adapting a multibar motif to thePythagorean tree. Fig. 14 shows the extended tri-baradapted to a 451 Pythagorean tree after 15 iterations, andFig. 15 shows it adapted to a balanced 301 Pythagorean

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

Fig. 16. A spiral tri-bar and hexagonally bound isometric spirals.

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tree after 10 iterations. Note that the perceptual effectremains evident for several iterations, until the bars becometoo fine to distinguish.

9. Spirals

Although spirals are not fractals, they are brieflymentioned here as area-filling curves that display aspectsof self-similarity and variable resolution. Fig. 16(top left)

shows a spiral version of the tri-bar; this is no longer animpossible figure but has a valid physical interpretation asa conical spiral.However, impossible spirals may be achieved by again

using the isometric grid. The remainder of Fig. 16 shows anisometric cube iteratively layered to produce impossiblespirals of arbitrary resolution. These are similar to multi-bar designs, except that concentric layers share bars withtheir inner and outer neighbours (four colours are

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required). Each iteration maintains a perfectly hexagonaloutline, making these isometric spirals suitable for tilingsand other artistic applications.

10. Conclusion

This paper demonstrates ways in which fractal techni-ques may be applied to impossible object designs, withmixed success. While some designs resist fractal develop-ment, others prove amenable and can even yield pleasantsurprises, such as the emergence of the second type ofillusion in the Hilbert meander.

The combination of the extended tri-bar with thePythagorean tree demonstrates that the perceptual effectcan be successfully maintained at continuously decreasingscales in the one design. Tree balancing provides a way torender such designs efficiently.

Practical applications for the techniques discussed abovemight include the production of large format art worksthat display the perceptual effect to many more levels of

recursion that the figures in this paper allow; the effectcould then be seen at multiple levels, depending onthe distance at which the picture is observed. Futurework might include the creation of such large formatpictures that include different types illusions at each levelof scale.

References

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[3] Ernst B. The eye beguiled: optical illusions. Berlin: Taschen; 1986.

[4] Peitgen H, Jurgens H, Saupe D. Chaos and fractals: new frontiers of

science. New York: Springer; 1992.

[5] Escher MC. The graphic work. Berlin: Taschen; 1989.

[6] Weisstein E. Peano–Gosper curve, Mathworld. /http://mathworld.

wolfram.com/Peano-GosperCurve.htmlS.

[7] Schuster DH. A new ambiguous figure: a three-stick clevis. American

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