Why Monotonous Repetition is Unsatisfying
Nikos A. Salingaros, The University of Texas at San Antonio.Conjectures on combinatorial complexity
When applying mathematics to interpret our world we invariably runinto formidable difficulties. Explaining human perception of oursurroundings and our reactions to the environment requires that weknow the mechanisms of our interaction with the world. Unfortunately,we don't -- not yet. Thus, explanations of why we react to differentforms in our environment tend to be conjectural.We know from observation that human beings crave structured variationand complex spatial rhythms around them, but notrandomness. Monotonous regularity is perceived as alien, withreactions ranging from boredom to alarm. Traditional architecturefocuses on producing structured variation within a multiplicity ofsymmetries. Contemporary architecture, on the other hand, advocatesand builds structures at those two extremes: either random forms, ormonotonously repetitive ones. Let us explore why human beings find thelatter unappealing, and propose what they do like instead, with theultimate aim of characterizing that mathematically.I present some ideas on design and the influence of certain structureson human perception. These questions arose in the context ofarchitecture and urbanism, yet the problem goes much deeper, intocombinatorics and human physiological response. Lacking a rigoroustheory that explains the phenomenon, I am offering some intuitiveresults in the hope that someone will pursue them further. I believethere exists a straightforward mathematical model for what is goingon.The observed effect concerns modules repeating in a geometricallyregular manner: an application of translational symmetry. For example,consider identical rectangles lined up straight. On the scale ofskyscrapers, many of those buildings simply repeat the floor design(as seen on its side) vertically, so that the whole building showsvertical translation symmetry (Figure 1).
Figure 1. A skyscraper shows vertical repetition. There are also countless examples of exact repetition of unitshorizontally, either identical structural elements within onebuilding's facade (Figure 2), or separate but identical buildingslined up straight along a road. A typical example is the repeatingmodular box making up an urban housing or office development (Figure3).
Figure 2. Non-traditional building showing horizontal repetition. 
Figure 3. Identical modular buildings repeat along a street. Many observers react negatively to such simplisticrepetition. And, apparently, our degree of unease increases as moreunits are seen to repeat. Identical objects perfectly lined upgenerate a feeling of discomfort in the viewer. At the very least, weexperience something mechanical to which we react negatively as humanbeings, since we are used to more natural structures with variationand complexity. Readers are encouraged to check thisassertion. Observations should be performed with the entire body in afull-scale environment: simply looking at pictures or at areduced-scale model fails to engage all of our perceptive apparatusand will not lead to useful results. The conjecture is that somethingfundamental is at play here that affects our perceptive mechanism,triggering a negative signal.So, what do people prefer? And why? I open the can of worms ofarchitectural fashion by illustrating an older example of tallbuilding typology, dating from the end of the 19th to the beginning ofthe 20th centuries (Figure 4). Monotonous vertical repetition isabsent. In addition, a richness of articulation that is part oftraditional tectonics and ornamentation provides a hierarchy ofdecreasing scales.
Figure 4. Neo-Gothic skyscraper. Even though architects since the 1920s advocate monotonous repetitionin tall buildings (Figure 1), it is not really appealing to mostpeople. I believe buildings with more complex yet ordered shapes makea far better city -- but then, they must also pay attention to thehuman scale (which is another topic for discussion).Steps toward an explanation
First, the mathematics points us to the algorithmic complexity of theconfiguration. The simplest complexity measure considers thegenerating code of a configuration. Complexity can be measured asbeing directly proportional to the length of code (though this is initself a simplistic measure that does not apply in more sophisticatedsituations, it is sufficient here). In this case, repetition in onedirection is trivially simple, since the code for generating it is:"define a unit A, then align n identical copies along one direction toget AAAAAAAAA...". The generating code is trivial.Second, why is human neurological response actually negative? Someinsight into the effect comes from the notion of Biophilia, whichasserts that our evolution formed our neurological system withinenvironments defined by a very high measure of a specific type ofcoherent complexity. That is, our neurological system was created(evolved) to respond directly and exquisitely to complex, fractal,hierarchical geometric environments. When placed in environments thathave opposite geometrical features, therefore, we feel ill atease. The theory is being verified by experiments: see the collectionof essays "Biophilic Design" edited by Stephen Kellert etal. Minimalist environments make us feel ill at ease. Simplisticrepetition is one such minimalistic geometrical setting in which wefind no algorithmic complexity, hence no visual and intellectualinterest. But our response is not simply to ignore it: we cannot, andit provokes anxiety in us. Questions immediately arise, such as why does the feeling of uneaseincrease with the number of repetitions? Here is evidently astraightforward Combinatoric problem, if only we knew what the humanbrain is measuring or counting when looking at repeating modules. Butwe don't. One guess is that the feeling of unease grows not linearlybut exponentially with the number n of repeating modules. If the brainis counting permutations among identical units, or trying to labeleach unit, then the possible combinations increase exponentially. Itcould also be true that the brain is frustrated by trying to identifydistinct modular units so as to fix a coherent picture of itsenvironment -- necessary for survival and deciding upon afight-or-flight response. If the units are identical they cannot becatalogued in memory.Some examples
In what environments does one encounter large-scale geometricalconfigurations with a lot of monotonous repetition? Actually, all suchexamples are human-made, being strictly the results of industrialproduction. I claim that simplistic repetition occurs neither innature, nor in pre-industrial human creations. In nature, we almost never find simplistic repetition on themacroscopic scale. (Yes, pure crystals do have microscopic regularitybut that structure is not visible -- furthermore, naturally-occurringpure crystals are quite rare.) Inanimate physical structures almostalways have some variations that prevent the unpleasant monotonouseffect. For example, the most regular repetition I can immediatelythink of occurs in the hexagonal cells of honeybees and solidifiedlava flows (the Giant's Causeway); but in each of those cases therepetition occurs with abundant minor imperfections (Figure 5). Thosegeometries therefore avoid the "industrial" feeling of beingmonotonous. Looking at wild honeycombs is fascinating, and walking oncrystallized lava is as well. And neither of these structures is acommon part of our living environment. Other physical geometriesdefined by repetitions occur with a great deal of variation and soescape monotony.
Figure 5. Natural honeycomb. Living structures with repetition show so much variation in therepetition that monotony is entirely avoided. Consider the leaves of atree: no two are identical, and their positioning combines adistribution based on the Fibonacci sequence with randomness due tothe growth of the tree branches as influenced by environmentalfactors. No simplistic repetition along a line here! Large-scale hexagons have been used in buildings. Where an additionalvariation is introduced to distinguish the modules, the resultsucceeds, but where the modules repeat monotonously, the overalleffect is felt negatively. The architects of those buildings appearunaware of the effect of monotonous repetition, but then so manybuildings from the past one hundred years blatantly display monotonousrepetition, so it seems to be something desired rather than arrived ataccidentally. As the experience is not yet rigorously documented itcould be dismissed as personal opinion or preference. Nevertheless, Ibelieve this is NOT personal preference but instead the reaction ofour bodies, and is thus felt by the general population. Avoiding Combinatorial Complexity
I introduced the notion of combinatorial complexity in the book"Twelve Lectures on Architecture". This is precisely the effect ofmonotonous repetition experienced in the environment. Two solutionswere given of how to avoid combinatorial complexity. Both solutionsinvolve breaking the translational symmetry in some way. The first solution is to introduce symmetry breaking by means ofvariety in the repeating modules (Figure 6). Symmetry breaking is akey notion that comes from theoretical physics: one adds smalldifferences to an otherwise perfect symmetry. The configuration is NOTsymmetric unless we ignore those minor differences. Therefore, thereis approximate symmetry on the global scale but it does not extend tothe smaller scales. In the present example, we maintain thetranslational symmetry on the larger scale (the repetition of amodular unit), but introduce variations within each module so everymodule is only approximately the same. Strictly speaking, it's nolonger a module. These small variations are sufficient to affect ourperception of the whole configuration, however, changing it from beingmonotonous to interesting.
Figure 6. Columns with variety, spaced symmetrically. The second solution is to group a few modules together into a clusterof no more than three or four (Figures 7 and 8). We somehow tie threeor four modules into a supermodule, which itself then repeats. What weare doing is in fact introducing a hierarchy where previously noneexisted. In the original repeating configuration of n modular units,the scales are only two: the module itself, and all the modules linedup filling the size of the entire configuration. By grouping modules,we define a new scale at the size of the grouping, not exactly threeor four modules large, because it includes modules plus anyintermediate spaces. 
Figure 7. Grouping columns into clusters of three. 
Figure 8. Grouping columns into clusters of four. These two groupings establish a strong informational relation betweenwhat we initially considered to be the module and any spacesurrounding it; grouping columns with spaces links alternatingcontrasting units. Rhythm on both the architectural and urban scalesdepends upon the intricate interweaving of space and material, whichare treated on an equal design footing. It is worth pointing out that these solutions come from traditionalarchitecture, and are seen to be re-invented repeatedly by differentcultures in history and in different geographical regions. Somethinginnate is driving humankind towards discovering and implementing thesesolutions, and it's not simply a matter of aesthetic preference. Alsonote that our modern industrial age (beginning, say, from the 1920s)is marked by its break with the architecture of the past by thedistinction of whether to pursue and celebrate monotonous repetition,versus avoiding it altogether. Since the effect produces unease in theuser, this raises serious questions about why architects and urbanistsmake it a point to generate it in their buildings. Christopher Alexander's explanations
Christopher Alexander is a pioneer in investigating environmentalcomplexity and developing techniques for generating living geometry inthe built environment. By "living geometry" we mean a particularcomplexity that embodies coherence and which is perceived asphysiologically and psychologically positive by humanbeings. Alexander refers to the environmental effect as "healing",confirming the independent line of investigation coming fromBiophilia. Although not expressed in the present manner, Alexander'swork offers fundamental insight into the problem of monotonousrepetition. Alexander has presented "Fifteen Fundamental Properties" that arefound in all coherent structures, comprising inanimate matter,biological structures, and especially in human-made objects andenvironments built before the industrial age. Those rules can beapplied to generate living environments today, and are clearly usefulin avoiding, or repairing, monotonous repetition so as to remove itsnegative effect. A reader can find descriptions of these properties inAlexander's "The Nature of Order. Book 1: The Phenomenon of Life", andmy own summary in "Twelve Lectures on Architecture". I describe threeof Alexander's properties here. 1. Levels of Scale postulates that stable structures contain ahierarchy of distinct scales, and those scales are carefully spaced sothat the scaling factor between two consecutive scales is very roughlyequal to three. This is a universal property satisfied, for example,by all fractals. (Whereas fractals exist with every scaling factor,Alexander postulates that hierarchies with scaling factor near threeare perceived as more natural). There should exist distinct scaleswell defined in the structure. The larger scales are related throughsome magnification (exact or approximate) to the smaller scales, usinga scaling factor. As described above, grouping repeating units intoclusters introduces intermediate scales where none existedinitially. As such, it is one solution to avoiding or repairingmonotonous repetition. For example, what makes a colonnade informationally comfortabledepends just as much on hierarchy as on repetition. Inter-columnarspacing ranges from two column widths in some Classical temples, tofour in the nave arcade of a Basilica and in Roman colonnades, to sixin many Medieval Cloisters, to eight in Far Eastern traditionalarchitecture (with variations for individual cases). In all theseinstances, the space between columns defines the next higher scale,and repetition links two consecutive hierarchical scales. As a result,the columns and spaces are perceived togethercoherently. Twentieth-century architects introduced extremely thincolumns (called "pilotis" or stilts) and widened their separation sothe intervening space is more than twelve times the width of a column.2. Alternating Repetition postulates that simple modules should notrepeat, but rather, it is paired contrasting modules that can doso. There are several consequences of this rule. The alternation ofunits leads to contrast, which introduces spatial rhythm (albeitprimitive but at least present, whereas monotonous repetition has nospatial rhythm at all) (Figure 9). Looking at natural, biological, andpre-industrial structures Alexander found alternating repetition to bewidespread, and noted that it was adopted as a technique for creatingstable configurations.
Figure 9. With smaller-scale insertions, windows become alternating.Alternating repetition is directly related to the groupings discussedearlier. An alternating pattern ABABABABA really includessymmetric groupings on many distinct scales: ABA, BAB, ABABA, BABAB,etc. defined naturally through their bilateral symmetry. This wealthof subsymmetries is not evident in configurations with monotonousrepetition. For a discussion, see "The Nature of Order. Book 1: ThePhenomenon of Life". Even so, a configuration with alternatingrepetition but no further variations or groupings on higher scaleswill show monotonous repetition if it is large enough. 3. Gradients occur when the size of similar components decreases inone direction. Here is a solution that was not mentioned earlier inthis essay, and which breaks monotonous repetition: make all the unitsof different size, not randomly, but in a carefully controlled mannerso as to create a gradient. Then, the ensemble is perceived asharmonious and not unsettling. Translational symmetry is brokenbecause the units have decreasing lengths. Again, Alexander foundcountless examples of gradients in natural, biological, andpre-industrial structures. Gradients prevent monotonous repetition, but in a different manner tosymmetry breaking. In the latter, the main symmetry is maintainedwhile symmetry is broken on smaller scales. Gradients, on the otherhand, break the symmetry on the original scale and do not necessarilydo anything on smaller scales. Here I mentioned only three of Alexander's fifteen fundamentalproperties, yet there are other ones that bear directly and indirectlyupon our problem of monotonous repetition. All of this discussion isbut a small part of a general theory of design that uses recursivealgorithms [a good topic for a future paper in this series]. Assumingas axiomatic several geometrical properties found in nature, Alexanderhas formulated a method for designing and constructing complexsystems. Interestingly, following Alexander's design method "The Theory ofCenters", one cannot get to monotonous repetition. It's not thatmonotonous repetition is forbidden in any ideological sense; ratherthe algorithmic design rules can never arrive at solutions thatdisplay monotonous repetition. That mechanical configuration, and mostother unnatural, anxiety-inducing geometries, resides outside thespace of solutions obtainable from the Theory of Centers. To reachmonotonous repetition, one has to abandon the design rules thatgenerate living structure. Therefore, since the Theory of Centers wasmost certainly followed by designers and builders of all traditionalcultures instinctively, this explains why we never see monotonousrepetition in traditional artifacts, buildings, and cities. Symmetry breaking prevents informational collapse
An identically repeating module generates simplistic translationalsymmetry. As explained previously, such a configuration hasalgorithmically trivial complexity. From the information theory pointof view, the configuration is collapsible to its single module plusthe rule for repetition. Thus, the configuration as a whole has noinformational stability: it is prone to collapse. Symmetry breakingchanges this because it is no longer possible to condense the wholeinto a single repeating module. There may be more to it than simple visual concerns about monotony indesign. A physical structure is only one of an enormous variety ofcomplex systems that run our universe. Each complex structure mustprotect itself against structural collapse, otherwise we will not findit around to observe. Does informational collapse parallel other, moresignificant mechanisms of systemic collapse? And do complex systemsfind analogous methods of avoiding systemic collapse? Since symmetry breaking through the creation of higher-order groupingsgenerates a hierarchy, this itself is one basic feature of a stablesystem. By introducing distinct levels in a scaling hierarchy, thecomplex system distributes itself on different levels, and thus it isnot dependent solely upon one or two levels. Symmetry breaking as seenin nature and in traditional artifacts, buildings, and cities is notrandom, but serves to define an irreducible hierarchicalstructure. This question is discussed further in "Twelve Lectures onArchitecture".Blending into the background and emotional nourishment
Natural environments are characterized by an enormous degree ofstructural complexity, yet for the most part, we perceive them asbackground. Our perceptive system is apparently wired to noticeanything that contrasts with a natural background. It signals alarmand makes us uneasy. Since natural environments are fractal, itfollows that non-fractal objects will stick out and be noticed byus. This includes pure Platonic forms (cubes, rectangular prisms,pyramids, spheres) that define just a single scale, the largestone. Usually, those repeating forms create the monotonous repetitioneffect discussed here. Andrew Crompton sent me some helpful suggestions on the topic of thisessay: Human creations are designed either as neutral or picturesque,and traditional products are designed to vanish into oursurroundings. When we inhabit an environment, or surround ourselveswith human-made objects, we don't want any individual object to botherus -- that is, not to disrupt the sense of visual coherence we can drawfrom our complex environment. Repetitive buildings or buildingcomponents may be computationally boring, but they do imposethemselves upon our cognition by not blending into thebackground. They stick out. They do not scale (as I discussed earlier)so they do not fit into a traditional structural hierarchy of a citythat has evolved over generations. Monotonous repetition disturbs us because it is unnatural; and is sobecause it fails to share geometrical features common in naturalcomplex structures. The phenomenon goes further, however, in that"blending into the background" is not a neutral effect, but definitelya regenerative one. Biophilia gives us emotional nourishment (withconcomitant physiological benefits) from a complex, coherentenvironment; therefore I am talking about positive effects.Thoughts about contemporary architecture
In contemporary architecture, many practitioners have rebelled againstmonotonous repetition and have come up with their ownsolutions. Invariably, those solutions inject randomness into thetranslational symmetry in a way that leads away from coherence. Thisis the opposite from the solutions outlined above and implemented bytraditional architecture and urbanism, which seek coherence. Someonewho is familiar with contemporary architects' philosophy of wishing tobreak with the past at all costs should not be surprised thattraditional evolved solutions are not adopted, but that the oppositeeffect is sought. The facades and plans of many contemporary buildings rely on modularunits that are for the most part monotonously repetitive. Thetranslational symmetry is sometimes broken by random changes, however,so that the overall effect is one of imbalance, irrationality, andlack of purpose (Figure 10). This negative impression is justifiedsince the architect simply introduces random changes for visualeffect, not for any structural or functional reason. The reaction ofthe user is not positive, because our body also reacts to randomnesswith alarm.
Figure 10. Randomness in a facade destroys coherence.While the topic of contemporary design lies outside the presentinvestigation, those examples of fashionable architecture thatrandomize symmetry contrast sharply with the solutions describedhere. In general, architectural symmetry breaking as practiced todayviolates perfect symmetries (which could be a good thing) but on thewrong scale, so that the coherence of the ensemble is reduced insteadof enhanced (Figure 10). Again the discussion goes back to our body's predilection for coherentcomplexity in our environment, and our negative reaction when builtforms deny it to us. In its search for design novelty, architecturalsymmetry breaking as seen in contemporary structures deliberatelyavoids creating the sought-for hierarchy of subsymmetries on distinctscales. Conclusions
I claimed a visual effect of monotonous repetition and suggested thatit induces unease and even anxiety in viewers experiencing such astructure at full scale. Hopefully, researchers in environmentalpsychology will perform the necessary rigorous testing in order toestablish any effect such structures have on our psychology andphysiology. Looking for an explanation of this effect from mathematicsled me to conjecture some sort of combinatorial analysis that ourbrain engages in, the details of which are as yet unknown. The processof analyzing our environment occurs automatically because we need toposition our bodies within it informationally, and subconsciouslyjudge our safety from environmental threats. If an environmentembodies monotonous repetition, it could tire our neurological system,and that is possibly what creates a negative effect on ourbodies. This essay concluded with suggestions for avoiding the effectof monotonous repetition. Altogether, I believe this is a pretty butnot well-defined, hence woefully under-investigated mathematicalproblem.References
Christopher Alexander (2001) The Nature of Order. Book 1: ThePhenomenon of Life, Center for Environmental Structure, Berkeley,California. Stephen R. Kellert, Judith Heerwagen, and Martin Mador, editors (2008)Biophilic Design: the Theory, Science and Practice of BringingBuildings to Life, John Wiley, New York. Nikos A. Salingaros (2010) Twelve Lectures on Architecture:Algorithmic Sustainable Design, Umbau-Verlag, Solingen, Germany.Related reading
Allen Klinger and Nikos A. Salingaros (2000) A Pattern Measure,Environment and Planning B: Planning and Design, volume 27, pages537-547. Availablehere.And there you have it. I hope you enjoyed this discussion.Nikos and I are hoping that someone out there will take these ideaseven further. For instance, the whole area of algorithmiccomposition (either of images or music) revolves around being able to convert ideas like those expressed above into algorithms for generating imagesand music. I would love to hear about any ideas readers might havealong these lines.If you enjoyed this article, you might also like Koch Snowflakes or Algorithmic Art, for colorful designs incorporating some of these ideas. As usual, please post questions, comments and other suggestions, or G-mail me directly. Remember you can sign up for email alerts about new posts by entering your address in the widget on the sidebar. Or follow@ingThruMath on Twitter toget a 'tweet' for each newpost. About the Blog has some additional pointers for newcomers, who mayalso want to look atthe Contents page for a complete list of previous articles. See younext time!
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