"Given the local stock run over the past 12 months, we believe the local stock may do corrective movement next week," said Affin Investment Bank Vice-President, Head of Retail Research, Dr Nazri Khan.
He told Bernama that despite the short-term dips, the medium-term looked promising as there were some resilience with the local stocks recouping earlier losses and was still holding above the 1,620 points support level.
"Investors are now waiting and wondering when China and other Group of Seven countries will act to boost the market.
"In fact, we view the latest boost of bigger-than-expected US$127 billion in asset purchases from the Bank of Japan as a positive surprise for investors and hence opens out a possibility of more coordinated central bank action to boost the global economy.
"We are currently pegging 1,620 and 1,600 as support levels while resistance stands at 1,640 and 1,650," he added.
For the week just-ended, the FBM KLCI finished 2.62 points lower at 1,640.33 against last Friday's 1,642.95.
The Finance Index lost 160.58 points to 14,602.62, Industrial Index declined 6.29 points to 2,812.71 and the Plantation Index erased 187.86 points to 8,279.28.
The FBM Emas Index trimmed 126.94 points to 11,048.67, the FBM Mid 70 Index fell 134.24 points to 11,982.13 and the FBM Ace dipped 25.34 points to 4,328.48.
The weekly volume fell to 3.85 billion units, valued at RM6.99 billion, from 4.73 billion shares, valued at RM8.52 billion, recorded last Friday.
Main market volume decreased to 2.73 billion units worth RM6.84 billion from 3.34 billion units valued at RM8.29 billion registered previously.
Volume on the ACE market declined to 911.53 million units, worth RM130.84 million, from 1.1 billion shares worth RM204.74 million transacted last week.
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Certainly mathematicians are not the only ones to appreciate suchdesigns. Accordingto Wikipedia, peoplehave been making quilts at least since Renaissance times. Similargeometric designs also show up in architecture, oriental carpets andother art forms around the world. In part this comes from a deepseated human aesthetic appreciation for symmetry.What is symmetry? Essentially it is a repetition in shape andcolor that our brains seem to enjoy, perhaps because it simplifieswhat would otherwise be a complicated jumble of unrelated visualelements. Most plants and animals have at least approximate left-rightmirror symmetry, so perhaps our brains evolved to notice symmetricforms because they help us interpret common scenes.Barbara's fabric quilt is quite beautiful: one can enjoy looking at itfor an extended period of time, the eye following various paths,enjoying the contrasts between the black diamonds and the colorfultriangles. You can click the photo to see more detail.What constitutes a visually "interesting" or "beautiful" pattern? Artists like my sister develop an intuitive understanding of what"looks" good, but is there a way to capture that intuition in moremathematical language?After much searching, I read an interesting article on the webby NikosA. Salingaros called The "Life" of a Carpet: an Application ofthe Alexander Rules. While his focus is on Oriental Carpets, histheory and design guidelines apply much more generally. He provides adetailed description of the characteristics that make interestingdesigns.We can find many of his ideas illustrated in Barbara's quilt. Some arepretty straightforward, like juxtaposing contrasts, e.g. a darkoutline around a light shape, such as her black diamonds and the outerblack border contrasting with the bands of colorful triangles.Some of his other guidelines are more subtle, like the idea that thereshould be interesting detail at every distance scale, from the entirework down to very fine details. To my mind, one reason real quilts(and paintings) are more visually interesting than purely computergenerated geometric patterns is that fabric has texture and pattern ofits own. A small piece of fabric that forms a single triangle is oftennot one homogeneous color, but in fact a complex mix of colors andtextures on a much smaller, finer scale.We see this too in Barbara's quilt. At the big-picture level, we havethe black diamonds against the colored diagonals. But within thecolored diagonals is a second layer of contrasts, this time mostly ofcolors (warm reds against cool blues) and values (dark against light),accentuated by the contrast in the orientation of the triangles.The Carpet article also recommends including some randomness,but not too much. This also makes sense. A plot of random dots, linesor polygons looks like the visual equivalent of audio noise. This isnot surprising, since both have similar statistical properties. Incontrast, a modest amount of randomness - occasional deviations fromperfect symmetry - can actually add interest to a pattern.Here too, the artist has done this intuitively. Part of the visualinterest in the quilt is that the colored triangles do not repeat inperfectly symmetric patterns, so that you keep coming back to them,looking for the pattern, catching glimpses of it but not quite fullyunderstanding its subtleties. There is symmetry at the high level; onecan imagine the pattern of black diamonds going on forever in alldirections; yet we would be hard pressed to say exactly what coloreach small triangle should be if we were to extend the design.My children, who are much more musically talented than I, tell me thatthe same is true in classical music, where the listener comes toexpect certain patterns and can be pleasantly surprised when the musicoccasionally does something different.A couple years ago, I decided to try out the design ideas inthe Carpet article by making a painting of a quilt-likepattern. I do not have the patience (or skill) to try actually sewinga quilt out of real fabric. As a prelude to painting, I wrote apattern generating program so I could experiment with some variations.There are a large number of traditional quilt patterns, often formedby assembling right triangles of various colors into a square "block",and then repeating identical copies of that block in a grid layout,optionally rotating them to create different kinds of symmetries. Onevery interesting mathematical question is to try to enumerate, or atleast characterize, a wide variety of such patterns. Mathematicianshave used group theory to classify repeating designs in theplane into 17 basic types calledWallpapergroups. However, when we allow patterns that do not repeatexactly (such as in our quilt picture) the possibilities are muchbroader, and I will not even begin to attempt a full description. Asjust the tip of the iceberg, let me point out that there are manyother ways to create geometric patterns besides using righttriangles. For instance, one can tile the plane using hexagons. Alsointeresting are patterns involving 5-sided shapes(e.g. PentagonalTesselations) and unusual configurations suchas Penrosetiling, in which two basic shapes can tile an arbitrarily largearea in a non-repeating (non-periodic) manner. The "traditional" patterns have the advantage of simplicity:right triangles and squares are easy to draw and cut out without a lotof special measuring equipment or wasted fabric. Some of the "new"patterns (e.g. Penrose tiling) are a little more complicated, butshould I think be straightforward if one were to make a cardboardtemplate for each shape and then just trace them onto fabric. I decided to stay within the traditional framework of righttriangles, but to try patterns that were not constrained by repeatingblocks, even though they should still involve some level ofsymmetry.Computer generated patterns often look somewhat mechanical, e.g.
To me, this is not very interesting, aside from the novelty aspect: itwould be difficult to create this image as a fabric quilt, or as aconventional painting, because the visual impact depends so stronglyon the very precise, very mechanical progression in the colors.My next step was to aim for images that looked more like traditionalquilts. I also experimented with adding randomness atvarious scales. Here are three example of designs I was able to create.
Ultimately, I decided that my painting would need morerandomness and less symmetry than these images. Actually making apainting is rather different from writing a computer program; you canmodify color as you go until it "looks right", and you can addtextures much like those of fabric. After a great deal ofexperimenting (and over-painting), the final result looked like this:
As you can see, I attempted to follow the guidelines of contrasts andrandomness at a variety of scales. You can judge for yourself whetherthe result is attractive.I hope you enjoyed this article. While most of my posts are moretechnical, with source code to experiment with, I think it is fun tosometimes step back and enjoy the aesthetic side of math. To me, evenwithout "formulas", the images in this post are deeply mathematicaland quite intriguing. I certainly enjoyed creating my painting - it isfun to use a different side of the brain for a while! If you areinterested in these kinds of patterns, the links mentioned above arejust the beginning: there are a huge variety of web pages out therewith fascinating examples of periodic and non-periodic geometricpatterns described as Tesselations or Tilings.As usual, please post questions, comments and other suggestions below,or email me directly at the address described at the end of theWelcome post. Remember you can sign up for email alerts about newposts by entering your address in the widget on the sidebar. Or follow@ingThruMath on Twitter to get a 'tweet' alerting you to each newpost. See you next time!
Let me describe the background a little more clearly. We areinterested here in what academics call market microstructure:the magic of how prices actually get established on the scale ofindividual transactions. The thing being traded could be a piece offruit, a gadget like an ipod, an hour of professional services, or 100shares of Apple Computer on the stock market. The key point is thatnot everyone wants these things equally badly: a hungry person wantsan apple, but a person who just ate might rather have cash, so theycan buy a gadget instead. Moreover, not everyone has the same startingposition (or endowment): some people start the game holding anapple, while others do not. As a result of this lack of symmetry, some people may wish tomake trades with other people. In our simple example, you can trademoney for apples and vice versa, but in a general equilibriummodel you can trade in a whole variety of different goods(e.g. oranges, lemons) simultaneously. In our simple example, everyoneis in the same place at the same time, but again, in a generalequilibrium model you can even have different locations and differenttime periods; you can interpret differences in price over time asinterest rates, and develop all kinds of deep insights into the realeconomy.When economists first started trying to analyze markets, roughly acentury ago, there were no computers to do simulations, so they neededto make simplifying assumptions to reduce the problem to simplealgebra. As with theorems in math, there are a lot of technicaldetails you can include in the assumptions, and mathematicaleconomists have studied how much you need to assume in order to provethat people will act a certain way. For example,the Arrow- Debreu - McKenzie model, which is at the heart of generalequilibrium models in economics, requires certain technicalassumptions (e.g. convexity) in order to prove the existence of aunique equilibrium.For today, though, let's keep it simple. Imagine that each person hasa secret reservation price at which they are indifferentbetween having an apple and having cash. For a hungry person, thisreservation price is higher than for someone who just ate. For astarving person, for whom this apple is their last hope of not dying,the price is essentially infinite: they will never give away theirapple if they start with one, and they will give away all of theirstarting cash to obtain one if they do not have one. For someone whois not hungry at all and who hates the taste of apples, the price isessentially zero: they will sell their original apple, if they haveone, for any positive price, and will never spend any of their moneybuying one.Call the people who start the game with an apple sellers, andthe others buyers. The names reflect what they could do,but not necessarily what they want to do, nor even whatthey will do.Let's walk through a version of the conventional theory. Imagine weline up all the sellers in increasing order of reservationprice. Also, line up the buyers in decreasing order ofreservation price. Now we walk down the line, exchanging apples,until the seller wants more than the corresponding buyer willoffer. For example, suppose the reservation prices look like this:
Notice that the equilibrium price is determined by where the linescross, not by any sort of "average" reservation price: changing thelast seller's reservation price from 16 to 32 would have no impact onthe equilibrium price or quantity. Economists refer to this phenomenonusing the word marginal: not in the usual English senses of"at the side of" or "unimportant", but rather tomean "incrementally" or "at the tipping point". The "marginalplayer" is the one in the center of the diagram, where thelines cross, who is essentially indifferent to buying or selling atthe equilibrium price. Why does the theory predict this particular outcome? Theconventional argument for the emergence of the equilibrium price underperfect competition goes like this. Should someone in the market shoutout "the price is 8 dollars", more sellers would be interested, butfewer buyers. Presumably some of the sellers (those with reservationprices less than 8) would lower their price in order to win back thebuyers. Similarly, should someone claim "the price is 2 dollars",additional buyers would come to the table, but sellers would walkaway. This would encourage those buyers with high reservation pricesto raise their offers. In both cases, prices far from equilibriumwould trigger supply-demand imbalances that would nudge the playersback toward the equilibrium price. In many ways, this is a very convincing story. However, it isa bit abstract. Economists use it to model all kinds of real worldmarkets; yet not all markets resemble an open-outcry auction. When yougo to a supermarket in the United States to buy groceries, postedprices are not subject to negotiation (although this is not the casewith automobiles, nor is it always true in othercountries). Supermarkets do compete with each other, and do advertiseprices in local newspapers, and shoppers can (if they have a car andenough spare time) drive to multiple stores to compareprices. Shoppers' reservation prices do vary - some people clipcoupons to ensure the best possible price, while others are relativelyprice insensitive. However, prices fluctuate a lot from day to day andstore to store. Prices in other markets can be even wilder: just thinkof the huge gyrations in the stock market in the last few weeks. It is hard to imagine that this is all based on purelyrational calculations or to see it as a stable equilibrium situation.In recent years, many economists have begun looking at alternativetheories (e.g. behavioral economics) that attempt tobetter describe how real people behave.This brings us to today's question: how well does the conventionalmodel fit reality? Or, to be more precise, since we will be doingcomputer simulations rather than experiments with live humans, we wantto know:
This first image comes from Google'sCrisis Response service, which overlays "Google maps" with data from the National Weather Service and other sites. The actual site is interactive, so you can toggle on and off features like the forecasted path of the storm, and you can zoom in to see the locations of nearby Red Cross shelters and local Evacuation routes.It turns out we can also get very detailed forecasts for things likewind speed. This next picture shows the National Weather Servicewind speed projections for Boston, hour by hour.
This image camefrom Boston.com,which has apparently extracted the National Weather Service forecastdata and put it into a nice graphical format where you can click toselect your city.High winds can be damaging, but hurricanes can produce damagingflooding as well.
This third image comes fromTides and Currents at the National Oceanic and Atmospheric Administration (NOAA), which runs the National Weather Service. Again, the actual site is interactive, with forecasts and past history for a variety of locations.This particular plot shows the actual water level versus the predictedlevel for this time of year (without a hurricane). The tides normallyproceed on a fairly smooth, roughly sinusoidal basis, since they aregoverned by the Moon revolving around the Earth. The red "observed"line is now more than a meter (3.5 feet) above the blue "predicted"line, and in fact that discrepancy has been rising steadily all day,as shown by the green line (observed minus predicted). In fact,although it is presently low tide, the water level is up whereit would normally be at high tide, so in a few hours when hightide naturally arrives, there could be flooding.So how does the National Weather Service come up with all of this?Well, of course, it is all done "by computers", these days, but whatdoes that really mean?Physicists long ago worked out a set of "partial differential equations"that describe fluid flow. Roughly speaking, think of a chunk of air,possibly encased inside an inflated balloon so you can visualizeit. Various forces act on the balloon, causing it to move around. Forinstance, gravity pulls it down, the wind can blow it, and soforth. The temperature matters as well, since hotair is less dense than cold air and hence will tend to rise. Themoisture content also matters, in part because rain and snow are verymuch in the category of things we want meteorologists toforecast.We can capture the impact of these various forces in "partialdifferential equations". They are "partial" because the quantitiesof interest (temperature, pressure, wind speed, and so forth) arefunctions of not one but several variables: latitude, longitude,elevation and time. They are "differential" because they describethe change in these quantities over time (in calculus terms,they involve derivatives). Unfortunately, partial differential equations (PDEs) are quitedifficult to solve, even with the aid of computers. A great deal ofacademic research in applied mathematics -including myown Ph.D. back in 1991 - goes into finding more accurate and efficient ways tosolve various kinds of PDEs using super-computers. The weather equations areparticularly difficult to solve, which is one of the reasons why it ishard to accurately forecast the weather more than a few days forward;this is connected to the fact that the weather equations describe aChaoticSystem.One big hurdle is data. To really solve the weather equationsaccurately, we need to know the initial conditions. That means we needto know the temperature, pressure, wind speed, and so forth,at every place on Earth, not to mention everywhere abovethe Earth as well, at least up to heights where the air becomes thinand no longer plays much role in determining the weather. Of course,we do not know all these numbers. But, applying the resources ofgovernments around the world, the human race is these days able tocollect measurements of these quantities at a large number oflocations around the planet, on a frequent, automated and accuratebasis. This helps a lot. Notice that even to forecast weather in the US, we needmeasurements from all over the globe, because weather systems movearound: the wind does not respect national boundaries. The more money we spend (on monitoring additional sites), themore accurate the forecasts can be. However, more data also meansmore computer processing time is needed just to do the calculations,which is why weather forecasters are amongst the largest users ofsuper-computing facilities (right up there with code breakers,protein-folding biologists, and quantum chemists). Think of a globe of the earth covered by a mesh of latitudeand longitude lines. If we use 24 longitude lines (one per time zone)and 24 latitude lines, we get a total of 576 regions: not squareshaped, since the surface of the Earth is curved, butquadrilaterals. Suppose we divide the atmosphere above us into 10layers; now we have 5760 three-dimensional "cells". Each of these islike a "balloon", in my earlier analogy, and the equations describewhat happens to the air inside it.This is fine, except that each of these 5760 cells covers a huge chunkof the Earth's surface, on the order of the size of Texas. We wouldprobably prefer a more specific forecast, one that can tell us aboutthe weather in our own town, not just the "average" weather forhundreds of miles around us. The solution is clear: we need a finermesh.So, suppose we double the mesh: we use twice as many longitude lines,twice as many latitude lines, and twice as many layersvertically. That gives us 8 times as many cells, which means thecomputer will take (at least) 8 times longer to crank through all thecalculations - and we will have to spend 8 times as much money tocollect all the data. But unfortunately, even the resulting 46,080cells are still rather large - much larger than typical cities, andlarger than many states. So, we need to repeat the "refinement"process and double the mesh again. Now we have 64 times as many cellsas we started with - with 64 times the cost and computer requirementsas well. Actually, the way the solution algorithms work, the computertime generally goes up faster than the problem size, so with 64 timesas many cells, the computer might need 128 times as long to work onit, or even more.You see where this is going. To get the "cells" down to a reasonablyhuman size scale requires a huge number of cells, and a vastbudget for data collection and calculations. But the result is veryhandy: extremely detailed forecasts that take into account all mannerof local conditions, as illustrated by the images at the start of thisarticle.Now consider what happens if it takes the computer 24 hours to crankout a forecast for 6 hours in the future.That is a forecast that is 18 hours obsolete even when it first comesout of the computer! This leads weather forecasters to use parallelcomputers. These are a special kind of super-computer thatessentially consists of a large number of ordinary computers ("nodes")connected together with a specially designed high speed network sothat they can communicate very rapidly. If you have 2 nodes, you mightlet one work on the equations for the cells in the NorthernHemisphere, and the other on the Southern. If you have 4 nodes, youcan also split East and West. And so on.Of course, there is a potential problem with splitting up thecalculations this way. The weather near the equator is impacted byboth Northern and Southern Hemisphere cells, so somehow the two nodesneed to periodically communicate, exchanging information about theboundary or interface region. And since the weather in a day or two orthree may be influenced by the current conditions far away, we cannotsimply ignore the Southern Hemisphere portion of the calculation.This exchange of data at the interfaces takes time, and the more nodesin our parallel super computer, the more time (on a percentage basis)gets "wasted" doing this communication, instead of doing the actualcalculations. That means that a parallel computer with 256 nodes willnot, in general, run 256 times faster than a single node: it mightonly achieve a "speedup" of 64, say, due to the overhead of constantlyhaving to exchange interface data between nearby nodes.Finally, there is an element of probability in all ofthis. Since we lack complete initial data for the entire Earth, wehave to "guess" what the temperature, pressure, etc. are in-betweenthe various measurement stations. To the extent that we guessincorrectly, the forecasts will also be incorrect. So, we can trymaking several different guesses and seeing how much difference itmakes. For instance, suppose I know the current temperature in Bostonis 60 degrees (Fahrenheit), and the temperature in New York is 70degrees, but I lack an observing station in Hartford (part way betweenNew York and Boston). I can guess that Hartford is 65 degrees, butperhaps it really is only 64, or 66, or something. So now, instead ofrunning my weather simulation once, I need to run it several times,with different guesses for Hartford, and then I can see how muchdifference it makes. This gets quite complicated, since I also have toguess the starting values for a lot of other places besides Hartford!Ultimately it is this sort of approach that enables the WeatherService to make quantitative predictions of form "there is a 30%chance of rain today in Hartford". So, as you can see, there is a great deal of complexity (andcost) hidden behind all the wonderful charts and graphs we saw above.As usual, we have only scratched the surface. My hope is that regularreaders of this blog are getting a sense for the vast range ofquestions that mathematics can help answer. I hope you enjoyed this discussion. As usual, please postquestions, comments and other suggestions, or G-mail me directly. Remember you cansign up for email alerts about new posts by entering your address inthe 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!
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).
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. 
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. 
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. 
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. 
First, you need a copy of R. This is the free, high quality,open source statistical programming language that has become astandard for statisticians in industry and academia because it is botheasy and powerful. Download the latest version for Windows, Mac, orLinux from The R Project forStatistical Computing. Click on "Download R" and select a mirror(meaning, pick a site located close to you, to speed up the downloadprocess - there are mirrors all over the world), then click "DownloadR for Windows" (or Mac or Linux). Then just double click the installerand accept the default selections. You should now have a desktop iconor Start menu entry for starting R. You can copy and paste the samplecode from this blog post right into the R console window, and it willprint answers and draw graphs right on your computer screen.We are going to answer the questions by running simulations. Not thegiant computer-game kind of simulations, with photo-realistic images ofblackjack tables, just a simple mathematical simulation of theessential elements of the process.What do we need to know to set up the simulation? Not very much. Wedon't even need to know the details of any particular casino game,just your probability of winning and your payoff if you dowin. These vary depending on the game you choose to play.So, let's assume the following situation:
You see a large peak near zero (you never really go negative, that'sjust an artifact of the smoothing process inherent in drawing thecurve), and a smaller peak at and above $200. If you win the first twobets, you walk away with $225, but other combinations of wins andlosses can lead to a variety of other winning outcomes between $200and $300.I've been saying you will get an answer "close" to $89, because eachtime you run the program, you will get different random numbers, andso get a slightly different final answer. That's why we simulate10,000 different nights: it helps average out the noise, so that youwind up with a pretty consistent analysis, regardless of the specificroll of the dice. If you want to always get the same answer each time,put 'set.seed(123)' at the start of the code instead.Mathematicians call this sort of situation a "random walk", becauseyour balance staggers randomly up and down over time, and it has"absorbing barriers" at $2 and $200, because once you reach (or pass)those values, you stop. Here is a picture of one particular night,showing your balance over time:
In this example, you won the first bet, but then lost the nexttwo. Then you won again, but then you lost 5 times in a row, whichforced you to stop.So far, so good (or bad). Whether you like the odds reflected in these pictures or not, they are the results of betting half your cash each time, in a double-or-nothing game with a 48% chance of winning, given that you stop if you double your initial cash. But the real question is, "compared to what?" We need to try some alternative strategies to see if they are better or worse. We assume you cannot change 'w' and 's', because those arefixed characteristics of the game you are playing. In reality, youcould go look for a different, more favorable game, but 48%double-or-nothing is about as favorable as typical casino games get,actually.So what can you change? You can change 'f', or you can modify the'nextBet' function to do something else, such as bet a fixed dollaramount each time, rather than a fixed percentage. This is easy enough:type in