Emergy Synthesis 10, Proceedings of the 10th Biennial Emergy Conference (2019)

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Toward the Thermodynamics and Emergy of Picture and Other Puzzle Solving

Dennis Glenn Collins ABSTRACT

This talk follows up on the Author's and Scienceman's paper "Clusters of High Transformity Individuals" Chapter 36 in Emergy Synthesis 9. Here instead of substrate being converted into product by a generalization of Michaelis-Menton enzyme kinetics, the interest is in pieces of a puzzle being converted into a finished picture. Other applications involve returning and re-connecting people to their homes after a storm or flood, or restoration of electrical grid after a hurricane, or assembling DNA in one dimension. At the start of putting together a, say 1000-piece puzzle, there are 1000 components, and clusters are gradually built up as pieces are fitted together, until if successful there is only one giant cluster or component with all 1000 pieces (or say the electrical grid is restored). Features of the puzzle, such as border, buildings, trees and sky, correspond to enzymes that aid in getting the puzzle done and their transformity can be measured by the jumps or fraction of the puzzle they help to complete. In doing a puzzle, pre-sorting into, say all the manmade structure pieces or border of a roof and then all the tree pieces leads to jumps in progress of doing the puzzle. Thermodynamically the problem involves completely distinguishable particles as perhaps a modification of Fermi-Dirac statistics, since each piece goes in exactly one place. Attempts to measure entropy can involve measuring the work required to add each piece, and topological properties, such as Betti numbers or number of holes left in each cluster, studied, sometimes in reference to Zipf's law with changing powers and the Author's previous paper. INTRODUCTION

This paper follows up on Giannantoni’s work on “emerging solutions,” how some problems can be solved faster than expected, in this case how a picture puzzle may be solved faster by reference to features of the puzzle. There is also the problem of relating puzzle solving to thermodynamics and emergy (Odum’s energy memory). In general puzzle solving is connected to simulated annealing, a wayof forming crystals as temperature decreases. At the start there are all the pieces floating around on the table, and as the pieces are added, clusters are formed until the picture emerges as a kind of giant crystal. Typically the border is done first. This process is the reverse of heating, whereby for example the defroster on a car melts a frozen windshield, first making a hole in the ice toward the center of the windshield, then causing clusters of ice patches, then clearing ice toward the border of the windshield as the melted water flows away. Thus puzzle solving is basically a temperature lowering process and temperature could be measured by how many pieces are left isolated, which number gradually goes to zero as the puzzle is solved, from a maximum of the number of pieces in the puzzle to start.

In comparison to the Author’s previous paper with David Scienceman “Clusters of High Transformity Individuals,” features of the picture on the puzzle may act as enzymes to help form product (completed parts of the puzzle) from the substrate (isolated pieces of the puzzle). For example one feature, say a part of a car, may occur on two pieces of the puzzle, binding them together. Two important differences are 1) that nature works somewhat in parallel, allowing products to form in different areas at the same time, whereas puzzle solving is typically done serially (unless there is more than one person working on the puzzle at the same tine) and 2) the particles in the chemical problem are generally

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indistinguishable, whereas the pieces of a puzzle are supposed to be completely distinguishable (one piece goes in exactly one place). These two factors mean that puzzle solving generally goes slower than chemical reactions, but allows a more perfect result (the puzzle piece can be lifted to put in the necessary hole, but the chemical particle may be blocked, leading to imperfections in the crystal.

Another consideration is that the water-to-ice transition takes place at a constant temperature 32 degrees F at standard pressure, somewhat contradicting the above view; nonetheless the puzzle solver can be considered as a kind of refrigerator, applying energy to lower temperature by the cycle of doing work to add pieces, thereby lowering overall temperature. It appears that metals have somewhat different properties from water and typically the phase boundary becomes smeared out (versus exact 32) if there are more components present.

Further calculations here are done with respect to one ”Vista” puzzle of a Kyle Petty NASCAR 45 racer by LeapYear Publishing 2003, which says 100 pieces on the front but actually has 108 (=12x9} interlocking pieces. Please see Photo 1. A property helpful for the analysis here is that the pieces are roughly rectangular and form a 12”x9” grid, so that it is relatively easy to find the number of exposed sides at any stage of putting the puzzle together. Here the word “sides” may be misleading because the “sides” of the puzzle are only about 1/64” high. “Interlocking” means a non-border edge of the puzzle has either a protruding flange or inset cavity that makes its length about 1 5/8” long versus 1”. The interlocking helps the enzyme or feature binding of the pieces. Please see photo. Remark: Putting a race car on a puzzle brings up questions of maximum emergy, why people use up scarce fossil fuels to drive around in circles and even more people spend even more fossil fuels to drive sometimes hundreds of miles to see people drive around in circles. Perhaps it is better than war. Actually business writer Eric January says going in circles is not that bad because it preserves momentum (cf. moon around earth).

THE MODEL

The Author put the puzzle together 4 times, with intervals to gather data: 1) three 30-min. intervals, 2) seven 10-min. intervals, 3) thirteen 5-min. intervals, and 4) fifteen 4-min. intervals, taking one hour to complete the puzzle itself. It was found that doing longer than 4 minutes at a time, some properties were missed, such as holes being formed and filled up during a 10-min. work interval. All the analysis is done on the last case. Possibly there was some learning during the repeats, reducing the time from 90 to 60 minutes. Table 1 shows the data of case 4, including the number and size of clusters at each stage, the number of remaining (isolated) pieces, the number of holes (Betti number), and the number of exposed sides and covered sides. At the start there are 12x9x4=432 exposed sides and at the 0.

Table 1. Puzzle Solving Data. time exposed covered sides cluster sizes entropy (x100) used up pieces left Betti (holes)

1 432 0 0 468 0 108 0 2 430 2 2 466 2 106 0 3 414 18 6+3+2 453 11 97 0 4 392 40 12+3+2+2 434 19 89 0 5 384 48 13+4+3+2+2 426 24 84 0 6 368 64 18+4+3+3+2+2 406 31 76 0 7 356 76 18+6+4+4+3+3 393 38 70 0 8 348 84 18+10+6+4+3 380 41 67 0 9 320 112 44+8 298 52 56 0 10 288 144 48+14 261 62 46 1 11 266 166 66 212 66 42 4 12 242 190 71 187 71 37 4 13 186 246 81 138 81 27 2 14 140 292 91 88 91 17 1 15 76 356 103 26 103 5 2 16 42 390 108 0 108 0 0

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Figure 1 shows a modified graph from the previous Clusters paper with three main enzyme peaks between 0 and 4 on the time scale, scaled to 0 to 10.8 on the vertical (product) scale Figure 2 shows the puzzle data from 1 to 16 on the time scale and 0 to 108 on the vertical (number of pieces completed) scale. Although both graphs show three jumps or “bulges,” presumably due to enzymes or features, the chemical graph shows more “logistic” pattern whereas the puzzle data illustrates more linear overall pattern, perhaps due to parallel activity versus serial activity as mentioned above. The puzzle features thus may contribute to departure from linear behavior of about 7 pieces added (total 7x15 = 105) per time interval. It remains to devise a system of differential equations which will illustrate this moderate jump departure from linear behavior.

Clear All kf=.1;kc=.5;kr=.001;k1=5;k2=3;k3=1;k4=4 s=NDSolve[{en'[t]==-kf*en[t]*S[t]+(kr+kc)*Es[t]+k1*en[t]-k2*en[t]*Es[t],S'[t]==-kf*en[t]*S[t]+kf*Es[t],Es'[t]==kf*en[t]*S[t]-(kr+kc)*Es[t]+k3*en[t]*Es[t]-k4*Es[t],P'[t]==kc*Es[t]*(10.8-P[t]),en[0]==4,S[0]==10,Es[0]==0,P[0]==0}, {en,S,Es,P},{t,0,12}] Plot[Evaluate[{en[t],S[t],Es[t],P[t]}/.s],{t,0,12}] T=Table[{{.1*k,en[.1*k],P[.1*k]}}/.s,{k,0,50}];MatrixForm[T] All Clear 4

{{en->InterpolatingFunction[ ],S->InterpolatingFunction[ ],

Es->InterpolatingFunction[ ],P->InterpolatingFunction[ ]}}

Figure 1. Modified graph from the previous Clusters paper with three main enzyme peaks between 0 and 4 on the time scale, scaled to 0 to 10.8 on the vertical (product) scale.

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T={0,2,11,19,24,32,38,41,52,62,66,71,81,91,103,108} ListPlot[T] V=Table[{T[[i]],i-1},{i,1,16}] ListPlot[V] Plot[15*t/108,{t,0,108}] {0,2,11,19,24,32,38,41,52,62,66,71,81,91,103,108}

{{0,0},{2,1},{11,2},{19,3},{24,4},{32,5},{38,6},{41,7},{52,8},{62,9},{66,10},{71,11},{81,12},{91,13},{103,14},{108,15}}

Figure 2. Puzzle data from 1 to 16 on the time scale and 0 to 108 on the vertical (number of pieces completed) scale.

Graph 1 illustrates the overall process, whereby at the start the 108 isolated pieces of the puzzle are strung out along the x-axis at height 1, and as the time progresses the pieces are formed into clusters along the x=0 line increasing gradually (sort of linearly) up to a 108 peak at the left rear as the pieces are used up, leaving 0’s along the back row. Unfortunately the surface as shown does not preserve conservation of puzzle pieces, and actually the surface should have height 1 (remaining isolated pieces) except for a few clusters along the left-hand border and 0’s to the right of the 1’s.

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Plot3D[(x+108-107*(15-t)/15-15*x/(15-t))*HeavisideTheta[(15-t)-15*x/107],{x,0,107},{t,0,14.99},PlotRange->All]

Graph 1.

Presumably serial computer solution, whereby pieces and clusters are matched at random to see if they fit, would be much slower (perhaps approaching exponentially slower) than taking advantage of features. It appears that as puzzle pieces with a certain feature (say car) are used up, the rate of puzzle solving goes down, leading to attempts to find another feature. Thus there are three rates: parallel nature (fast), working with features (maybe linear) and random matching (maybe very slow). If the rate of puzzle solving goes way below linear (so many pieces put together per time interval spent), there is a tendency for people to give up on the puzzle.

Photo 1

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ANALYSIS OF TIME SLICES

Among other problems, it remains to study the statistics of clusters of pieces at the various time slices. There are at least three candidates: 1) there is an apparent proof that for completely distinguishable particles, the distribution should be Boltzmann (a Exp[-bx]). 2) Looking at formation of clusters, say populations forming into cities or meanings attaching to words, there is an argument the distribution should be Zipfian (a x^(-b)). 3) Considering that there is the reverse of a heating effect, the distribution could be normal (a Exp[-b x^2]) as heat spreads out sometimes according to a normal or Gaussian density. Trying these three distributions by themselves, there is much error, so that further analysis is necessary. Here it is observed that a cluster of size 1 is for free, that is no effort is required to obtain the row of 108 isolated puzzle pieces at the start or the first element of any cluster thereafter. Thus the first element should not be counted in cluster size analysis. Thus the distributions or densities should have the form

f(x) + UnitStep[c-x] where UnitStep[x] is a function that is zero if x is negative and becomes positive 1 from x = 0

onwards for all positive x, where c is the cutoff for used up puzzle pieces, and clusters are listed along the x-axis in decreasing size from left to right, followed by 1’s for the isolated pieces followed by 0’s

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for the pieces used up in forming clusters. The -x in the UnitStep(c-x) means that actually the values go from 1’s to 0’s. “c” is the dividing point between 1’s and 0’s. With this modification, nearly all the sum-of-squares errors go down to small amounts and the three cases can be compared.

The results by the FindFit command of Mathematica are presented in Chart 1. At the start all methods have zero error. Of the remaining 15 cases, the Zipf method apparently has least error for 11 cases, the Boltzmann for 2 cases and the normal for 2 cases (7th slice tied for the largest number 6 of clusters and 12th slice for largest Betti number 4). A couple of wrongly fitted pieces had to be redone along the way, possibly contributing to some error.

Thus although the results are somewhat inconclusive, the Zipf distribution seems best. More test puzzles and more cases of working out the puzzles, as well as more theoretical study, may shed more light on this matter. A conclusion is that the hypotheses of the Boltzmann proof for distinguishable particles do not generally hold for some reason. Clearly the decisions of the puzzle solver can make a difference; for example trying to get the last border pieces that will combine to create a very large cluster and make the distribution more Zipf-like versus spending more effort on adding to smaller but do-able clusters.

Program 1. Boltzmann program for 9th time slice. The third cluster is about 2 instead of 1, leading to about 1 unit square error and “normalization” 108.9 versus 108. The second cluster is slightly too small at 7.66 versus 8. T=Table[{i,0}, {i,1,108}];T[[1]]={1,44};T[[2]]={2,8}; Table[T[[j]]={j,1},{j,3,58}]; z=FindFit[T,a*Exp[-b*x]+UnitStep[58-x],{a,b},x] U=Table[{x,a*Exp[-b*x]+UnitStep[58-x]/.z},{x,1,108,1}] Sum[U[[k,2]],{k,1,108}] Sum[(T[[m,2]]-U[[m,2]])^2,{m,1,108}] {a->277.639,b->1.86451} {{1,44.0261},{2,7.66781},{3,2.03332},{4,1.16014},{5,1.02482},{6,1.00385},{7,1.0006},{8,1.00009},{9,1.00001},{10,1.},{11,1.},{12,1.},{13,1.},{14,1.},{15,1.},{16,1.},{17,1.},{18,1.},{19,1.},{20,1.},{21,1.},{22,1.},{23,1.},{24,1.},{25,1.},{26,1.},{27,1.},{28,1.},{29,1.},{30,1.},{31,1.},{32,1.},{33,1.},{34,1.},{35,1.},{36,1.},{37,1.},{38,1.},{39,1.},{40,1.},{41,1.},{42,1.},{43,1.},{44,1.},{45,1.},{46,1.},{47,1.},{48,1.},{49,1.},{50,1.},{51,1.},{52,1.},{53,1.},{54,1.},{55,1.},{56,1.},{57,1.},{58,1.},{59,4.65946*10-46},{60,7.22083*10-

47},{61,1.11902*10-47},{62,1.73417*10-48},{63,2.68746*10-49},{64,4.1648*10-

50},{65,6.45425*10-51},{66,1.00023*10-51},{67,1.55006*10-52},{68,2.40216*10-

53},{69,3.72266*10-54},{70,5.76906*10-55},{71,8.94039*10-56},{72,1.38551*10-

56},{73,2.14714*10-57},{74,3.32745*10-58},{75,5.1566*10-59},{76,7.99127*10-

60},{77,1.23842*10-60},{78,1.9192*10-61},{79,2.97421*10-62},{80,4.60917*10-

63},{81,7.1429*10-64},{82,1.10695*10-64},{83,1.71545*10-65},{84,2.65846*10-

66},{85,4.11985*10-67},{86,6.3846*10-68},{87,9.8943*10-69},{88,1.53334*10-

69},{89,2.37623*10-70},{90,3.68248*10-71},{91,5.7068*10-72},{92,8.84391*10-

73},{93,1.37055*10-73},{94,2.12397*10-74},{95,3.29154*10-75},{96,5.10095*10-

76},{97,7.90502*10-77},{98,1.22505*10-77},{99,1.89848*10-78},{100,2.94211*10-

79},{101,4.55943*10-80},{102,7.06581*10-81},{103,1.095*10-81},{104,1.69694*10-

82},{105,2.62977*10-83},{106,4.07539*10-84},{107,6.31569*10-85},{108,9.78752*10-86}} 108.917 1.20505

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Chart 1. Time Slice Analysis.

“ Boltzmann” a*Exp[-b*x] + UnitStep[c-x] “ Zipf” a*x^(-b) + UnitStep[c-x]

Time a b sum square error a b sum square error 1 0 1 108 0 10^-17 .74 108 0 2 892.13 6.79 108.001 10^-6 1 54.038 108 10^-220 * 3 12.67 .926 108.303 .161 * 5.0977 1.755 109.763 .7496 4 51.44 1.546 106.927 1.166 10.988 2.338 108.508 .5061 * 5 36.951 1.137 108.449 2.411 11.998 1.897 109.832 .9460 * 6 74.067 1.478 103.877 7.124 16.905 2.120 108.034 2.9301 * 7 40.074 .8878 104.028 12.517 16.968 1.614 112.579 5.3662 * 8 31.399 .6154 108.93 1.305 * 17.760 1.404 120.07 15.4066 9 277.639 1.864 108.917 1.205 42.067 2.894 110.732 5.5518 10 193.606 1.411 110.396 10.636 47.315 2.335 114.845 32.2755 11 50980.4 6.664 108.074 .0069 65 59.164 108.0 10^-191 * 12 56079.3 6.686 108.077 .0077 70 60.975 108 10^-191 * 13 70954.6 6.787 108.081 .0082 80 61.398 108 10^-176 * 14 74762.5 6.722 108.095 .0119 90 61.952 108 10^-155 * 15 82419.7 6.694 108.112 .0161 102 60.066 108 10^-98 * 16 88236.2 6.714 108.125 .0168 107 192.201 108 10^-112

Chart 1. (cont’d)

“Normal (Gaussian)” aExp[-b*x^2]+ UnitStep[c-x]

Time a b sum square error 1 10^(-17) .4308 108 0 2 76.4 4.336 108 10^(-12) 3 6.79 .2602 107.91 .230 4 18.88 .5420 106.28 1.75 5 17.71 .3990 104.99 4.58 6 29.17 .5426 102.51 9.25 7 23.20 .3291 100.24 23.09 8 18.72 .1450 106.20 6.46 9 78.92 .6071 108.3 .1135 * 10 73.06 .4397 109.11 2.13 * 11 1573.29 3.186 108.003 .000022 12 1677.94 3.176 108.004 .000027 13 1954.29 3.195 108.005 .000030 14 2181.88 3.188 108.005 .000040 15 2414.75 3.164 108.006 .000062 16 2475.35 3.141 108.007 .000077

• = Best Method (Least error)

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Remark: The error can be reduced by mixing methods. For example taking a1 Exp[-b1 x]+ a2 x^(-b2) +a3 Exp[-b3 x^2]+ UnitStep[48-x] yields error .0004 in slice 10 with a1=-10.9,b1=2.72, a2=-199.5, b2= 6.65, a3= 628.9, b3=.933. However there remains the interpretation of negative densities.

EMERGY ANALYSIS

There seems to be little doubt that emergy will be maximized when the puzzle is completed, i.e. the picture on the puzzle has emerged, representing the used-up energy memory of fitting each piece into the puzzle. The Author’s previous paper “Emergy and Econophysics” suggests five procedures may be equivalent in some cases: 1) minimize Fisher information (gradients), 2) maximize covered sides (survival of fittest), 3) maximize entropy (2nd law of thermodynamics), 4) maximize emergy and 5) find result of generalized gravity (Newton dynamics). Here 1) occurs when the number of exposed sides (gradients, not counting top and bottom of puzzle pieces) is reduced from 432 to a minimum 42 around the border with no exposed sides on the interior. 2) occurs at the same time as the number of covered sides goes from 0 to 390 (=432-42). 3) occurs in the limited sense that, although the disorder of the puzzle has gone to zero, the disorder of the puzzle solver/puzzle system (cf. refrigerator analogy) has reached a maximum since the puzzle solver will not expend more energy once the puzzle is done (an “equilibrium” is reached), and 5) occurs in the sense that the puzzle pieces are pulled together to their maximum extent without overlapping (Pauli exclusion). Thus the other four cases can be studied as clues to finding 4) maximum emergy.

As far as 1) minimize Fisher information goes, each time two pieces of the given puzzle are put together at least two sides are no longer exposed, and in the case of filling in a small hole, eight sides will no longer be exposed, so that 1) can be calculated as the puzzle is worked on. As the Betti number is the number of holes, a large Betti number means rapid progress can be achieved by filling in the holes and reducing the number of exposed sides as the Betti number goes down. Similarly 2) can be calculated as 432 minus number of exposed sides. According the Giananntoni formula of Transformity as a product of generative transformity Tg and dissipative transformity Td, the overall transformity at a given stage could be measured as Tg x Td = SYM x ( # of covered sides), where SYM can be measured from the number of equal pairs of distances in the grid that represents puzzle pieces completed (cf. Author’s previous paper). For example if the border is completed, SYM is at least that of an empty rectangle. If another row from the bottom border, or a line of puzzle pieces representing the stripe on the pavement is completed, SYM goes up by the formula for a line, and so on. SYM reaches an “artificial symmetry” maximum of a solid rectangle when the 12x9 grid is finished. Please see later section on artificial symmetry. Supposing the isolated pieces are randomly positioned, they will not contribute to SYM as the puzzle is solved.

In this case, since SYM and (# of covered sides) both go up at the same time, it is probably unnecessary to calculate SYM. Taking this view, the transformity can be measured by number of covered sides, namely total 390, with about 130 due to border, 130 due to car features, and 130 due to pavement/stripe feature.

REMARK ON DISSIPATIVE TRANSFORMITY

Prigogine worked with “dissipative structures,” which are structures that dissipate energy while building order. Generally more order correlates with more energy dissipated, so that “dissipative transformity” could be measured by the energy dissipated. On the other hand dissipative transformity could be measured by the order created or built up, as by covered sides. Thus “dissipative transformity” can be measured by seemingly opposite quantities. For example in puzzle solving the amount of energy dissipated or spent piece.by.piece to put the puzzle together could be measured. Or on the other hand, the order created piece-by-piece as the picture emerges could be measured by the sides covered up.

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ANOMALY

The following calculation does not seem to fit the above analysis, although otherwise interesting. A previous paper raised the question of whether or not temperature could be measured as a product of a generative factor and a dissipative factor. Since here temperature goes down by simulated annealing as the emergy goes up, a temperature could be measured as T= (1/SYM) x (1/# covered sides) as the puzzle is solved. Another possible formula would be (1/SYM) x (# exposed sides), where the number of exposed sides represents randomness or dissipative factor (# of loose or isolated pieces). SYM reaches a maximum when there is maximum symmetry, as in a perfect crystal or perfect 12x9 grid, contributing to minimum temperature. The fact that water changes to ice at the same temperature 32 degrees F. may limit the value of this approach in the case of water, but maybe the relative importance of the two factors could change, leading to the same product value 32 as ice forms. For example, water could have Tg =1 and Td = 32 and ice could have Tg=32 and Td =1. Further, the product formula can be reduced to a sum formula by taking logarithms. Log[T] = Log[Tg*Td] =Log[Tg] + Log[Td], which could become a simple formula Log[T] = #covered sides + #isolated sides =const. or 0 + 5 at the start (say water) and {5+ 0) at the end (say ice) or all pieces in puzzle. Here Log[32] base 2 is 5 since 2^5 =32. More scientifically at freezing point of water base e, Log [273.2] =5.6102, so that with water at 273.2 degrees Kelvin, Log[Tg]=0 and Log[Td]=5.6102 and after conversion to ice with decrease of 22 units of entropy, Log[Tg]=5.6102 and Log[Td]=0 as the ice “grid” gets filled up. A problem is that there are about a dozen types of ice crystal. SUPERSYMMETRY CONSIDERATION

Puzzles are generally not left forever, but crunched up and put back into a smaller size box until possibly taken out later. The scale change in the case of this puzzle from 12x9 down to 9.5 x 6.5 inches (size of box) decreases the number of covered sides as the puzzle is broken up, and can be considered as a supersymmetry transformation, and a cycling can occur as the puzzle is taken out and put back in the box repeatedly, apparently at the whim of the owner of the box. This procedure can model death and the afterlife, as the pieces contain the information to re-construct the puzzle, or make it part of a larger puzzle. The completed puzzle can be considered as an equilibrium position between having pieces out on a table and closed in a box. In physics there is a speculated transition between bosons (light) and fermions (matter). Again there is little known about when or why these transitions may occur, except mostly in periods of high stress. CLUSTER-TO-STACKING ARTIFICIAL SYMMETRY

Although the preceding analysis is helpful for analyzing puzzles, it is based on taking clusters as

stacks, which may be called artificial symmetry, a name the Author doesn’t like but may be appropriate. It was discussed in the Author’s Emergy Synthesis 8 Chapter 3 talk “Emergy/Symplexity” but not included in the paper except for the Table 4 listing “4x4 Checker 2D” on page 25 with SYM 2384. In the case of a checker pattern, taking the distance between white and black squares as 1 unit can cause the symmetry to “pop” or become greater than it actually is by considering all squares of the same color to be stacked in their distance from a different color, a kind of “racism.” In the case of beauty this kind of cluster-to-stacking symmetry can be achieved somewhat by cosmetics. This procedure creates a kind of optical illusion. In the case of a puzzle the illusion is that the maximum symmetry MAXSYM is attained when the grid is completed. In fact the symmetry SYM is far from maximum when the grid is completed and could only be attained by taking 108 unit vectors in 108-dimensional space, for example, or all 108 points (pieces) stacked up at exactly the same point.. When the puzzle is completed, the pieces are not at the same distance from each other or physically “stacked” on top of each other. Also the entropy calculated could be considered an artificial or “puzzle” entropy, since the actual entropy would go to zero only if the puzzle “points” were in the same bin.

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GENERATIVE TRANSFORMITY

It is possible to make a program that computes SYM for a rectangular grid (or pillar in 3D) and compare it with actual maximum symmetry MAXSYM (= (n+1)*n*(n-1)*(n-2)/8 for n points), which is done in Table 2. Thus it might appear the SYMPLEXITY = SYM*(MAXSYM – SYM) is zero when the puzzle is completed, whereas in fact SYMPLEXITY is only on the increasing side of the graph and a few percent toward its maximum. Stated otherwise, the program makes a table of the distance between every two points; if those distances are set equal to say (cluster #) *e (instead of the true distance, thereby making more pairs of equal distance) if the two points (pieces of the puzzle) are in the same cluster, there is a case of artificial symmetry. This procedure has the effect of stacking up pieces that are in the same cluster.

The entry of interest here is n=108 with full 9x12 grid, whereby SYM is 468,282 versus MAXSYM of 16,689,753, giving symplexity = 7596222882822, far from the 0 of artificial “puzzle entropy” of Table. 1. For example the SYM of full rectangular 9x12 grid could be increased by taking a full 3x3x4 pillar. The grid entropy is based on distribution of distance between points and is extensive, generally increasing with n. It is an interesting math question to compute SYM of a square or rectangular grid or pillar, or whether any curve fitting program could find it. Previous work by the Author has calculated some things like this, which likely would involve gamma functions.

It is believed if a grid of gaussians were set up according to the number of pieces of the puzzle completed, and the maximum number of covered sides computed versus scale, it would correlate closely with the number of covered sides as counted in Table 1. Table 2. Results of computing SYM for a rectangular grid (or pillar in 3D) and comparing it with actual maximum symmetry MAXSYM (= (n+1)*n*(n-1)*(n-2)/8 for n points).

n # points Grid dim SYM grid ent MAXSYM SYM*(MAXSYM-SYM)=SYMPLEXITY

1 1x1 0 0 0 4 2x2 7 .636 15 7*8=56 5 1x5 10 45 10*35=350 9 3x3 138 1.493 630 138*492=67896 16 4x4 976 2.037 7140 976*6164=6016064 25 5x5 4242 2.446 44,850 172259136 27 3x3x3 8953 2.012 61,425 469781816 36 6x6 14,075 2.741 198.155 2000590926 49 7x7 38,248 3.016 690,900 24962633696 64 8x8 90,304 3.245 2,031,120 175263448064 81 9x9 192,316 3.446 5,247,180 972131225024 100 10x10 377,679 3.627 12,248,775 4483463666184 108 9x12 468,282 3.758 16,689,753 7596222882822 121 11x11 693,382 3.795 26,350,170 17789954977016 400 20x20 30,549,616 4.832 3,183,980,100 96336090368894144 500 20x25 60,595,164 5.069 7,781,218,875 467832459950333604 500 5x10x10 137,690,203 4.263 7,781,218,875 1052439014484000416

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WORLD LARGEST PUZZLE The world’s largest puzzle according to Guiness had 551,232 pieces on Sept, 24, 2011 in Ho Chi Minh City, takin 17 hours with 3132 sections of 176 pieces each and 1600 students. This problem apparently was some kind of take-off on the Towers of Hanoi problem. Dividing into sections as pre-sorting “enzymes,” as dealing with features, greatly simplifies the problem. It might be interesting to calculate the work entropy per piece versus the 108 piece case that took one hour for one person, the Author. ELECTRICAL GRID

It appears the above analysis may be relevant to repairing an electrical grid in terms of trying to form borders around the problem and look for features, such as regions that can be redone by similar procedures. For example in the case of restoring the grid in Puerto Rico after hurricane Maria, there were parts of the grid that could be restored based on various features, such as rebuilding substations, availability of cable and telephone poles, mountain work required, or special salt-resistant cable along seashores. Table 3 from Ferris (2018, time slices not uniformly spaced) shows per cent grid recovery in Puerto Rico after hurricane Maria. Recall per cent recovery can go down due to backlash from previous clusters in the clusters paper (Collins, 2017) or to misplaced or accidentally messed up pieces in the puzzle solving, or further outages in the electric grid case.

Romm (2017) also shows recovery after 2017 Hurricane Irma in the U.S. mainline, which looks very much like the top line in Figure 1. Table 3. Percent Grid Recovery in Puerto Rico After Hurricane Maria. Date Time slice % recovery Date Time slice % recovery 9/20 0 0 11/30 11 65 9/29 1 5 12/5 12 68 10/4 2 5 12/13 13 62 10/11 3 10 12/20 14 65 10/18 4 19 12/27 15 70 10/26 5 26 1/3/17 16 69 11/2 6 36 1/10 17 80 11/9 7 42 1/17 18 82 11/16 8 42 1/24 19 82 11/20 9 46 1/31 20 80 11/27 10 58 2/7 21 80 2/14 22 83

A FIRST THERMODYNAMIC MODEL

The Author’s 2001 paper “Thermodynamic Modeling of Moral Codes,” presented February 24 at the 16th SIDIM Conference in Humacao, PR, contained an optimization problem with linear cost and Boltzmann solution, which can be considered as a simulated annealing solution of a ten-piece puzzle. Within the paper, the program showed that a “Garden of Eden” solution with all probability concentrated at 0 (saint) level was possible at temperature T=1 if there were no constraint on average cost. As T increased by doubling to 2,4,8,16,32,64,128,256,512,1024, the distribution tended to the uniform (or drama) limit of .1, i.e. the starting point of puzzle solving. The next two pages repeat pages 18 and 22 of the paper. The filling out of levels as T increases correspond to higher crime levels (sin), the reverse of puzzle solving, which puzzle solving is like a return to the Garden of Eden.

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Remark: The author presented the basic paper at the MAA Tri-State Conference in Valparaiso, IN on Fri, March 23, 2018 and at the SIDIM XXXIII Conference at UPR-Rio Piedras, PR on March 24, 2018 and as a Sigma Xi Chapter 511 lecture at the Chemistry Building Q-124 on April 5, 2018, besides at the Emergy Synthesis 10 Conference in Gainesville, FL January 26, 2018.

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CONCLUSION

Thermodynamics can contribute to puzzle solving and puzzle solving can contribute to emergy analysis. Much work needs to be done, including working to obtain a partial differential equation for the process, and working with larger systems (puzzles) and calculating SYM for the given system. ACKNOWLEDGMENTS

Thanks are due Dr. Sharlynn Sweeney if this paper is included, and Glenn Collins and Dr. Leyda Almodovar for support. REFERENCES

Allis, William P and Herlin, Melvin A. 1952. Thermodynamics and Statistical Mechanics, McGraw-Hill, NY, pp.225-226.

Auberson, G. and Escoubes,B. 1965. Production of Light Bosons by Central Collisions at Very High Energies, El Nuevo Cimento, Vol XXXVI, N. 2, 16 Marzo, pp. 628-653 (Appendix B pp. 651-653).

Cheng, Chi-Ho 2009. Thermodynamics of the System of Distinguishable Particles, Entropy 11, pp.326-333.

Collins, Dennis G. 2015. Emergy/Symplexity. Chapter 3, pp.13-27. In Brown, M.T., Sweeney, S., Emergy Synthesis 8, Proceedings of the 8th Biennial Emergy Conference, Center for Environmental Policy, Univ. of Florida, Gainesville, FL.

Collins,`Dennis G. and Scienceman, David 2017. Clusters of High Transformity Individuals. Chapter 36 in Brown, M.T., Sweeney, S., Emergy Synthesis 9, Proceedings of the 9th Biennial Emergy Conference, Center for Environmental Policy, Univ. of Florida, Gainesville, FL

Ferris, David 2018. Puerto Rico’s grid recovery, by the numbers, E&E News Reporter Feb 20, 2018, recovered from Internet 8-1-18.

Hagedorn, Rolf, 1964. Thermodynamics of Distinguishable Particles: A Key to High-Energy Strong Interactions, Chapter 19, CERN-TH-483 Oct. 1964, preprint, pp. 183-222.

January, Eric O. 2005 Mo’ Sense Part I, Mo’ Sense Publications, pp.28-32. Nagle, John E. 2004. Regarding the Entropy of Distinguishable Particles, Journal of Statistical Physics

Vol 117, Nos 5/6 Dec., pp.1047-1062. Romm, Joe 2017. Stunning chart shows how badly Trump has bungled Puerto Rico’s grid recovery, Oct.

11, 2017, recovered from Internet 8-1-18. Swendsen, Robert H., 2002. Statistical Mechanics of Classical Systems with Distinguishable Particles,

Journal of Statistical Physics, Vol 107, Nos. 5/6 June, pp.1143-1166.

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