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Statistical Geometry aka Random Tiling

Explore the statistical properties of geometric shapes generated using a random tiling algorithm. Focus on circles and squares placed within a boundary and analyze the halting behavior.

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Statistical Geometry aka Random Tiling

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  1. STATISTICAL GEOMETRYakaRANDOM TILINGJohn ShierApril 2016 Shier -- Statistical Geometry and Art

  2. An Example 1000 circles are placed within a circle, filling 95% of it. The area of circle i is proportional to i-c where c = 1.46. The circles are placed with the largest first. The rule for placement of a circle is random search for a position which does not overlap any previously placed circle or the bounding circle. This resembles “Apollonian circles”, but the circles are all nontouching, and the gasket is a single connected whole. The area of the largest few circles is a large fraction of the whole area. Shier -- Statistical Geometry and Art

  3. Area Rule We wish to fill an area A with shapes. We define a sequence of areas Ai as a power law in isuch that where the Hurwitz zeta function is defined by This has the consequence that • Thus we have defined an infinite sequence of areas Ai that • Obey a negative-exponent power law in i with exponent c • Is space-filling in the limit i  • Has two parameters c and N (c > 1, N > 0) The rule also applies in 1D (lengths) or 3D (volumes). Shier -- Statistical Geometry and Art

  4. The Placement Algorithm There is no STOP in this chart because as far as is known the algorithm does not halt within the allowed ranges of c and N. Does not halt if cmax > c > 1 N > Nmin > 0 cmax depends on the shape and is highest for simple compact shapes This algorithm is SLOW Shier -- Statistical Geometry and Art

  5. Summary: The boundary can have any shape provided the N and c values allow placement of shapes 0 and 1. The placed areas can have any shape or sequence of shapes and the algorithm will run without halting as long as the area rule is obeyed and cmax > c > 1. Shier -- Statistical Geometry and Art

  6. The algorithm for circles. The first 25 circle placements (red) and the associated trials (black). Constrained randomness. The unused space is called the “gasket”. Shier -- Statistical Geometry and Art

  7. Halting It has been observed in thousands of runs that there is a wide range of c values for which halting is not seen. For sufficiently high c values it becomes impossible to place shape 1. Thus there is a maximum c value. The maximum c value depends upon the shape and N, and is highest for compact shapes (circle, square) and lowest for sparse ones (e.g., snowflakes). When halting is seen, it is usually in the first 100 placements. Shier -- Statistical Geometry and Art

  8. DOES THE ALGORITHM HALT? The question of whether the algorithm halts is of central interest for understanding it. Recently a formal proof of nonhalting has been found for the special case of circles fractalized within a circle. The circle-circle algorithm is unconditionally nonhalting with 1.0965 > c > 1 when N = 1. Christopher Innes Math Horizons, Feb. 2016 issue, pp. 8-12 Shier -- Statistical Geometry and Art

  9. Is This True? a busy page This gives data from 2000 repeated runs per c value. The raw data is given, as well as data sorted by the number of placements at which halting occurred (top region). It can be seen that the points make a good approximation to a continuous s-shaped curve. The data in the table shows that if placement fails (i.e., the algorithm halts) it occurs for a rather low number of placements. This provides good statistical support for the idea of a range of c in which halting does not occur, but it is not a formal proof. Shier -- Statistical Geometry and Art

  10. Run-Time Behavior – log-log plots another busy page (cumulative trials nc versus placed shapes n) Log-log plots (log10) of the cumulative number nc of trials (y axis) needed to place n shapes versus n (x axis). This data is well fitted by a straight line, i.e., nc(n) obeys a power law in n. On this basis, one can conclude that the process never halts. The different groups of curves have different c values (largest slope = largest c) and N=1. There is a largest c value, which depends on the shape. The data gets much noisier for large c values. The exponent f in the power law nc(n) obeys f  c For the circles the c values are (from the bottom): 1.20, 1.25, 1.30, 1.35, 1.40. For the squares the c values are (from the bottom): 1.20, 1.25, 1.30, 1.35, 1.40, 1.45. Data from five runs is shown for each c value. In all cases N = 1. log10(placements) Shier -- Statistical Geometry and Art

  11. Squaring the Circle Here 5000 squares are fractalized within a circular boundary. The fill is nearly 98% and illustrates the high fill factors achievable with squares. The gasket is white, but with this high fill factor you don’t see it. Shier -- Statistical Geometry and Art

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  14. 1D Example In one dimension we have a Cantor-like random fractal that can have a fractal dimension as low as about D = .38-.39 (versus D = .6309… for the Cantor set). If one wishes to reach rigorous conclusions about these fractals this is probably the place to start. Shier -- Statistical Geometry and Art

  15. 3D Example Courtesy of Paul Bourke, U. of Western Australia. Here we see 10000 toroidal rings with random orientations fractalized within a cube. The rings are colored by size, with the largest red and the smallest blue and green. The rings are shaded to give a better view. It can be seen that the randomness results in a few of the rings being chain-linked to other rings, i.e., random fractal topology. (4D works too) Shier -- Statistical Geometry and Art

  16. A real gasket (black). A key to the halting problem is whether the gasket can always accommodate another shape. Its width as seen “by eye” is bounded. Shier -- Statistical Geometry and Art

  17. Average Gasket Width as a Concept area = A length = L The blue blob stands for the (shrinking) gasket area at some stage of the algorithm, and the line for the (growing) perimeter. You are asked to tug and knead the blob until it has the perimeter of the line. You cannot have big blobs and long thin gerrymanders. You must cut most of the line into segments and use them to create holes inside the blob. Mental sumo wrestling with this will show that area/perimeter is a plausible measure of average gasket width. The simplest solution is just widen the line to width A/L. Shier -- Statistical Geometry and Art

  18. The Dimensionless Average Gasket Width (numerical – circles – N=1) This data assumes N = 1. The values do not depend on the random numbers. b(c,1,n) may go to a limit as n  although this data does not rigorously prove it. The available space shrinks in lock-step with the next-to-be-placed circle size. Shier -- Statistical Geometry and Art

  19. The Dimensionless Average Gasket Area This quantity gives a value for the average gasket area relative to the area of the next-to-be placed shape. It is valid for any shape. Useful for proofs? When c approaches 1, there is a very large amount of room for the next shape. values of gasket_area/(n_place*next_area) N=1 c=1.2000000 c=1.2500000 c=1.3333333 n= 1 21.937114 17.958609 13.995080 2 13.413474 10.921224 8.434314 4 9.179358 7.431837 5.686002 8 7.078779 5.704498 4.330698 16 6.035766 4.848462 3.661286 32 5.516818 4.423121 3.329454 64 5.258117 4.211258 3.164403 128 5.128981 4.105550 3.082116 256 5.064469 4.052754 3.041035 512 5.032227 4.026372 3.020510 1024 5.016109 4.013184 3.010251 2048 5.008051 4.006591 3.005122 4096 5.004022 4.003295 3.002556 8192 5.002006 4.001647 3.001273 16384 5.000998 4.000823 3.000629 32768 5.000493 4.000411 3.000306 65536 5.000240 4.000204 3.000142 Shier -- Statistical Geometry and Art

  20. Fractal Dimension The allowable c values are quite different depending upon the number d of Euclidean dimensions. For 1D, c can range up to about 2.7; for 2D it can go up to around 1.5, for 3D it is difficult to go beyond c values around 1.2. (c must always be > 1) Physical Review papers on random Apollonian packing lead to a very simple formula for the fractal dimension D: D = d/c It agrees with numerical “box counting” estimates of the fractal D for 1D and 2D done by Prof. Paul Bourke (U. of Western Australia). References: P. S. Dodds, J. S. Weitz, Phys. Rev. E, 65, 056108-1, (2002). P. S. Dodds, J. S. Weitz, Phys. Rev. E, 67, 016117-1, (2003). G. W. Delaney et al., Phys. Rev. Letters, 101, 120602, (2008). Shier -- Statistical Geometry and Art

  21. Order and Disorder Randomly fractalized equilateral triangles. In all cases there are 1200 triangles and N = 4. (a) c = 1.24; (b) c = 1.21; (c) c = 1.18; (d) c = 1.15. In (a), which has a c value close to the upper limit, there is strong short-range order, with most triangles having a triplet of near neighbors (see the red group). In (d) the arrangement is much more random. The average number of trials needed for a placement is very high in (a) and much lower in (d). We thus see that the exponent c acts as an “order parameter”, which allows one to choose the degree of order or disorder in the arrangement of the shapes. It is also seen that the fill factor is much lower in (d) (with a fixed number of placed shapes). Shier -- Statistical Geometry and Art

  22. Connection with Sierpinski The Sierpinski construction is in a sense the high-c limit for fractalized triangles. Its fractal D falls nicely in place with the sequence of D values for random triangle fractals. This is another way of looking at the existence of an upper limit for c. (Of course this could all be just be a coincidence.) Shier -- Statistical Geometry and Art

  23. Chaos and Order for Circles This is much the same as for triangles, showing how the Apollonian circles make an end point for tighter and tighter random circle fractals. Shier -- Statistical Geometry and Art

  24. The Unexplored What does “near neighbor” mean here? With circles one can easily compute the edge-to-edge distances for every circle pair. If you draw a line from center to center for every pair with spacing less than some fixed value, you get this – a rather elaborate graph of the “nearby” circles. There are connected nets of every size, with one of the largest ones shown in blue. Shier -- Statistical Geometry and Art

  25. Hollow Shapes When the shape has a hole in it, the gasket breaks up into many pieces defined by the shape of the hole. It is possible to define a hierarchy of ranks for the shapes. A shape with nothing inside it is rank 0, a shape with only shapes of rank 0 inside it has rank 1, etc. If the highest rank inside a shape is m, its rank is m+1. There is 1 shape of rank 3, 13 of rank 2, 36 of rank 1, and 101 of rank 0. Rank tracks closely with size, and changes at size boundaries. The number of shapes of rank m follows a negative-exponent power law in m. Shier -- Statistical Geometry and Art

  26. Squares and Diamonds White squares and black diamonds; red gasket. Alternating squares and diamonds “don’t mix” at a high c value (c=1.40). The black and white “islands” have “coastlines” with bays and promontories on all length scales, so that the coastline is also a fractal with its own fractal dimension (like the coastline of England). Clustered filling is a quite general feature with two or more shapes. Shier -- Statistical Geometry and Art

  27. Mutually Avoiding Circles We have seen that when c is “small” (only a little more than 1) there is a lot of vacant space in the pattern. Consider circles. Can one impose an additional constraint that the circles must have a minimum separation from each other? YES. We introduce a new parameter b and insist that circle i with radius ri must be at least bri from any previously-placed circle. When such a place is found by random search the new circle is placed in the data base with radius ri. Statistical studies show that the algorithm continues to be nonhalting. If b > 1 the maximum c is lowered. Shier -- Statistical Geometry and Art

  28. The b value increases (clockwise from upper left) from 1.0 to 2.0. Note the “grouted” look of the lower left image (b =2). The circle areas follow the same sequence of values in all cases. Shier -- Statistical Geometry and Art

  29. Tessellations – Wallpaper Groups One can use the algorithm to fill the primitive unit cell of a tessellation, and then expand the pattern according to the rules for the particular wallpaper group of interest. Here we have arrows in a 30º triangle, with the p6mm wallpaper group. There is an interesting blend of randomness and order. Shier -- Statistical Geometry and Art

  30. Boundary Circles p2mm In the previous example all shapes were entirely inside the boundary. Here the algorithm is modified so that any circle which crosses the boundary is moved perpendicular to the boundary and centered on the boundary before testing for overlap with previous circles. This is another way to implement mirror lines. The random positions and random colors conceal the underlying symmetry. Shier -- Statistical Geometry and Art

  31. Natural Power Laws Many natural power-law statistical distributions are known, including >> Magnitudes of earthquakes (Gutenberg-Richter circa 1950) >> Sizes of wars (by lives taken) >> Large incomes (Pareto, circa 1890) >> Citations of scientific papers >> Sales of books >> Word usage (Zipf, 1949) >> Populations of cities >> … many more … (all of these are statistics of random events) Irreversible If the placement sequence is viewed as progression in time, the process is irreversible – it only runs biggest-to-smallest and not smallest-to-biggest. The “arrow of time” runs only one way. It shares this property with random walk. Shier -- Statistical Geometry and Art

  32. Computation and Mathematics Thus far, the subject has been explored as “computer experiments” and conclusions reached by induction. I have looked for counterexamples (shapes that can’t be fractalized) and thus far have not found any. I am not aware of any other mathematically-defined structure which so intimately mixes ideas from function theory (power law, zeta function), statistics (randomness), and geometry (shapes). Most recursive geometric fractals have a single specific fractal dimension D. This construction can have any fractal Dyou wish over a quite wide range. How many loose ends does this subject have? Dozens. Shier -- Statistical Geometry and Art

  33. Conjectures (1) The algorithm does not halt within the allowed range of c and N for a given shape. This may be the “central point” of this mathematical structure. It is evident that this property requires a rather delicate balance in the sequence of areas versus n. Even a proof for a specific shape (e.g., circle) would be of much interest. This property is supported by the observed power law seen in trials versus placements. (2) The algorithm only works for an area sequence obeying a power law in placement number. Are there any alternative area rules which would also work? (3) The algorithm can “fractalize” any shape or sequence of shapes obeying the area rule for some cmax > c > 1 and N > Nmin > 0. On the face of it this is a remarkable statement and not at all intuitive or obvious. It is not (yet?) contradicted by any of the available evidence despite substantial searching. Compact shapes (circle, square) work up to a high c value (1.5), while sparse, sprawling shapes have lower maximum c values. What form would proofs of these claims actually take? Had I not read Mandelbrot’s book, none of this would have been found. Shier -- Statistical Geometry and Art

  34. Uses in Teaching The 2D algorithm could be an interesting problem for computer science students. For circles and squares the code should be within the grasp of most students near the end of the first course or in a later course. The main iterative loops take only 40-50 lines of code in C for circles or squares. A C-code example can be downloaded from my web site. It could provide an interesting student problem for computer graphics courses. In calculus courses one always passes to the limit and (usually) gets a neat result. In this case the computations can only approach an infinite limit, but can never get there. It illustrates “rate of convergence” for students. The algorithm gets slower and slower as huge numbers of shapes are fractalized. (More efficient search algorithms could ease this.) Shier -- Statistical Geometry and Art

  35. This is not an invention so much as a discovery. If Pythagoras had had a high speed computer with good graphics he would have found it first. Much more info at the web site john-art.com Shier -- Statistical Geometry and Art

  36. The Meaning of Probability … all problems in probability are ultimately about quantum mechanics. “Every single time we use probability successfully, that use actually comes from quantum mechanics” says Albrecht. (and I thought it was just about numbers) -- cosmology researcher Andreas Albrecht taken from The New Scientist, 5-Jan-2013, p. 8. Shier -- Statistical Geometry and Art

  37. Have powerpoint, will travel. I am prepared to present this material to any student or faculty group within the greater Twin Cities. End Ende Fin Fine Finis Конец Shier -- Statistical Geometry and Art

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