Drawing of algorithm
0001-1233.webm
Animation of algorithm
- Weave animated
- Graph object as artist's muse
- Digital Discocubes
- Command Line Usage
- Plotting the Solution
- Dependencies
- Running Times up to 8 billion
- Licensing
- â
Planar embedding of a cube and a discocube. From the set of all graphs G, where the order of G is of the Uncentered octahedral numbers A130809, only the first two instances shown above; n[0] and n[1] are planarly embeddable i.e., it can be represented on a two-dimensional surface without any of its edges crossing.
⊠Graph object as artist's muse
âA great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.â
George PÃģlya: How to Solve It: A New Aspect of Mathematical Method
A deterministic and linear-time algorithm for constructing a Hamiltonian cycle on all instances of discocube graphs (tested for graphs with over 8 billion vertices, the world's population). Discocube graphs are 3-dimensional grid graphs derived from: a polycube of an octahedron | a Hauy construction of an octahedron with cubes as identical building blocks | the accretion of cubes around a central cube forming an octahedron at the limit | the subgraph of the infinite 3-dimensional square grid graph consisting only of points contained within an octahedron | a 3d L1-norm unit ball.
This algorithm is an artist's rendering of his muse, a graph object, using programming as a language (instead of painting flowers and apples or singing hymns about angels) and a means by which to describe his muse's body as an endless contour drawing or, in graph theory terms, a Hamiltonian cycle.
Pablo Picasso: Trois Danseuses (The Three Dancers)
The graphs from platonic solids are all Hamiltonian.
What links an endless contour line drawing and a Hamiltonian cycle? Both entail tracing a path without interruption, using a continuous line to describe the subject/object to be depicted. An endless contour line drawing is one in which the artist uses a single, uninterrupted line to describe a subject's form and shape. Similarly, a Hamiltonian cycle describes a graph by tracing a path along edges of the graph, visiting vertex precisely once before returning to the beginning vertex. The path connects the vertices in a continuous line, describing the graph object.
Named after Sir William Rowan Hamilton, the Hamiltonian cycle problem is a classic graph theory problem solved by finding a closed loop in a graph that visits every node exactly once and ending at the starting point. Hamilton formulated the problem in the rules of his Icosian game, in which players insert numbered pegs into holes on a wooden board to represent steps in a path. The objective is to insert the pegs in order along a path to form a closed loop, much like the Hamiltonian cycle problem where the pegs represent nodes in a graph and the path of inserted pegs represents the cycle.
After pages of studies, drawings, and a little math, this algorithm is the result of using the artistic process to solve a mathematical problem without having the means by which to solve it mathematically. A love-child of banging your head against the wall until a hole appears (learning something) and bending the wall with your mind through pure will and imagination (winging it until you fly). When a graph becomes an artist's muse, how does the artist go about rendering their vision as a painter would paint a portrait, making it their own? Can one claim authorship by merely copying a form? If my new material is programming, would functions fill the studio of my mind otherwise filled with machines and material? Will I find patterns in the music I make from finding solutions?
dcbau.mp4
An artist manipulates their medium to create forms, using brush strokes to describe how the curve of the neck disappears behind the back, or playing with colors and contrasts to subtly bring the skin of a subject living 500 years ago back to life. For many, the medium is the end to itself, like painting a painting or photographing a photo, rather than the medium being wholly dependent on the concept or idea to be executed. In my artistic practice the medium used for any particular project is dependent on the first finding the right language and then developing it into a visual language.
In this project, I studied the discocube visually as a body, imagining each turn not as a discrete mathematical object but as a series of possible movements, as an endlessly iterated dance captured in a long camera exposure, resulting in less equating and more doodling, and wishing I knew more math. The result is a family of algorithms for finding Hamiltonian cycles with varying degrees of refinement (edge distribution), with the weave algorithm producing the least refinement. The other algorithms are concerned with finding an initial Hamiltonian cycle with a higher mutation rate and whose edges are more uniformly spread across the three axes x, y, and z, and through subsequent processes, the solution is polished like a diamond, i.e., the initial tour improved upon until the edges are evenly distributed across all three axes, culminating in an always-turning Hamiltonian cycle. Owing to the regularity and consistency of the solution produced by this determined and predictive algorithm, using weave() to obtain a diamond-grade discocube would take an inconceivable period of time, necessitating the development of other algorithms capable of accomplishing this reasonably.
dc_clseup.mp4
Why weave? Finding the solution to the problem reminded me of macramÃĐ, of tying knots, weaving, and spinning yarn. I thought of how patterns in handwoven fabric are actually unwitting recordings of a knitter's hand movements, like how a piano roll is a recording of the pianist's finger hitting ebony, or how a seismograph records the motion of the earth, or how our skin is a type of recording of our life... I followed the thought further and asked myself: was there a pattern to expose and use to construct the discocube, level by level, similar to how one would knit a scarf, row by row, until the desired result is reached? To illustrate the intention of the code succinctly, I've structured the code to mimic the process of weaving a piece of tapestry, from spinning the yarn to incorporating the weft into the warps.
The first eleven discocubes and their respective orders (number of nodes)
To paraphrase Hauy:
When solving problems that involve analyzing how nature progresses, we are led by very rapid methods to results that are not immediately obvious. These results may appear paradoxical and surprising. However, if we take the time to carefully examine the steps we took to reach these results, we will begin to understand the underlying principles that led to these outcomes. By going back over the process step by step, we can better understand the logic behind the final results.
What started as a hack-your-own version of a depth-first search with shortcuts for the discocube graph (solving up to 960 vertices) metastasized into pages of overgrown mixin classes mysteriously coupled to each other like overgrown vines, pushing me deeper and deeper into the underbelly of its mutant tentacles. Although able to solve instances with over a million vertices, the algorithm had the clarity of primordial soup. So, as a sadistic gardener, I painstakingly pruned my own pubicity (my unescapable web of thorny vines) into presentable tiny bonsai trees. So what is a bonsai if not a tree in intimate scope?
dc_thomas.mp4
The result of this creative process is a family of algorithms developed specifically to solve various graph problems on dodecahedron graphs, 3D grid graphs, and hexprism honeycomb diamond graphs. The algorithm presented in this repository is the least complex, making it the fastest. It does the job, solving the Hamiltonian cycle problem for over millions of vertices in seconds and graphing with over a billion vertices in less than an hour and a graph with over 8 billion vertices in less than 5 hours, while other algorithms in the family take longer but also have other objectives, like forming an always-turning cycle with even edge distribution across all axes. But that's beyond the scope of this repository.
Detail for a Hamiltonian cycle for a graph with 79,040 nodes.
This algorithm has no while loops and will terminate after a definitive set of steps. The strength of this algorithm is knowing exactly when, where, and what is to happen, thereby reducing the amount of calculations needed (which is surprising as the creative process in creating this was anything but deterministic). It is a construction algorithm, constructing the path layer by layer until loops are produced, which are then joined using cycle merging. Further optimizations of the algorithm have also discarded the memory-heavy adjacency list, choosing instead to perform individual calculations where needed. Making and solving a graph with over a billion vertices, where n = 1000, won't require a distributed graph engine on the cloud anymore, and it takes a little over ten minutes.
⊠Links
Discocubes 8 - 1760
⊠digital discocubes
Play around and inspect a digital discocube here
As each solution is as unique as a fingerprint, or a diamond it allows one to have their own digital version of a discocube, which is also an instruction for building your own.
see confetti cube digital here
Discocubes as glb, using different mirrored texture yields personalized results and unique reflections meaning each discocube has its own reflection/shadow fingerprint! With millions of combinations available (glass texture/image/color, mirror texture/image/color, edge texture/image/color), the possibilities are endless!
The always turning hamiltonian cycle digital discocubes are not produced by the algorithm in this repository, but by another polynomial-time algorithm.
⊠Command line usage
To use the package via the command line, navigate to the root directory of the project in your terminal and run the following command:
cargo run --release [Graph start instance] [Graph end instance] [steps] [repeats]
cargo run --release 1 100 1 100
For each graph starting from 32 to 1.373 million vertices solve each graph order in steps of one and running each 100x to get the best time.
⊠Plotting the solution
The solution can be plotted using this python module to visualize and check the solution.
⊠Dependencies
This repository uses the following crates (ordered by most to least used) for which it is grateful.
For iterator traits, ndarrays, matrix operations on ndarrays, and parallelizing sequential computations:
itertoolsExtra iterator adaptors, functions and macros.rayonData-parallelism library for parallelizing sequential computations whilst guaranteeing data-race freedom.ndarrayProvides an n-dimensional container for general elements and for numerics.
For serializing and writing to .csv file:
csvFast and flexible CSV reader and writer, with support for Serde.serdeFramework for serializing and deserializing Rust data structures efficiently and generically.
For timestamping:
chronoA feature-complete superset of the time library.
⊠Running times
Running times from 8 to over 8 billion vertices
⊠Running times for graphs with 8 to over 8 billion vertices (solved in under 1 hour)
| ðģ 1 | âïļ 8 | ð 0.0000121670
| ðģ 2 | âïļ 32 | ð 0.0000336670
| ðģ 3 | âïļ 80 | ð 0.0000556250
| ðģ 4 | âïļ 160 | ð 0.0000860830
| ðģ 5 | âïļ 280 | ð 0.0001728340
| ðģ 6 | âïļ 448 | ð 0.0002152500
| ðģ 7 | âïļ 672 | ð 0.0002383330
| ðģ 8 | âïļ 960 | ð 0.0002807920
| ðģ 9 | âïļ 1320 | ð 0.0003124580
| ðģ 10 | âïļ 1760 | ð 0.0003534590
| ðģ 11 | âïļ 2288 | ð 0.0003796250
| ðģ 12 | âïļ 2912 | ð 0.0004421670
| ðģ 13 | âïļ 3640 | ð 0.0004606250
| ðģ 14 | âïļ 4480 | ð 0.0005024590
| ðģ 15 | âïļ 5440 | ð 0.0005727920
| ðģ 16 | âïļ 6528 | ð 0.0006797920
| ðģ 17 | âïļ 7752 | ð 0.0006532500
| ðģ 18 | âïļ 9120 | ð 0.0006986670
| ðģ 19 | âïļ 10640 | ð 0.0008152920
| ðģ 20 | âïļ 12320 | ð 0.0008502500
| ðģ 21 | âïļ 14168 | ð 0.0009908330
| ðģ 22 | âïļ 16192 | ð 0.0009650000
| ðģ 23 | âïļ 18400 | ð 0.0011169170
| ðģ 24 | âïļ 20800 | ð 0.0011144590
| ðģ 25 | âïļ 23400 | ð 0.0011674589
| ðģ 26 | âïļ 26208 | ð 0.0013020000
| ðģ 27 | âïļ 29232 | ð 0.0013477500
| ðģ 28 | âïļ 32480 | ð 0.0014275840
| ðģ 29 | âïļ 35960 | ð 0.0014958750
| ðģ 30 | âïļ 39680 | ð 0.0014919169
| ðģ 31 | âïļ 43648 | ð 0.0015422500
| ðģ 32 | âïļ 47872 | ð 0.0016930420
| ðģ 33 | âïļ 52360 | ð 0.0017866250
| ðģ 34 | âïļ 57120 | ð 0.0018867080
| ðģ 35 | âïļ 62160 | ð 0.0020575831
| ðģ 36 | âïļ 67488 | ð 0.0020389170
| ðģ 37 | âïļ 73112 | ð 0.0022446250
| ðģ 38 | âïļ 79040 | ð 0.0022678750
| ðģ 39 | âïļ 85280 | ð 0.0024716251
| ðģ 40 | âïļ 91840 | ð 0.0025920840
| ðģ 41 | âïļ 98728 | ð 0.0026535830
| ðģ 42 | âïļ 105952 | ð 0.0027707920
| ðģ 43 | âïļ 113520 | ð 0.0029962501
| ðģ 44 | âïļ 121440 | ð 0.0029682911
| ðģ 45 | âïļ 129720 | ð 0.0033047090
| ðģ 46 | âïļ 138368 | ð 0.0034581251
| ðģ 47 | âïļ 147392 | ð 0.0036917501
| ðģ 48 | âïļ 156800 | ð 0.0038126251
| ðģ 49 | âïļ 166600 | ð 0.0039234171
| ðģ 50 | âïļ 176800 | ð 0.0039170841
| ðģ 51 | âïļ 187408 | ð 0.0042002499
| ðģ 52 | âïļ 198432 | ð 0.0043289168
| ðģ 53 | âïļ 209880 | ð 0.0043575000
| ðģ 54 | âïļ 221760 | ð 0.0045866668
| ðģ 55 | âïļ 234080 | ð 0.0048543750
| ðģ 56 | âïļ 246848 | ð 0.0049461662
| ðģ 57 | âïļ 260072 | ð 0.0054035839
| ðģ 58 | âïļ 273760 | ð 0.0055786660
| ðģ 59 | âïļ 287920 | ð 0.0055797920
| ðģ 60 | âïļ 302560 | ð 0.0060125832
| ðģ 61 | âïļ 317688 | ð 0.0062666670
| ðģ 62 | âïļ 333312 | ð 0.0064859171
| ðģ 63 | âïļ 349440 | ð 0.0067297500
| ðģ 64 | âïļ 366080 | ð 0.0070810001
| ðģ 65 | âïļ 383240 | ð 0.0073842080
| ðģ 66 | âïļ 400928 | ð 0.0075188749
| ðģ 67 | âïļ 419152 | ð 0.0080193747
| ðģ 68 | âïļ 437920 | ð 0.0091587501
| ðģ 69 | âïļ 457240 | ð 0.0094685405
| ðģ 70 | âïļ 477120 | ð 0.0086249169
| ðģ 71 | âïļ 497568 | ð 0.0091080833
| ðģ 72 | âïļ 518592 | ð 0.0094844159
| ðģ 73 | âïļ 540200 | ð 0.0098849582
| ðģ 74 | âïļ 562400 | ð 0.0103974165
| ðģ 75 | âïļ 585200 | ð 0.0107163750
| ðģ 76 | âïļ 608608 | ð 0.0108824996
| ðģ 77 | âïļ 632632 | ð 0.0113353329
| ðģ 78 | âïļ 657280 | ð 0.0114379581
| ðģ 79 | âïļ 682560 | ð 0.0119424583
| ðģ 80 | âïļ 708480 | ð 0.0127124172
| ðģ 81 | âïļ 735048 | ð 0.0130524170
| ðģ 82 | âïļ 762272 | ð 0.0137428343
| ðģ 83 | âïļ 790160 | ð 0.0139735406
| ðģ 84 | âïļ 818720 | ð 0.0146117499
| ðģ 85 | âïļ 847960 | ð 0.0145710828
| ðģ 86 | âïļ 877888 | ð 0.0154153341
| ðģ 87 | âïļ 908512 | ð 0.0159500837
| ðģ 88 | âïļ 939840 | ð 0.0162739586
| ðģ 89 | âïļ 971880 | ð 0.0171253756
| ðģ 90 | âïļ 1004640 | ð 0.0177017916
| ðģ 91 | âïļ 1038128 | ð 0.0178309605
| ðģ 92 | âïļ 1072352 | ð 0.0184610840
| ðģ 93 | âïļ 1107320 | ð 0.0192627925
| ðģ 94 | âïļ 1143040 | ð 0.0196390003
| ðģ 95 | âïļ 1179520 | ð 0.0203737915
| ðģ 96 | âïļ 1216768 | ð 0.0208999999
| ðģ 97 | âïļ 1254792 | ð 0.0215984602
| ðģ 98 | âïļ 1293600 | ð 0.0222403761
| ðģ 99 | âïļ 1333200 | ð 0.0230612494
| ðģ 100 | âïļ 1373600 | ð 0.0237264577
| ðģ 101 | âïļ 1414808 | ð 0.0241850000
| ðģ 102 | âïļ 1456832 | ð 0.0250747073
| ðģ 103 | âïļ 1499680 | ð 0.0259533338
| ðģ 104 | âïļ 1543360 | ð 0.0269267503
| ðģ 105 | âïļ 1587880 | ð 0.0275232922
| ðģ 106 | âïļ 1633248 | ð 0.0281232502
| ðģ 107 | âïļ 1679472 | ð 0.0291454587
| ðģ 108 | âïļ 1726560 | ð 0.0296714995
| ðģ 109 | âïļ 1774520 | ð 0.0304882079
| ðģ 110 | âïļ 1823360 | ð 0.0312354155
| ðģ 111 | âïļ 1873088 | ð 0.0320182070
| ðģ 112 | âïļ 1923712 | ð 0.0330021679
| ðģ 113 | âïļ 1975240 | ð 0.0342445411
| ðģ 114 | âïļ 2027680 | ð 0.0350919999
| ðģ 115 | âïļ 2081040 | ð 0.0360104553
| ðģ 116 | âïļ 2135328 | ð 0.0369302928
| ðģ 117 | âïļ 2190552 | ð 0.0381915830
| ðģ 118 | âïļ 2246720 | ð 0.0385576226
| ðģ 119 | âïļ 2303840 | ð 0.0401081257
| ðģ 120 | âïļ 2361920 | ð 0.0406793319
| ðģ 121 | âïļ 2420968 | ð 0.0421322919
| ðģ 122 | âïļ 2480992 | ð 0.0432262495
| ðģ 123 | âïļ 2542000 | ð 0.0448335856
| ðģ 124 | âïļ 2604000 | ð 0.0458601229
| ðģ 125 | âïļ 2667000 | ð 0.0467119589
| ðģ 126 | âïļ 2731008 | ð 0.0481842496
| ðģ 127 | âïļ 2796032 | ð 0.0490408763
| ðģ 128 | âïļ 2862080 | ð 0.0506184176
| ðģ 129 | âïļ 2929160 | ð 0.0517759174
| ðģ 130 | âïļ 2997280 | ð 0.0529624149
| ðģ 131 | âïļ 3066448 | ð 0.0550655387
| ðģ 132 | âïļ 3136672 | ð 0.0554944165
| ðģ 133 | âïļ 3207960 | ð 0.0574050397
| ðģ 134 | âïļ 3280320 | ð 0.0590833314
| ðģ 135 | âïļ 3353760 | ð 0.0595314987
| ðģ 136 | âïļ 3428288 | ð 0.0616207533
| ðģ 137 | âïļ 3503912 | ð 0.0627114177
| ðģ 138 | âïļ 3580640 | ð 0.0649086237
| ðģ 139 | âïļ 3658480 | ð 0.0664635450
| ðģ 140 | âïļ 3737440 | ð 0.0669075847
| ðģ 141 | âïļ 3817528 | ð 0.0695139617
| ðģ 142 | âïļ 3898752 | ð 0.0711619630
| ðģ 143 | âïļ 3981120 | ð 0.0727650821
| ðģ 144 | âïļ 4064640 | ð 0.0747104138
| ðģ 145 | âïļ 4149320 | ð 0.0756006241
| ðģ 146 | âïļ 4235168 | ð 0.0783675835
| ðģ 147 | âïļ 4322192 | ð 0.0801399201
| ðģ 148 | âïļ 4410400 | ð 0.0825362056
| ðģ 149 | âïļ 4499800 | ð 0.0837670788
| ðģ 150 | âïļ 4590400 | ð 0.0853706226
| ðģ 151 | âïļ 4682208 | ð 0.0879387930
| ðģ 152 | âïļ 4775232 | ð 0.0900958702
| ðģ 153 | âïļ 4869480 | ð 0.0924538299
| ðģ 154 | âïļ 4964960 | ð 0.0951462910
| ðģ 155 | âïļ 5061680 | ð 0.0967189595
| ðģ 156 | âïļ 5159648 | ð 0.0988397151
| ðģ 157 | âïļ 5258872 | ð 0.1019583791
| ðģ 158 | âïļ 5359360 | ð 0.1027188748
| ðģ 159 | âïļ 5461120 | ð 0.1059616208
| ðģ 160 | âïļ 5564160 | ð 0.1081095040
| ðģ 161 | âïļ 5668488 | ð 0.1102593392
| ðģ 162 | âïļ 5774112 | ð 0.1126611307
| ðģ 163 | âïļ 5881040 | ð 0.1150461286
| ðģ 164 | âïļ 5989280 | ð 0.1181682497
| ðģ 165 | âïļ 6098840 | ð 0.1205536276
| ðģ 166 | âïļ 6209728 | ð 0.1237108335
| ðģ 167 | âïļ 6321952 | ð 0.1253125370
| ðģ 168 | âïļ 6435520 | ð 0.1278068274
| ðģ 169 | âïļ 6550440 | ð 0.1313734204
| ðģ 170 | âïļ 6666720 | ð 0.1344302148
| ðģ 171 | âïļ 6784368 | ð 0.1370944977
| ðģ 172 | âïļ 6903392 | ð 0.1395644099
| ðģ 173 | âïļ 7023800 | ð 0.1428508759
| ðģ 174 | âïļ 7145600 | ð 0.1446470916
| ðģ 175 | âïļ 7268800 | ð 0.1485457122
| ðģ 176 | âïļ 7393408 | ð 0.1515042037
| ðģ 177 | âïļ 7519432 | ð 0.1547876298
| ðģ 178 | âïļ 7646880 | ð 0.1581035405
| ðģ 179 | âïļ 7775760 | ð 0.1609414220
| ðģ 180 | âïļ 7906080 | ð 0.1643581241
| ðģ 181 | âïļ 8037848 | ð 0.1674534231
| ðģ 182 | âïļ 8171072 | ð 0.1702348739
| ðģ 183 | âïļ 8305760 | ð 0.1744350940
| ðģ 184 | âïļ 8441920 | ð 0.1764109582
| ðģ 185 | âïļ 8579560 | ð 0.1822682619
| ðģ 186 | âïļ 8718688 | ð 0.1858128011
| ðģ 187 | âïļ 8859312 | ð 0.1893977076
| ðģ 188 | âïļ 9001440 | ð 0.1925365478
| ðģ 189 | âïļ 9145080 | ð 0.1966986209
| ðģ 190 | âïļ 9290240 | ð 0.2020906210
| ðģ 191 | âïļ 9436928 | ð 0.2058481574
| ðģ 192 | âïļ 9585152 | ð 0.2081409991
| ðģ 193 | âïļ 9734920 | ð 0.2135929614
| ðģ 194 | âïļ 9886240 | ð 0.2167537063
| ðģ 195 | âïļ 10039120 | ð 0.2204166204
| ðģ 196 | âïļ 10193568 | ð 0.2253544927
| ðģ 197 | âïļ 10349592 | ð 0.2297920436
| ðģ 198 | âïļ 10507200 | ð 0.2340660542
| ðģ 199 | âïļ 10666400 | ð 0.2374149561
| ðģ 200 | âïļ 10827200 | ð 0.2429190129
| ðģ 201 | âïļ 10989608 | ð 0.2486662567
| ðģ 202 | âïļ 11153632 | ð 0.2522604764
| ðģ 203 | âïļ 11319280 | ð 0.2565264702
| ðģ 204 | âïļ 11486560 | ð 0.2610530257
| ðģ 205 | âïļ 11655480 | ð 0.2666850388
| ðģ 206 | âïļ 11826048 | ð 0.2696547210
| ðģ 207 | âïļ 11998272 | ð 0.2757692039
| ðģ 208 | âïļ 12172160 | ð 0.2815547585
| ðģ 209 | âïļ 12347720 | ð 0.2862318456
| ðģ 210 | âïļ 12524960 | ð 0.2915490866
| ðģ 211 | âïļ 12703888 | ð 0.2969746590
| ðģ 212 | âïļ 12884512 | ð 0.3020814955
| ðģ 213 | âïļ 13066840 | ð 0.3068147600
| ðģ 214 | âïļ 13250880 | ð 0.3128221631
| ðģ 215 | âïļ 13436640 | ð 0.3169069588
| ðģ 216 | âïļ 13624128 | ð 0.3231490850
| ðģ 217 | âïļ 13813352 | ð 0.3292649388
| ðģ 218 | âïļ 14004320 | ð 0.3355272114
| ðģ 219 | âïļ 14197040 | ð 0.3442309499
| ðģ 220 | âïļ 14391520 | ð 0.3481214046
| ðģ 221 | âïļ 14587768 | ð 0.3518816829
| ðģ 222 | âïļ 14785792 | ð 0.3584341705
| ðģ 223 | âïļ 14985600 | ð 0.3655003309
| ðģ 224 | âïļ 15187200 | ð 0.3700094819
| ðģ 225 | âïļ 15390600 | ð 0.3760042489
| ðģ 226 | âïļ 15595808 | ð 0.3842642903
| ðģ 227 | âïļ 15802832 | ð 0.3881231546
| ðģ 228 | âïļ 16011680 | ð 0.3951343596
| ðģ 229 | âïļ 16222360 | ð 0.4000105858
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| ðģ 236 | âïļ 17749088 | ð 0.4497081339
| ðģ 237 | âïļ 17974712 | ð 0.4586576819
| ðģ 238 | âïļ 18202240 | ð 0.4644054174
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| ðģ 240 | âïļ 18663040 | ð 0.4793971479
| ðģ 241 | âïļ 18896328 | ð 0.4870021641
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| ðģ 249 | âïļ 20833000 | ð 0.5512285233
| ðģ 250 | âïļ 21084000 | ð 0.5612413883
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| ðģ 252 | âïļ 21592032 | ð 0.5786451101
| ðģ 253 | âïļ 21849080 | ð 0.5857163668
| ðģ 254 | âïļ 22108160 | ð 0.5941022635
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| ðģ 259 | âïļ 23434320 | ð 0.6519679427
| ðģ 260 | âïļ 23705760 | ð 0.6610835791
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| ðģ 263 | âïļ 24532640 | ð 0.6937137246
| ðģ 264 | âïļ 24812480 | ð 0.7061291337
| ðģ 265 | âïļ 25094440 | ð 0.7181665301
| ðģ 266 | âïļ 25378528 | ð 0.7214239240
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| ðģ 270 | âïļ 26536320 | ð 0.7776065469
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| ðģ 277 | âïļ 28646232 | ð 0.8595604300
| ðģ 278 | âïļ 28956480 | ð 0.8712547421
| ðģ 279 | âïļ 29268960 | ð 0.8897573352
| ðģ 280 | âïļ 29583680 | ð 0.9022457004
| ðģ 281 | âïļ 29900648 | ð 0.9105177522
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| ðģ 293 | âïļ 33882520 | ð 1.0750583410
| ðģ 294 | âïļ 34229440 | ð 1.0891497135
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| ðģ 296 | âïļ 34930368 | ð 1.1190025806
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| ðģ 299 | âïļ 35999600 | ð 1.1700984240
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| ðģ 306 | âïļ 38578848 | ð 1.2775118351
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| ðģ 308 | âïļ 39337760 | ð 1.3126298189
| ðģ 309 | âïļ 39720920 | ð 1.3313826323
| ðģ 310 | âïļ 40106560 | ð 1.3432358503
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| ðģ 312 | âïļ 40885312 | ð 1.3776762486
| ðģ 313 | âïļ 41278440 | ð 1.3859341145
| ðģ 314 | âïļ 41674080 | ð 1.4143201113
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| ðģ 320 | âïļ 44101120 | ð 1.4987275600
| ðģ 321 | âïļ 44514568 | ð 1.5415856838
| ðģ 322 | âïļ 44930592 | ð 1.5574114323
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| ðģ 327 | âïļ 47049632 | ð 1.6434062719
| ðģ 328 | âïļ 47481280 | ð 1.6657049656
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| ðģ 330 | âïļ 48352480 | ð 1.6988587379
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| ðģ 333 | âïļ 49679160 | ð 1.7440986633
| ðģ 334 | âïļ 50126720 | ð 1.7606859207
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| ðģ 336 | âïļ 51029888 | ð 1.7924468517
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| ðģ 338 | âïļ 51943840 | ð 1.8152449131
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| ðģ 340 | âïļ 52868640 | ð 1.8608152866
| ðģ 341 | âïļ 53335128 | ð 1.8804984093
| ðģ 342 | âïļ 53804352 | ð 1.8971538544
| ðģ 343 | âïļ 54276320 | ð 1.9238594770
| ðģ 344 | âïļ 54751040 | ð 1.9492731094
| ðģ 345 | âïļ 55228520 | ð 1.9652727842
| ðģ 346 | âïļ 55708768 | ð 1.9872024059
| ðģ 347 | âïļ 56191792 | ð 2.0084447861
| ðģ 348 | âïļ 56677600 | ð 2.0305397511
| ðģ 349 | âïļ 57166200 | ð 2.0602569580
| ðģ 350 | âïļ 57657600 | ð 2.0797111988
| ðģ 351 | âïļ 58151808 | ð 2.0990145206
| ðģ 352 | âïļ 58648832 | ð 2.1228535175
| ðģ 353 | âïļ 59148680 | ð 2.1440532207
| ðģ 354 | âïļ 59651360 | ð 2.1542353630
| ðģ 355 | âïļ 60156880 | ð 2.1813373566
| ðģ 356 | âïļ 60665248 | ð 2.2163894176
| ðģ 357 | âïļ 61176472 | ð 2.2290546894
| ðģ 358 | âïļ 61690560 | ð 2.2577862740
| ðģ 359 | âïļ 62207520 | ð 2.2764096260
| ðģ 360 | âïļ 62727360 | ð 2.3054440022
| ðģ 361 | âïļ 63250088 | ð 2.3209156990
| ðģ 362 | âïļ 63775712 | ð 2.3474292755
| ðģ 363 | âïļ 64304240 | ð 2.3677115440
| ðģ 364 | âïļ 64835680 | ð 2.4003891945
| ðģ 365 | âïļ 65370040 | ð 2.4228780270
| ðģ 366 | âïļ 65907328 | ð 2.4527673721
| ðģ 367 | âïļ 66447552 | ð 2.4714524746
| ðģ 368 | âïļ 66990720 | ð 2.5030930042
| ðģ 369 | âïļ 67536840 | ð 2.5241811275
| ðģ 370 | âïļ 68085920 | ð 2.5526947975
| ðģ 371 | âïļ 68637968 | ð 2.5826325417
| ðģ 372 | âïļ 69192992 | ð 2.6038544178
| ðģ 373 | âïļ 69751000 | ð 2.6332132816
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| ðģ 375 | âïļ 70876000 | ð 2.6879386902
| ðģ 376 | âïļ 71443008 | ð 2.7146666050
| ðģ 377 | âïļ 72013032 | ð 2.7563600540
| ðģ 378 | âïļ 72586080 | ð 2.7757461071
| ðģ 379 | âïļ 73162160 | ð 2.7977271080
| ðģ 380 | âïļ 73741280 | ð 2.8362705708
| ðģ 381 | âïļ 74323448 | ð 2.8553814888
| ðģ 382 | âïļ 74908672 | ð 2.8915631771
| ðģ 383 | âïļ 75496960 | ð 2.9218170643
| ðģ 384 | âïļ 76088320 | ð 2.9435520172
| ðģ 385 | âïļ 76682760 | ð 2.9808762074
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| ðģ 387 | âïļ 77880912 | ð 3.0382304192
| ðģ 388 | âïļ 78484640 | ð 3.0813434124
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| ðģ 390 | âïļ 79701440 | ð 3.1379735470
| ðģ 391 | âïļ 80314528 | ð 3.1679635048
| ðģ 392 | âïļ 80930752 | ð 3.1992635727
| ðģ 393 | âïļ 81550120 | ð 3.2300934792
| ðģ 394 | âïļ 82172640 | ð 3.2542500496
| ðģ 395 | âïļ 82798320 | ð 3.2980310917
| ðģ 396 | âïļ 83427168 | ð 3.3268754482
| ðģ 397 | âïļ 84059192 | ð 3.3620347977
| ðģ 398 | âïļ 84694400 | ð 3.3948915005
| ðģ 399 | âïļ 85332800 | ð 3.4242134094
| ðģ 400 | âïļ 85974400 | ð 3.4592323303
| ðģ 401 | âïļ 86619208 | ð 3.4899778366
| ðģ 402 | âïļ 87267232 | ð 3.5227410793
| ðģ 403 | âïļ 87918480 | ð 3.5635776520
| ðģ 404 | âïļ 88572960 | ð 3.5935065746
| ðģ 405 | âïļ 89230680 | ð 3.6327414513
| ðģ 406 | âïļ 89891648 | ð 3.6607322693
| ðģ 407 | âïļ 90555872 | ð 3.6933164597
| ðģ 408 | âïļ 91223360 | ð 3.7418606281
| ðģ 409 | âïļ 91894120 | ð 3.7638649940
| ðģ 410 | âïļ 92568160 | ð 3.8051760197
| ðģ 411 | âïļ 93245488 | ð 3.8393759727
| ðģ 412 | âïļ 93926112 | ð 3.8652687073
| ðģ 413 | âïļ 94610040 | ð 3.9112775326
| ðģ 414 | âïļ 95297280 | ð 3.9450783730
| ðģ 415 | âïļ 95987840 | ð 3.9859733582
| ðģ 416 | âïļ 96681728 | ð 4.0158762932
| ðģ 417 | âïļ 97378952 | ð 4.0553226471
| ðģ 418 | âïļ 98079520 | ð 4.0899710655
| ðģ 419 | âïļ 98783440 | ð 4.1247377396
| ðģ 420 | âïļ 99490720 | ð 4.1694235802
| ðģ 421 | âïļ 100201368 | ð 4.2037181854
| ðģ 422 | âïļ 100915392 | ð 4.2537355423
| ðģ 423 | âïļ 101632800 | ð 4.2907476425
| ðģ 424 | âïļ 102353600 | ð 4.3410701752
| ðģ 425 | âïļ 103077800 | ð 4.4085803032
| ðģ 426 | âïļ 103805408 | ð 4.4271612167
| ðģ 427 | âïļ 104536432 | ð 4.4607510567
| ðģ 428 | âïļ 105270880 | ð 4.5109295845
| ðģ 429 | âïļ 106008760 | ð 4.5306105614
| ðģ 430 | âïļ 106750080 | ð 4.5749878883
| ðģ 431 | âïļ 107494848 | ð 4.6500787735
| ðģ 432 | âïļ 108243072 | ð 4.6717743874
| ðģ 433 | âïļ 108994760 | ð 4.7238435745
| ðģ 434 | âïļ 109749920 | ð 4.7592077255
| ðģ 435 | âïļ 110508560 | ð 4.7970252037
| ðģ 436 | âïļ 111270688 | ð 4.8226242065
| ðģ 437 | âïļ 112036312 | ð 4.8673853874
| ðģ 438 | âïļ 112805440 | ð 4.9161777496
| ðģ 439 | âïļ 113578080 | ð 4.9609446526
| ðģ 440 | âïļ 114354240 | ð 5.0242443085
| ðģ 441 | âïļ 115133928 | ð 5.0692362785
| ðģ 442 | âïļ 115917152 | ð 5.1170597076
| ðģ 443 | âïļ 116703920 | ð 5.1668162346
| ðģ 444 | âïļ 117494240 | ð 5.1868915558
| ðģ 445 | âïļ 118288120 | ð 5.2363333702
| ðģ 446 | âïļ 119085568 | ð 5.2728896141
| ðģ 447 | âïļ 119886592 | ð 5.3395667076
| ðģ 448 | âïļ 120691200 | ð 5.3689684868
| ðģ 449 | âïļ 121499400 | ð 5.4454574585
| ðģ 450 | âïļ 122311200 | ð 5.4567236900
| ðģ 451 | âïļ 123126608 | ð 5.5140542984
| ðģ 452 | âïļ 123945632 | ð 5.5848412514
| ðģ 453 | âïļ 124768280 | ð 5.6180467606
| ðģ 454 | âïļ 125594560 | ð 5.7027635574
| ðģ 455 | âïļ 126424480 | ð 5.7007894516
| ðģ 456 | âïļ 127258048 | ð 5.7667279243
| ðģ 457 | âïļ 128095272 | ð 5.8025741577
| ðģ 458 | âïļ 128936160 | ð 5.8558149338
| ðģ 459 | âïļ 129780720 | ð 5.9396958351
| ðģ 460 | âïļ 130628960 | ð 5.9576253891
| ðģ 461 | âïļ 131480888 | ð 5.9992723465
| ðģ 462 | âïļ 132336512 | ð 6.0377497673
| ðģ 463 | âïļ 133195840 | ð 6.0959258080
| ðģ 464 | âïļ 134058880 | ð 6.1883983612
| ðģ 465 | âïļ 134925640 | ð 6.2206907272
| ðģ 466 | âïļ 135796128 | ð 6.2725119591
| ðģ 467 | âïļ 136670352 | ð 6.2992658615
| ðģ 468 | âïļ 137548320 | ð 6.3347177505
| ðģ 469 | âïļ 138430040 | ð 6.4146580696
| ðģ 470 | âïļ 139315520 | ð 6.4513297081
| ðģ 471 | âïļ 140204768 | ð 6.5426158905
| ðģ 472 | âïļ 141097792 | ð 6.5523233414
| ðģ 473 | âïļ 141994600 | ð 6.6255145073
| ðģ 474 | âïļ 142895200 | ð 6.6785917282
| ðģ 475 | âïļ 143799600 | ð 6.7110738754
| ðģ 476 | âïļ 144707808 | ð 6.8051223755
| ðģ 477 | âïļ 145619832 | ð 6.8102507591
| ðģ 478 | âïļ 146535680 | ð 6.8606562614
| ðģ 479 | âïļ 147455360 | ð 6.9208254814
| ðģ 480 | âïļ 148378880 | ð 6.9970793724
| ðģ 481 | âïļ 149306248 | ð 7.0476222038
| ðģ 482 | âïļ 150237472 | ð 7.1175599098
| ðģ 483 | âïļ 151172560 | ð 7.1777501106
| ðģ 484 | âïļ 152111520 | ð 7.2448754311
| ðģ 485 | âïļ 153054360 | ð 7.2952923775
| ðģ 486 | âïļ 154001088 | ð 7.3523969650
| ðģ 487 | âïļ 154951712 | ð 7.4165315628
| ðģ 488 | âïļ 155906240 | ð 7.4655795097
| ðģ 489 | âïļ 156864680 | ð 7.5236325264
| ðģ 490 | âïļ 157827040 | ð 7.5819687843
| ðģ 491 | âïļ 158793328 | ð 7.6839604378
| ðģ 492 | âïļ 159763552 | ð 7.7059221268
| ðģ 493 | âïļ 160737720 | ð 7.7312030792
| ðģ 494 | âïļ 161715840 | ð 7.8410520554
| ðģ 495 | âïļ 162697920 | ð 7.8767709732
| ðģ 496 | âïļ 163683968 | ð 7.9313440323
| ðģ 497 | âïļ 164673992 | ð 8.0396604538
| ðģ 498 | âïļ 165668000 | ð 8.0553207397
| ðģ 499 | âïļ 166666000 | ð 8.1096744537
| ðģ 500 | âïļ 167668000 | ð 8.1465568542
| ðģ 600 | âïļ 289441600 | ð 17.5048580170
| ðģ 700 | âïļ 459295200 | ð 35.5125579834
| ðģ 800 | âïļ 685228800 | ð 116.5416107178
| ðģ 900 | âïļ 975242400 | ð 174.7076110840
| ðģ 1000 | âïļ 1337336000 | ð 240.5564362585
| ðģ 1100 | âïļ 1779509600 | ð 324.5367092784
| ðģ 1200 | âïļ 2309763200 | ð 434.1448342025
| ðģ 1300 | âïļ 2936096800 | ð 527.8912534102
| ðģ 1400 | âïļ 3666510400 | ð 660.1129210170
| ðģ 1500 | âïļ 4509004000 | ð 816.2368340823
| ðģ 1600 | âïļ 5471577600 | ð 1001.091214897
| ðģ 1700 | âïļ 6562231200 | ð 1332.292241911
| ðģ 1800 | âïļ 7788964800 | ð 1638.822858899
| ðģ 1900 | âïļ 9159778400 | ð 1987.633434514
⊠Licensing
This package is licensed under the MIT license.
Thanks for making it this far!