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i or π can be computed to an arbitrary high precision n pi 60 3 14159265358979323846264338327950288419716939937510582097494 let us compare it with a continued fraction approximation for example using the first sequence line of on line encyclopedia of integer sequences oeis a001203 produces π with precision 100 s 3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2 1 84 2 1 1 15 3 13 1 4 2 6 6 99 1 2 2 6 3 5 1 1 6 8 1 7 1 2 3 7 1 2 1 1 12 1 1 1 3 1 1 8 1 1 2 1 6 1 1 5 2 2 3 1 2 4 4 16 1 161 45 1 22 1 2 2 1 4 1 2 24 1 2 1 3 1 2 1 n fromcontinuedfraction s 120 3 14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706808656553677697242141 here we verify the precision using wolfram language n pi 200 1 0441745026369011477 10 100 more details can be found in wolfram mathworld page pi continued fraction ew1 2 continued fraction terms plots it is interesting to consider the plotting the terms of continued fraction terms of pi first we ingest the more pi terms from oeis a001203 20k terms terms partition toexpression stringsplit import https oeis org a001203 b001203 txt whitespace 2 all 2 length terms 20000 here is the summary resourcefunction recordssummary terms here is an array plot of the first 128 terms of the continued fraction approximating pi mat integerdigits terms 1 128 2 maxdigits max length mat mat map join table 0 maxdigits length mat arrayplot transpose mat mesh true imagesize 1000 next we show the pareto principle manifestation of for the continued fraction terms first we observe that the terms a distribution similar to benford s law tallypi keysort associationthread transpose tally terms 1 16 length terms termsb randomvariate benforddistribution 10 2000 tallyb keysort associationthread transpose tally termsb length termsb data tallypi tallyb data keysort map keymap toexpression data data2 first values merge data 1 join associationthread keys data 1 null data 2 list barchart data2 plotlabel pi continued fraction terms vs benford s law chartlegends pi belford s law plottheme detailed imagesize large here is the pareto principle plot 5 of the unique term values correspond to 80 of the terms resourcefunction paretoprincipleplot terms plotlabel pareto principle for pi continued fraction terms imagesize large 3 classic infinite series many ways to express π as an infinite sum some converge slowly others surprisingly fast leibniz gregory series 1671 madhava earlier wl implementation pileibniz n_integer block 4 total table 1 i 2 i 1 i 0 n 1 n pileibniz 1000 20 3 1405926538397929260 verify with wolfram language again n pi 100 n pileibniz 1000 1000 0 000999999750000312499046880410106913745780068595381331607486808765908475046461390362862493334446861 nilakantha series faster convergence wl pinilakantha n_integer block 3 4 total table 1 i 1 2 i 2 i 1 2 i 2 i 1 n n pinilakantha 1000 20 3 1415926533405420519 n pi 40 n pinilakantha 1000 40 2 49251186562514647026299317045 10 10 3 beautiful products wallis product 1655 elegant infinite product wl running product p 2 0 do p 2 n 2 2 n 1 2 n 1 if divisible n 100 print row n p pi relative error n 1 1000 100 0 9975155394660373 relative error 200 0 998753895533088 relative error 300 0 9991683995999036 relative error 400 0 9993759752213087 relative error 500 0 9995006243131489 relative error 600 0 9995837669635674 relative error 700 0 9996431757700326 relative error 800 0 9996877439728793 relative error 900 0 9997224150056372 relative error 1000 0 9997501561641055 relative error 4 very fast modern series chudnovsky algorithm one of the fastest converging series used in record computations each term adds roughly 14 correct digits 5 spigot algorithms digits drip one by one spigot algorithms compute decimal digits using only integer arithmetic no floating point errors accumulate the classic rabinowitz wagon spigot based on a transformed wallis product produces base 10 digits sequentially simple but bounded version outline in wl clearall pispigotdigits pispigotstring pispigotdigits n_integer positive module len a q x i j nines 0 predigit 0 digits len quotient 10 n 3 1 a constantarray 2 len for j 1 j n j q 0 for i len i 1 i x 10 a i q i a i mod x 2 i 1 q quotient x 2 i 1 a 1 mod q 10 q quotient q 10 which q 9 nines q 10 appendto digits predigit 1 digits join digits constantarray 0 nines predigit 0 nines 0 true appendto digits predigit digits join digits constantarray 9 nines predigit q nines 0 appendto digits predigit take digits n pispigotstring n_integer positive module d pispigotdigits n tostring d 2 stringjoin tostring rest rest d pispigotstring 40 3 14159265358979323846264338327950288419 n pi 100 toexpression pispigotstring 40 0 10 38 6 bbp formula hex digits without predecessors bailey borwein plouffe 1995 formula lets you compute the n th hexadecimal digit of π directly without earlier digits very popular for distributed π hunting projects the best known digit extraction algorithm wl snippet for partial sum base 16 sense clearall bbpterm bbppipartial bbppi bbpterm k_integer nonnegative 16 k 4 8 k 1 2 8 k 4 1 8 k 5 1 8 k 6 bbppipartial n_integer nonnegative sum bbpterm k k 0 n bbppi prec_integer positive module n n ceiling prec 4 5 n bbppipartial n prec bbppipartial 10 n 20 3 1415926535897931296 bbppi 50 3 1415926535897932384626433745157614859702375522676 7 instead of conclusion π contains almost surely every finite sequence of digits your birthday appears infinitely often the feynman point six consecutive 9s starting at digit 762 memorization world record 100 000 digits π appears in the normal distribution quantum mechanics random walks buffon s needle problem probability 2 π let us plot a random walk using the terms of continued fraction of pi the 20k or oeis a001203 to determine directions listlineplot reverse anglepath terms plotstyle absolutethickness 0 6 imagesize large axes false here is a bubble chart based on the path points and term values pathwithvalues flatten thread reverse rest path terms gr bubblechart pathwithvalues chartstyle darkrainbow aspectratio automatic frame false axes false plottheme minimal rasterize gr imagesize 900 imageresolution 72 remark the bubble sizes indicate that some of terms are fairly large but the majority of them are relatively small here is a 3d point plot with of moving average of the 3d path modified to have the logarithms of the term values pathwithvalueslog mapat n log10 pathwithvalues all 3 listlineplot3d movingaverage pathwithvalueslog 200 aspectratio automatic plotrange all imagesize large ticks none references ew1 eric weisstein pi continued fraction wolfram mathworld posted in generative art mathematica number theory visualization wolfram language tagged graphics and visualization math mathematica mathematics number theory pi day leave a comment maze making using graphs posted on december 26 2025 by anton antonov antonov introduction this document notebook describes three ways of making mazes or labyrinths using graphs the first two are based on rectangular grids the third on a hexagonal grid all computational graph features discussed here are provided by the graph functionalities of wolfram language tl dr just see the maze pictures below and try to solve the mazes procedure outline the first maze is made by a simple procedure which is actually some sort of cheating a regular rectangular grid graph is generated with random weights associated with its edges the minimum spanning tree for that graph is found that tree is plotted with exaggeratedly large vertices and edges so the graph plot looks like a maze this is the cheat the maze walls are not given by the graph the second maze is made properly two interlacing regular rectangular grid graphs are created the second one has one less row and one less column than the first the vertex coordinates of the second graph are at the centers of the rectangles of the first graph the first graph provides the maze walls the second graph is used to make paths through the maze in other words to create a solvable maze again random weights are assigned to edges of the second graph and a minimum spanning tree is found there is a convenient formula that allows using the spanning tree edges to remove edges from the first graph in that way a proper maze is derived the third maze is again made properly using the procedure above with two modifications two interlacing regular grid graphs are created one over a hexagonal grid the other over a triangular grid the hexagonal grid graph provides the maze walls the triangular grid graph provides the maze paths since the formula for wall removal is hard to derive a more robust and universal method based on nearest neighbors is used simple maze in this section we create a simple cheating maze remark the steps are easy to follow given the procedure outlined in the introduction randomseed 3021 n m 10 25 g gridgraph n m gweighted graph vertexlist g undirectededge edgelist g edgeweight randomreal 10 1000 edgecount g information gweighted find the spanning tree of the graph mazetree findspanningtree gweighted shortest path from the first vertex bottom left to the last vertex top right path findshortestpath mazetree 1 n m length path out 46 graph plot simplemaze graph vertexlist mazetree edgelist mazetree vertexcoordinates graphembedding g simplemaze2 edgeadd simplemaze start 1 max vertexlist simplemaze end clear vf1 ef1 vf1 col_ colorq xc_ yc_ name_ w_ h_ col edgeform none rectangle xc w yc h xc w yc h ef1 col_ colorq pts_list e_ col opacity 1 absolutethickness 22 line pts grcheat graphplot simplemaze2 vertexsize 0 8 vertexshapefunction vf1 white edgeshapefunction ef1 white background darkblue imagesize 800 imagepadding 0 range minmax transpose flatten list cases grcheat _rectangle infinity all all 1 show grcheat plotrange range the maze above looks like a maze because the vertices and edges are rectangular with matching sizes and they are thicker than the spaces between them in other words we are cheating to make that cheating construction clearer let us plot the shortest path from the bottom left to the top right and color the edges in pink salmon and the vertices in red gpath pathgraph path vertexcoordinates graphembedding g path legended show grcheat graph gpath vertexsize 0 7 vertexshapefunction vf1 red edgeshapefunction ef1 pink background darkblue imagesize 800 imagepadding 0 plotrange range swatchlegend red pink darkblue shortest path vertices shortest path edges image background proper maze a proper maze is a maze given with its walls not with the space between walls remark for didactical reasons the maze in this section is small so that the steps outlined in the introduction can be easily followed make two regular graphs one for the maze walls and the other for the maze paths n m 6 12 g1 gridgraph n m vertexlabels name g1 vertexreplace g1 thread vertexlist g1 map w quotientremainder 1 n vertexlist g1 g2 gridgraph n 1 m 1 g2 vertexreplace g2 thread vertexlist g2 map quotientremainder 1 n 1 vertexlist g2 g2 graph g2 vertexlabels name vertexcoordinates map 1 1 2 graphembedding g2 grid wall graph paths graph information g1 information g2 see how the graph interlace show g1 highlightgraph g2 g2 imagesize 800 maze path graph mazepath graph edgelist g2 edgeweight randomreal 10 10000 edgecount g2 mazepath findspanningtree mazepath vertexcoordinates thread vertexlist g2 graphembedding g2 information mazepath combined graph g3 graph join edgelist g1 edgelist mazepath vertexcoordinates join thread vertexlist g1 graphembedding g1 thread vertexlist mazepath graphembedding mazepath vertexlabels name information g3 plot the combined graph highlightgraph g3 mazepath imagesize 800 remove wall edges using a formula g4 graph g3 vertexlabels none do i j e 1 i2 j2 e 2 if i2 i j2 j i2 j2 i j i j i2 j2 horizontal if i i2 j j2 g4 edgedelete g4 undirectededge w i2 j2 w i2 1 j2 vertical if j j2 i i2 g4 edgedelete g4 undirectededge w i2 j2 w i2 j2 1 e edgelist mazepath information g4 plot wall graph and maze paths maze space graph highlightgraph g4 mazepath imagesize 800 fancier maze presentation with rectangular vertices and edges with matching sizes g5 subgraph g4 vertexlist g1 g5 vertexdelete g5 w 0 0 w m 1 n 1 g6 graph g5 vertexshapefunction none edgeshapefunction opacity 1 darkblue absolutethickness 30 line 1 imagesize 800 here is how a solution can found and plotted solution findpath vertexlist 1 vertexlist 1 mazepath show g6 highlightgraph subgraph mazepath solution subgraph mazepath solution here is a more challenging to solve maze generated with n 12 and m 40 hexagonal version let us create another maze based on a hexagonal grid here are two grid graphs the first is a hexagonal grid graph representing the maze s walls the second graph is a triangular grid graph with one fewer row and column and shifted vertex coordinates n m 6 14 2 g1 resourcefunction hexagonalgridgraph m n g1 vertexreplace g1 thread vertexlist g1 w 1 vertexlist g1 g2 resourcefunction https www wolframcloud com obj antononcube deployedresources function triangularlatticegraph n 1 m 1 g2 graph g2 vertexcoordinates 1 sqrt 3 1 graphembedding g2 information g1 information g2 show g1 highlightgraph g2 g2 imagesize 800 maze path graph mazepath graph edgelist g2 edgeweight randomreal 10 10000 edgecount g2 mazepath findspanningtree mazepath vertexcoordinates thread vertexlist g2 graphembedding g2 information mazepath combine the walls maze and the maze path graphs i e make a union of them and plot the resulting graph g3 graphunion g1 mazepath vertexcoordinates join thread vertexlist g1 graphembedding g1 thread vertexlist mazepath graphembedding mazepath information g3 highlightgraph g3 mazepath imagesize 800 make a nearest neighbor points finder functor finder nearest thread graphembedding g1 vertexlist g1 take a maze edge and its vertex points e first edgelist mazepath amazepathcoords association thread vertexlist mazepath graphembedding mazepath points list e amazepathcoords find the edge s midpoint and the nearest wall graph vertices print middle edge point mean points print edge to remove undirectededge finder mean points loop over all maze edges removing wall maze edges g4 g1 do points map amazepathcoords list e m mean points vs finder m g4 edgedelete g4 undirectededge vs e edgelist mazepath information g4 here is the obtained graph show g4 imagesize 800 the start and end points of the maze avertexcoordinates association thread vertexlist g4 graphembedding g4 start end keys sort avertexcoordinates 1 1 out w 1 w 752 finding the maze solution out sequence 1 240 solution findshortestpath mazepath sequence keys sort amazepathcoords 1 1 solution pathgraph solution vertexcoordinates lookup amazepathcoords solution plot the maze g5 graph g4 vertexshapefunction none edgeshapefunction opacity 1 darkblue absolutethickness 8 line 1 imagesize 800 g5 vertexdelete g5 start end here is the solution of the maze show g5 highlightgraph solution solution additional comments the notebook can be us...