{"version":9,"randomSeed":"c3fd1dca6e03f9c93a70e0604b5d57de","graph":{"viewport":{"xmin":-2.1001008974054964,"ymin":-2.6044049277625216,"xmax":6.758030590483838,"ymax":5.655484096431433}},"expressions":{"list":[{"type":"text","id":"46","text":"define a sort of super-nth-root of x: SR_n(x) = y, such that y^y^y^...^y = x, where there are n y's in that power tower"},{"type":"text","id":"50","text":"then define a sort of a fractional tetration of a: a^^b = s^a^a^...^a, where there are floor(b) a's in that power tower, and s is derived by considering b as a rational p/q: s = SR_q(a)^p"},{"type":"text","id":"52","text":"then y_1 = x_1^^x_1, z_1 = e^^x_1"},{"id":"23","type":"table","columns":[{"values":["1","2","1.5","\\frac{1}{3}","\\frac{2}{3}","\\frac{4}{3}","\\frac{5}{3}","\\frac{7}{3}","\\frac{1}{2}","\\frac{1}{4}"],"hidden":true,"id":"21","color":"#c74440","latex":"x_{1}"},{"values":["1","4","1.569","0.216","0.727","1.339","1.751","22.936","",""],"hidden":true,"id":"22","color":"#2d70b3","latex":"y_{1}"},{"values":["2.718","15.154","4.672","1.601","2.125","3.595","5.43","1252.3","1.763","1.54"],"hidden":true,"id":"24","color":"#fa7e19","latex":"z_{1}"}]},{"type":"text","id":"61","text":"okay, here's a better approach"},{"type":"text","id":"63","text":"consider typical integer-suffix tetration x^^n := x^x^x^...^x, n times"},{"type":"text","id":"65","text":"for y in R, between 0 and 1, define extended tetration x^^y := lim (n -> inf) of s^^(round(ny)) where s^^n = x"},{"type":"text","id":"67","text":"below is y_2, where y_2^^x_2 = 2 (convergent values of s)"},{"id":"71","type":"table","columns":[{"values":["1","2","3","4","10","30","20"],"hidden":true,"id":"69","color":"#000000","latex":"x_{2}"},{"values":["2","1.55961","1.47668","1.4466","1.41619018","1.41421471","1.41425876"],"hidden":true,"id":"70","color":"#c74440","latex":"y_{2}"}]},{"type":"expression","id":"94","color":"#6042a6","latex":"s_{30}=1.41421471"},{"type":"expression","id":"96","color":"#c74440","latex":"s_{30}^{s_{30}}","labelSize":"medium"},{"type":"text","id":"81","text":"below is y_3 = 2^^x_3 (using approximation n = 30)"},{"id":"92","type":"table","columns":[{"values":["1","\\frac{1}{2}","\\frac{1}{3}","\\frac{1}{5}","\\frac{1}{6}","\\frac{1}{10}","\\frac{1}{15}","\\frac{1}{30}"],"hidden":true,"id":"90","color":"#c74440","latex":"x_{3}"},{"values":["2","1.9974","1.9837","1.927","1.8927","1.7608","1.6325","1.4142"],"hidden":true,"id":"91","color":"#2d70b3","latex":"y_{3}"}]},{"type":"expression","id":"98","color":"#388c46","latex":"y_{3}\\sim-\\frac{9}{10}z_{3}^{-x_{3}}+z_{2}","hidden":true,"residualVariable":"e_{3}","regressionParameters":{"z_{3}":603223.8617079114,"z_{2}":1.99584082555858}},{"type":"text","id":"101","text":"argh, just compute fractions LIKE ACTUAL FRACTIONS, that'll work right? (y_4 = 2^^x_4)"},{"id":"105","type":"table","columns":[{"values":["0","\\frac{1}{4}","\\frac{1}{3}","\\frac{1}{2}","\\frac{2}{3}","1","\\frac{4}{3}","\\frac{3}{2}","2","3"],"hidden":true,"id":"103","color":"#2d70b3","latex":"x_{4}"},{"values":["1","1.4466","1.4767","1.5596","1.7222","2","2.5861","2.7454","4","16"],"id":"104","color":"#388c46","latex":"y_{4}"}]},{"type":"folder","id":"111","title":"functiony stuff that i need to move around"},{"type":"expression","id":"26","folderId":"111","color":"#000000","latex":"f_{0}\\left(x,X\\right)=x^{x^{x^{x^{x^{X}}}}}","hidden":true,"labelSize":"medium"},{"type":"expression","id":"76","folderId":"111","color":"#c74440","latex":"f\\left(x\\right)=x^{x^{x^{x}}}","hidden":true},{"type":"expression","id":"37","folderId":"111","color":"#c74440","latex":"g\\left(x,a\\right)=a-\\frac{f\\left(a\\right)-x}{f'\\left(a\\right)}"},{"type":"expression","id":"43","folderId":"111","color":"#2d70b3","latex":"s=2","labelSize":"medium"},{"type":"expression","id":"40","folderId":"111","color":"#6042a6","latex":"k=g\\left(s,g\\left(s,g\\left(s,1.4466\\right)\\right)\\right)"}]}}