the logistic map at 3.2, iterated from 0.2

• T2: bin 1 matches the arcsine share ×90
• bin 1 holds the share the density says it should ×60
• at r = 4 the 0-th iterate has 2^0 folds ×21
• T2 composed 1 times has 2 fixed points ×16
• T2 composed 1 times is T2 ×16
• step 0 is cos(3^0 θ) ×13
• returns after 1 steps against the trace ×10
• the orbits of exact period 1, walked and by inversion ×10
• the points of exact period 1 make whole orbits ×10
• walks of length 1 from the start state are the admissible strings ×10
• sampling finds every admissible word of length 1 (the golden ratio φ) ×9
• sampling finds every admissible word of length 1 (β = 1.8) ×9
• sampling finds every admissible word of length 1 (β = 5/2) ×9
• at level 1 the admissible intervals number F(3) = 2 ×8
• the number of steps is a whole number between 2 and 400 ×7
• every point over 1 comes back after 1 doublings ×6
• the trace is 1 + (1 + √2)ⁿ + (1 − √2)ⁿ at n = 1 ×6
• 0.00000 returns to itself after 2 steps ×5
• a full two-branch fold has 2^1 points fixed by its 1th power ×5
• after 0 doublings the overlap is within 1/2^2 of a sixth ×5
• both maps have the same number of points of period dividing 1 ×5
• 0.6154 is fixed at r = 2.6 ×4
• after 0 doublings the lowest frequency present is 1 ×4
• T2(cos θ) = cos 2θ ×4
• the random point visits quarter 1 a quarter of the time ×4
• the total energy of f composed 0 times is a quarter ×4
• in the window of three, at 3.8300, the estimate is log of the golden ratio ×3
• the folds of f^6 at r = 3.5, sampled and from the preimages of 1/2 ×3
• the longest word counted is a whole number between 3 and 10 ×3
• the numerator of the starting point is a whole number between 1 and 10000000 ×3
• the parameter is between 0 and 2 ×3
• at r = 2.4 the error shrinks by the slope at the fixed point each step ×2
• the longest iterate counted is a whole number between 6 and 22 ×2
• the number of bins is a whole number between 8 and 80 ×2
• the number of steps drawn is a whole number between 3 and 20 ×2
• the number of times the map is applied is a whole number between 2 and 5 ×2
• the orbit at r = 2.4 converges far enough to measure a rate ×2
• √2 is fixed by the Newton step ×1
• 0, 1/3 and 1 are a cycle of three ×1
• A covers A and B, B covers all three, C covers B and C ×1
• a new crossing has a period above one that divides 2 ×1
• a point with no two 1s together never enters the top quarter ×1
• after the computed orbit is dead the exact one is still moving ×1
• and carries the right end to the right end ×1
• and it never reaches zero, because an odd denominator cannot be halved away ×1
• and never grows ×1
• and not before ×1
• and shrinks every distance by a factor under one ×1
• and stays there, because zero is fixed ×1
• and the constant the ratio approaches is 1/(2√2) ×1
• and the repeat found really is a repeat ×1
• and then separate visibly, at a step the figure marks ×1
• at 1,000 digits more than 99% of strings are within 0.05 of a half ×1
• at r = 4 the folds double and the estimate is log 2 ×1
• below the end of the doublings the deeper estimate is the smaller ×1
• between 6 and 60 steps are taken ×1
• between one and six logistic parameters from 3 to 4 ×1
• between one and three logistic parameters, each with an attracting fixed point ×1
• deleting the first digit of an admissible word leaves it admissible ×1
• each density has total mass one ×1
• each letter of the itinerary is the next binary place of the starting point ×1
• each Newton error is the square of the last over twice the current guess ×1
• each point of the cycle is carried to another ×1
• each point of the cycle lies in its letter's piece ×1
• each step of a closed word's point lands in that word's piece ×1
• every bin is visited ×1
• every corner of the staircase lies on the curve or on the diagonal ×1
• every crossing of the map is also a crossing of the composed map ×1
• every exact iterate is a whole numerator over the same denominator ×1
• every orbit's share settles on the predicted frequency ×1
• every piece is stretched ×1
• every start reaches the same point ×1
• every word the map writes passes Parry's test ×1
• five Newton steps from 1.9 reach twelve decimal places ×1
• how many steps of the orbit are drawn is a whole number between 3 and 12 ×1
• in base the tribonacci constant, 1 = .111 ×1
• in base φ no two 1s are ever consecutive ×1
• in base φ the allowed words are counted by the Fibonacci numbers ×1
• in base φ the digit 1 appears with frequency 1/(1 + φ²) ×1
• in base φ, 1 = .11 ×1
• its denominator is a whole number between 3 and 10000000 ×1
• no point is named by two closed words ×1
• one point drawn per step, and the start ×1
• T₂∘T₃, T₃∘T₂ and T₆ agree everywhere ×1
• the 2 new crossings of period 2 are whole orbits of 2 points ×1
• the average of one long orbit is the average over the whole interval ×1
• the base has a finite expansion of 1, so its graph is finite ×1
• the base is one the family knows ×1
• the change of coordinate fixes the left end ×1
• the change of coordinate is increasing, so it is reversible ×1
• the column count is a whole number between 20 and 241 ×1
• the computed orbit reaches exactly zero inside the drawn window, having run out of binary places ×1
• the contraction is one the family knows ×1
• the coordinate change carries the tent map to the logistic map ×1
• the coordinate is one whose invariant density this figure knows ×1
• the counts grow by a factor of β per letter ×1
• the cycle closes after the word's length ×1
• the cycle is one the family draws ×1
• the deeper estimate never falls as r rises ×1
• the degree is a whole number between 2 and 5 ×1
• the denominator is odd, or the exact orbit reaches zero as surely as the float one does ×1
• the digits of 1 add back up to 1 ×1
• the digits written reconstruct the start to within β to the minus the number of steps ×1
• the error after n steps is inside k to the n times the error at the start ×1
• the exact orbit closes up inside the drawn window ×1
• the exact shares add to one ×1
• the fixed point attracts, so there is a rate to measure ×1
• the folds grow by the golden ratio at the last step ×1
• the frequency of nonzero digits is the density's mass on the right piece ×1
• the graph's growth rate is the base ×1
• the highest frequency drawn is a whole number between 16 and 96 ×1
• the left piece covers the right, and the right covers both ×1
• the longest cycle counted is a whole number between 2 and 8 ×1
• the longest word counted is between two and eight ×1
• the longest word is a whole number between 6 and 14 ×1
• the lower parameter is between 3 and 4 ×1
• the map is one the family knows ×1
• the map is one this figure knows ×1
• the map itself brings the point back ×1
• the map itself has a fixed point ×1
• the map sends the interval into itself ×1
• the number of binary places the starting point has is a whole number between 3 and 8 ×1
• the number of drawn steps is a whole number between 2 and 400 ×1
• the number of levels is a whole number between 4 and 10 ×1
• the number of stages is a whole number between 3 and 7 ×1
• the number of times the map is composed is a whole number between 2 and 4 ×1
• the orbit approaches the fixed point exactly when the slope is shallow ×1
• the orbit at r = 3.83 settles on a cycle of three ×1
• the orbit of 1/3 lives in the two middle quarters ×1
• the orbit of 1/3 never enters [0, ¼) ×1
• the orbit starts inside the interval, or at 1 ×1
• the orbit stays inside the unit interval ×1
• the orbit's histogram matches Parry's density away from the steps ×1
• the point a repeating word names has that word as its itinerary ×1
• the point found by bisection is fixed ×1
• the points returning after 4 steps are as many as the trace of the matrix power ×1
• the quadratic curve is on or off ×1
• the random point's running average reaches a quarter ×1
• the range runs upward ×1
• the share near a half grows with the length ×1
• the slope at the fixed point is 2 − r ×1
• the solved point comes back after exactly the word's steps ×1
• the starting point has between three and eight binary places ×1
• the starting point is between 0 and 1 ×1
• the starting point is inside the interval ×1
• the steps drawn is a whole number between 3 and 14 ×1
• the straight pieces of the composed map are the allowed words of 4 letters ×1
• the transported orbit is the orbit of the transported point ×1
• the transported orbit obeys the tent map's own recurrence ×1
• the two agree exactly for at least eight steps ×1
• the upper parameter is between 3 and 4 ×1
• the view is one the family draws ×1
• the word is not a shorter word repeated ×1
• the word is two to eight letters of L and R ×1
• the word, read round, never puts L after L ×1
• the words of each length and the points of that period are equally many ×1
• two depths, the deeper at most 20 ×1
• two logistic parameters above one ×1
• two points are at most k times as far apart after the map as before ×1
• x² and x² − 1 composed in the two orders differ ×1

The whole library · What the figures prove