the logistic map at 3.2, iterated from 0.2

• bin 1 holds the share the density says it should ×60
• every point over 1 comes back after 1 doublings ×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
• both maps have the same number of points of period dividing 1 ×5
• 0.6154 is fixed at r = 2.6 ×4
• the number of steps is a whole number between 2 and 400 ×4
• the numerator of the starting point is a whole number between 1 and 10000000 ×3
• at r = 2.4 the error shrinks by the slope at the fixed point each step ×2
• the number of steps drawn is a whole number between 3 and 20 ×2
• the orbit at r = 2.4 converges far enough to measure a rate ×2
• the parameter is between 0 and 2 ×2
• √2 is fixed by the Newton step ×1
• a new crossing has a period above one that divides 2 ×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 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
• between 6 and 60 steps are taken ×1
• between one and three logistic parameters, each with an attracting fixed point ×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
• 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 start reaches the same point ×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
• its denominator is a whole number between 3 and 10000000 ×1
• one point drawn per step, and the start ×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 change of coordinate fixes the left end ×1
• the change of coordinate is increasing, so it is reversible ×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 denominator is odd, or the exact orbit reaches zero as surely as the float one does ×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 longest cycle counted is a whole number between 2 and 8 ×1
• the longest word counted is a whole number between 2 and 8 ×1
• the longest word counted is between two and eight ×1
• the map is one the family knows ×1
• the map is one this figure knows ×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 bins is a whole number between 8 and 80 ×1
• the number of drawn steps is a whole number between 2 and 400 ×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 stays inside the unit interval ×1
• the point a repeating word names has that word as its itinerary ×1
• the point found by bisection is fixed ×1
• the quadratic curve is on or off ×1
• the slope at the fixed point is 2 − r ×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 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 view is one the family draws ×1
• the words of each length and the points of that period are equally many ×1
• two logistic parameters above one ×1
• two points are at most k times as far apart after the map as before ×1

The whole library · What the figures prove