I have a lattice that has infinite length (directions) in $+x$ (leftwards) and $+y$ (downwards). What is the slope-value of the line shown on the graph when the magnitude of the line increases?

enter image description here

My guess is that there is a fraction there, but I dont know how to find it. Ok, after counting the cells of the interior rectangle of $3\times5$ I found out it might be $3/5 = 0.6$. But dunno if its correct.


closed as off-topic by Namaste, Arnaud Mortier, Xander Henderson, max_zorn, Taroccoesbrocco Aug 1 '18 at 11:27

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  • 1
    $\begingroup$ What exactly is the repeating pattern? $\endgroup$ – Arnaud Mortier Jul 31 '18 at 22:07
  • $\begingroup$ Why are not happy just considering the ratio of the sides of your rectangle? $\endgroup$ – Arnaud Mortier Jul 31 '18 at 22:10
  • $\begingroup$ Ok, I tried and found $3/5 = 0.6$, but is there another fraction for this value? $\endgroup$ – Natural Number Guy Jul 31 '18 at 22:13
  • $\begingroup$ @Arnaud Mortier the left edge on-bits (binary expansion) (that is iterations) of $f(2^n-1)$ in Odd Collatz function: $(3n+1)/2$ without the even results. So this is line is just some sub-section of the outputs. $\endgroup$ – Natural Number Guy Jul 31 '18 at 22:20
  • $\begingroup$ Then there is definitely no way to answer from so little information. You are probably after the asymptotics of some function but it's hard to tell. $\endgroup$ – Arnaud Mortier Jul 31 '18 at 22:28

We have

  • $\Delta x =10$

  • $\Delta y =15$


  • slope $= \tan \theta =\frac{\Delta y}{\Delta x}=\frac{15}{10}=\frac32$
  • $\begingroup$ The OP shows only a small part of the line, there is no way to answer the actual question being asked. $\endgroup$ – Arnaud Mortier Jul 31 '18 at 23:09
  • $\begingroup$ @ArnaudMortier I’m assuming that the pattern will repeat and at an infinite length the slope is equal to 3/2. I cannot see anybother interpretation of the OP. $\endgroup$ – gimusi Jul 31 '18 at 23:11
  • $\begingroup$ I tried asking additional details and although the answer wasn't clear, I'm almost certain that this is not the case. $\endgroup$ – Arnaud Mortier Jul 31 '18 at 23:13
  • $\begingroup$ Sorry for being unclear. The line is a straight line. I've updated the diagram for more clarity hopefully. Gimusi is close to what I want. $\endgroup$ – Natural Number Guy Jul 31 '18 at 23:16
  • $\begingroup$ @NaturalNumberGuy gimusi's answer is correct if the pattern repeats identically outside the visible box, which you haven't made clear. $\endgroup$ – Arnaud Mortier Jul 31 '18 at 23:19

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