Constructing a differential equation for hyperbolic crochet There is plenty of information about hyperbolic geometry and its melding with crochet, however I have yet to find an exact equation for determining the number of stitches in each row. I will try to ask my question without including crochet terms.
Let's say you have a row with a certain number of stitches, fox example 6. For every 2 stitches, you add 3 stitches on top. So the second row will have 9 stitches, and the third row will have 13.5 (ignore the decimals; in crochet you can't have half a stitch), then 20.25, 30.375, etc. Each row will be 1.5 longer than the previous. The circumference grows exponentially.
Heuristically (plotting and adding a trendline in Excel) I know the equation to be s(r) = 4*Exp(0.405*r). 4 comes from 6/1.5, or initial stitches divided by rate. 0.405 is a constant of integration, solved using s(2) = 9. So I know I have to construct a differential equation, but the only form I can think of is s'(r) = 6*1.5*s(r). The change of stitches equals the number of initial stitches times the rate of increase times the current number of stitches of the previous row. How does the constant of integration find itself in the exponent when most solutions have the form s(r) = A*Exp(r)?
Thanks for your help!
edit: Changed 4 = 6*1.5 to 4 = 6/1.5
 A: The general solution to $s'(r) = cs(r)$ with initial condition $s(0) = b$, is $s(r) = be^{cr}$. I suspect you have $b = 4$ because your Excel model starts at $r = 1$ rather than $r = 0$. $c = 0.405$ is presumably something to do with the aspect ratio of the stitches.
The differential equation describes the continuous behaviour that your crochet pattern is approximating discretely. The discrete behaviour is modelled by what is called a recurrence equation or recursion equation, $S(r+1) = cS(r)$, with initial condition $S(1) = b$ (starting from $1$ as you did). The solution is $S(r) = bc^{r-1}$ as you have discovered for the specific numbers you are interested in.
[Aside: I had never heard of hyperbolic crochet. Fascinating!]
A: The final equation is stitches(row) = 6*(1.5)^(row - 1). The first row will have 6 stitches, the second row will have 9, etc. The generic equation is stitches(row) = (Initial Stitches)*(Rate of increase)^(Row - 1). No need for a differential equation, it's actually quite simple.
