I want to find parametric equations for a sine-wave ('2-D helix') of the form $x, y=\sin(t), f(t)$, where $f(t)$ causes the expression to:
1. Start at (0, 1) (for positive and negative numbers).
2. Sets the distance $γ$ between loops of the 'helix' to powers of the golden ratio constant (the first such distance being $φ^{-1}$ for positive numbers and proceeding $φ^{0}, φ^{1}, φ^{2}…$ for positive numbers) at multiples of the golden angle input into the equations (i.e., $2·π·φ^{-1}$). The first such distance defined wholly by positive numbers is actually at: $2(2·π·φ^{-1})$ (The pattern must also apply to negative numbers in the same way, i.e., $φ^{-2}, φ^{-3}, φ^{-4}…$ (Note the special cases of $γ=φ^{-2}$ and $γ=φ^{-3}$ which are not defined entirely in terms of either positive or negative values for $t$.)
It must be noted that; the term '2-D helix' was used here because the problem is probably best grasped by thinking in 3-D for a moment. However, there's absolutely no need to solve this in 3-D: 3-D thinking is just a tool for clearer thought, in this case: See a graph here to gain greater insight into what I'm talking about. Also see my fig.
(Note that my fig. and graph show the concepts, obviously, they don't depict the graph of the expression I want as I don't have it. So, don't, for instance, try to match your graph up to my fig.'s actual proportions….)
I suspect that my graph and fig. and the somewhat informal description presented above, will give most readers the insights needed to address this problem, so long as they read everything carefully. However, if you don't find this to be true, see the notes beneath (the fig.) for more formal definitions, further context, more exact stipulations, and more:
A: Equation Requirements:
The equations must:
1. Be written $\sin(t), f(t)$: In other words, you can't do anything to $x(t)$; $x(t)$ must be just $\sin(t)$; the answer can't involve manipulating $x(t)$ at all.
2. Start at (0, 1) for positive and negative numbers.
3. Be a sine-wave / 2-D helix / sine-like curve NOT any random curve that can hit the correct points; it must be smooth, continuous, and, in 3-D, be a curve obtained by (metaphorically) winding a string up a cylinder.
4. Have $γ$ equal $φ^{-1}$ at $2·(2·π·φ^{-1})$ and go on $φ^{0}, φ^{1}, φ^{2}..$ for every positive, whole number multiple of the golden angle. Similarly for negative whole number multiples, noting the special cases of $γ=φ^{-2}$ and $γ=φ^{-3}$.
5. Use $\sin(t)$ Not $\cos(t)$.
B: Definition of where $γ$ should equal powers of $φ$:
1. $γ$ must equal powers of $φ$ at 'Golden points,' $g_1$ and $g_2$ the first such points defined wholly by positive values of $t$ yielding $γ=φ^{-1}$.
2. $g_1$ points are: Given by setting $t$ (for an expression of the kind under discussion) to some whole number multiple of the golden angle.
3. A $g_1$ point's corresponding $g_2$ point has $x$ given by its $g_1$ point. Its $y$ is given by the first (Lower-down) point where $x=$ the $x$ coordinate of $g_1$ intersects the 'side' of the wave on which $g_1$ is situated. (By side the following is meant: For the aforementioned equations, $0≤t≤\left(\frac{π}{2}\right)$ is the back side $\left(\frac{π}{2}\right)≤t≤\left(\frac{π}{2}\right)+π$ is the front side, and $\left(\frac{π}{2}\right)+π≤t≤\left(\frac{π}{2}\right)+π+π$ is the back side etc… Momentarily thinking in terms of 3-space may help.)
C: Other Definitions:
1. $φ=\left(\frac{1+5^{1/2}}{2}\right)$
2. $γ=$ (in 3-D) the horizontal distance between a point on a loop of a cylindrical helix and another point on a loop (the 'next' loop) above or beneath it, with the same $x,z$ as the first.
3. The Golden Angle is: $2·π·φ^{-1}$
D: Context:
1. an answer to this question will be equivalent to answering this question. But in this case, I took all extraneous, distracting factors out of the fundamental question leaving only what's really important.
2. I've been trying to solve this question for a very long time (more than half a year): I'm willing to award a large bounty (as soon as I can) to anyone who can solve it. Or help them however else I can. So, I would be extremely thankful for some help.
3. Please ask me if you don't understand some aspect of this!
4. See HERE and HERE for similar answers that may give clues! For the first link, see the update to the answer.
5. If you can't answer this but find the idea cool, I'd be thankful if you'd share it with someone who can.
Thank you all! Let's solve this!
Best Answer
I think you can describe your curve parametrically by $(x(t),y(t))=\left(\sin(t),f(t)\right)$, where $$f(t)=\frac{\phi^{\phi-3}}{\phi^\phi-1}\left(\phi^{\frac{\phi t}{2\pi}}-1\right)+1.$$
To verify, pick time $t_k=\frac{2\pi k}{\phi}$. The difference between $f(t_k)$ and $f(t_k-2\pi)$ should be $d_k=\phi^{k-3}$. So
$$\begin{align} f(t_k)-f(t_k-2\pi)&=\frac{\phi^{\phi-3}}{\phi^\phi-1}\left(\phi^{\frac{\phi t_k}{2\pi}}-\phi^{\frac{\phi(t_k-2\pi)}{2\pi}}\right)\\ &=\frac{\phi^{\phi-3}}{\phi^\phi-1}\left(\phi^{k}-\phi^{k-\phi}\right)\\ &=\frac{\phi^{k-3}}{\phi^\phi-1}\left(\phi^\phi-1\right)\\ &=\phi^{k-3}=d_k \end{align}$$ as required.
I came up with this by considering that decreasing $t_k$ by $2\pi k/\phi$ decreases the pitch by a factor of $\phi$, so we needed something like $\phi$ to the power of $k$, and at times $t_k$, $k=\phi t_k/(2\pi)$, so set the power as $\phi t/(2\pi)$. Then it was just a matter of finding the right coefficient in front of the $\phi^{\phi t/(2\pi)}$, which I found by doing the reverse of the verification above to find the appropriate $c$ in $c\phi^{\phi t/(2\pi)}$. Finally it had to be shifted vertically a bit to ensure $f(0)=1$.