Spirals generated from the center (GRG)
Inside view. In GRG every curve is generated from the center by discrete steps on the quarter turn.
We fix a turns step $\,\Delta u=\tfrac{1}{4N}\,$ and update the radius additively (Archimedean) or multiplicatively (golden/logarithmic).
We use period-1 trigonometry $s(u)=\sin(2\pi u)$, $c(u)=\cos(2\pi u)$; sample points are $P_k=\bigl(r_k\,c(u_k),\ r_k\,s(u_k)\bigr)$.
No $\pi$ is assumed — only quarter-turn partitions.
We present two canonical spirals built “from inside”: the Archimedean spiral and the golden spiral (a logarithmic spiral whose radius grows by the golden ratio per quarter turn). Both are sampled at uniform $\Delta u$ and drawn with chord polylines.
1) Archimedean spiral (additive radius)
Definition (GRG sampling). Fix $\Delta u=\tfrac{1}{4N}$ and a base radius $r_0>0$. Set
$$
u_k=k\,\Delta u,\qquad
r_k=r_0+k\,\Delta r,\qquad
P_k=\bigl(r_k\,c(u_k),\ r_k\,s(u_k)\bigr).
$$
The parameter $\Delta r$ controls the spacing between turns. The polyline through $P_k$ approximates the spiral.
2) Golden spiral (logarithmic, $\varphi$ per quarter turn)
Definition (GRG sampling). Let $\varphi=\tfrac{1+\sqrt5}{2}$ be the golden ratio. With $\Delta u=\tfrac{1}{4N}$ and base radius $r_0>0$:
$$
u_k=k\,\Delta u,\qquad
r_k=r_0\,\varphi^{\,4u_k},\qquad
P_k=\bigl(r_k\,c(u_k),\ r_k\,s(u_k)\bigr).
$$
Each quarter turn ($\Delta u=\tfrac14$) multiplies the radius by $\varphi$. The polyline through $P_k$ approximates the logarithmic spiral.
3) GRG operators and quick estimates
- Discrete perimeter along the spiral. For step $\Delta u=\tfrac{1}{4N}$, the piecewise-chord length is $L\approx \sum_k \|P_{k+1}-P_k\|$. For smooth spirals the local error is $O(\Delta u^2)$.
- Radial bands for area. The region between radii $r_k$ and $r_{k+1}$ can be accumulated by “sector-like” trapezoids on the quarter and scaled by $4$, reusing GRG’s area operator.
- π-free spirit. Everything is driven by quarter-turn partitions and finite sums; $s(u),c(u)$ are a compact shorthand for ruler-and-compass similarity.
4) Parameters used in the figures
- Archimedean: $\Delta u=\tfrac{1}{24}$ (so $N=6$ on the quarter), $r_0=5$, $\Delta r=5$, 24 steps.
- Golden: $\Delta u=\tfrac{1}{24}$, $r_0=6$, $r(u)=r_0\,\varphi^{4u}$, 36 steps.