Extension from 'Circular arrow bundle(Wonjeon, 圓箭)' and Centered Hexagonal Numbers to the Complex Plane and Differentiation

1. Structure and Combinatorial Analysis of Wonjeon (圓箭)

In Gusuryak (《九數略》) by Choi Seok-jeong (崔錫鼎, 1646–1715), wonjeon (圓箭, circular arrow bundle) describes a hexagonal packing where 6 arrows surround a central one, expanding outward in rings of multiples of 6.

• Edge connection combinations: choosing 2 out of 6 vertices gives \binom{6}{2} = 15.
• Edge constraint: the geometric perimeter condition restricts connections to 6 adjacent pairs.
• Vertex overlap correction: each side of ring k contains (k+1) points. Summing 6 sides gives 6(k+1), but each vertex is shared by two sides, so subtract 6. 6(k+1) - 6 = 6k \quad (k \geq 1)

2. Derivation of the Centered Hexagonal Number Formula

Define S_k as the number of points in ring k.

• k=0: S_0 = 1
• k=1: S_1 = 2 \times 6 - 6 = 6
• k=2: S_2 = 3 \times 6 - 6 = 12
• k=3: S_3 = 4 \times 6 - 6 = 18
• k \geq 1: S_k = (k+1) \times 6 - 6 = 6k

The cumulative total up to ring n:

g(n) = \sum_{k=0}^{n} S_k = 1 + 6\sum_{k=1}^{n} k = 1 + 6 \cdot \frac{n(n+1)}{2} = 3n^2 + 3n + 1

• g(0) = 1
• g(1) = 7
• g(2) = 19
• g(100) = 30301

3. Subtraction Method (減中數一) from Gusuryak and the Symmetry of f(n)

The original Gusuryak algorithm states: "first subtract the central 1 (先減中數一), then process the outer rings." Following this rule, define f(n) as the pure peripheral cumulative sum excluding the center.

f(n) = g(n) - 1 = 3n^2 + 3n

Extending f(n) to negative integers eliminates the constant term and produces exact symmetry:

• f(0) = 0 \longleftrightarrow f(-1) = 0
• f(1) = 6 \longleftrightarrow f(-2) = 6
• f(2) = 18 \longleftrightarrow f(-3) = 18
• f(3) = 36 \longleftrightarrow f(-4) = 36

With symmetry axis n = -\dfrac{1}{2}:

\boxed{f(n) = f(-(n+1))}

4. Geometric Meaning of the Derivative f'(n)

Differentiating f(n) = 3n^2 + 3n with respect to the continuous variable n:

f'(n) = 6n + 3

Since ring k has perimeter S_k = 6k, the derivative equals the arithmetic mean of adjacent ring perimeters:

f'(n) = \frac{6n + 6(n+1)}{2} = \frac{S_n + S_{n+1}}{2} \quad (n \geq 1)

f'(n) is the arithmetic mean of the perimeters (point counts) of ring n and ring n+1.

5. Extension to Real and Complex Numbers

The symmetry identity f(z) = f(-(z+1)) holds for all z \in \mathbb{C}:

f(-(z+1)) = 3(-(z+1))^2 + 3(-(z+1)) = 3(z^2+2z+1) - 3z - 3 = 3z^2 + 3z = f(z)

• Real example (z = 0.5): f(0.5) = 3(0.25) + 1.5 = 2.25, \quad f(-1.5) = 3(2.25) - 4.5 = 2.25

• Complex example (z = i): f(i) = -3 + 3i, \quad f(-i-1) = -3 + 3i

6. Trajectories on the Imaginary Axis and Derivative Mapping

Substitute z = ki (k \in \mathbb{R}).

6.1 Trajectory of f(ki)

f(ki) = 3(ki)^2 + 3(ki) = -3k^2 + 3ki

With real part X = -3k^2 and imaginary part Y = 3k:

\boxed{X = -\frac{1}{3}Y^2}

A parabola opening to the left (\text{Re} \leq 0).

6.2 Trajectory of f'(ki)

f'(ki) = 6(ki) + 3 = 3 + 6ki

With real part X = 3 and imaginary part Y = 6k:

\boxed{X = 3}

A vertical line fixed at real part +3.

7. Grid Transformation over the Full Plane (z = x+iy)

Substitute z = x+iy into f(z) and separate real and imaginary parts:

w = f(x+iy) = 3(x^2 - y^2) + 3x + i(6xy + 3y)

u = 3x^2 + 3x - 3y^2, \quad v = 3y(2x+1)

7.1 Image of Vertical Lines x = c: Family of Left-Opening Parabolas

From y = \dfrac{v}{3(2c+1)} (c \neq -\frac{1}{2}), substitute into u:

u = (3c^2 + 3c) - \frac{1}{3(2c+1)^2}v^2

A left-opening parabola with vertex (3c^2+3c, 0).

• c = 0 and c = -1: vertex (0,0), u = -\dfrac{1}{3}v^2. Identical parabolas; geometric manifestation of f(0) = f(-1).
• c = 0.5 and c = -1.5: vertex (2.25, 0), u = 2.25 - \dfrac{1}{12}v^2

7.2 Image of Horizontal Lines y = d: Family of Right-Opening Parabolas

From x = \dfrac{v}{6d} - \dfrac{1}{2}, substitute into u:

u = \frac{1}{12d^2}v^2 - \left(\frac{3}{4} + 3d^2\right)

A right-opening parabola with vertex \left(-\dfrac{3}{4}-3d^2, 0\right).

8. Conformal Mapping and the Cauchy–Riemann Equations

Vertical lines (x=c) and horizontal lines (y=d) in the z-plane map to two orthogonal families of parabolas in the w-plane because f(z) is complex-differentiable (analytic).

Partial derivatives of u = 3x^2 + 3x - 3y^2 and v = 6xy + 3y:

\frac{\partial u}{\partial x} = 6x + 3, \quad \frac{\partial v}{\partial y} = 6x + 3

\frac{\partial u}{\partial y} = -6y, \quad \frac{\partial v}{\partial x} = 6y

The Cauchy–Riemann equations:

\boxed{\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}}

When these hold, f(z) is conformal (angle-preserving) wherever f'(z) \neq 0.

9. Double-Angle Behavior at the Critical Point z = -\dfrac{1}{2}

f'(z) = 6z + 3 = 0 \quad \Longrightarrow \quad z = -\frac{1}{2}

Function value:

f\left(-\frac{1}{2}\right) = 3\left(\frac{1}{4}\right) - \frac{3}{2} = -\frac{3}{4}

Express the neighborhood in polar coordinates z = -\dfrac{1}{2} + re^{i\theta}:

f\left(-\frac{1}{2} + re^{i\theta}\right) = -\frac{3}{4} + 3r^2 e^{i(2\theta)} + \mathcal{O}(r^3)

• Distance r from -\frac{1}{2} maps to distance 3r^2 from -\frac{3}{4}.
• Angle \theta rotates to 2\theta.
• Orthogonal lines through -\frac{1}{2} collapse onto the real axis in the w-plane.

10. Generating Function

10.1 Ordinary Generating Function of S_k

F_S(t) = \sum_{k=0}^{\infty} S_k t^k = 1 + \sum_{k=1}^{\infty} 6k \cdot t^k = \frac{1 + 4t + t^2}{(1-t)^2} \quad (|t| < 1)

10.2 Ordinary Generating Function of g(n)

G(t) = \sum_{n=0}^{\infty} g(n) t^n = \frac{F_S(t)}{1-t} = \frac{1 + 4t + t^2}{(1-t)^3} \quad (|t| < 1)

The denominator (1-t)^3 encodes the fact that g(n) is a quadratic polynomial. The pole of order 3 at t=1 implies g(n) \sim 3n^2 growth.

11. Summary