Documentation

Technical notes, mathematical background and rendering details.

Controls

Movement: `W`, `A`, `S`, `D` keys.
Elevation: `Space` (Double-press to reset height).
Sprint: Hold `Shift`.
Slow Walk: Hold `Ctrl`.
Menu: `Esc` to toggle settings.
Critical Walking: `Ctrl + C` (when viewing the Zeta function) to walk along the critical line.
Newton Walking: `Ctrl + Z` to walk to a zero in the context of the current function.

Functions

The explorer supports various standard complex functions, including trigonometric, exponential, and logarithmic functions. However, the centerpiece of the project is the Riemann zeta function \( \zeta(s) \).

The Riemann zeta function

Defined as \( \zeta(s) = \sum_{n=1}^\infty n^{-s} \) for \( \operatorname{Re}(s) > 1 \), it is analytically continued to the rest of the complex plane. Our implementation uses the Dirichlet eta representation for numerical stability when \( \operatorname{Re}(s) > 0.5 \):

\zeta(s) = \frac{1}{1 - 2^{1-s}} \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}

For \( \operatorname{Re}(s) \le 0.5 \), the explorer utilizes the reflection formula to achieve analytical continuation to the entire complex plane:

\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)

In this calculation, the term \( \zeta(1-s) \) is evaluated using the Dirichlet Eta representation, since for \( \operatorname{Re}(s) \le 0.5 \), the reflected point \( 1-s \) has a real part greater than or equal to \( 0.5 \). This allows evaluation across the critical strip and beyond.

Note on Precision: Calculations are performed in GPU shaders using float32 arithmetic. This introduces numerical limitations and potential artifacts as the magnitude of the imaginary part \( |t| \) increases, due to the rapid oscillation and large intermediate values of the functions involved. These effects are especially pronounced in the \( \sin \) and \( \Gamma \) terms of the reflection formula, both of which can grow rapidly in magnitude.

Borwein's Algorithm

To efficiently and accurately compute the Riemann zeta function, we also utilize Borwein's algorithm. This method provides an accelerated approximation for the Zeta function, ensuring rapid convergence and numerical stability. We specifically use it in our zeros finder to refine the positions of potential zero candidates, achieving high precision in locating the non-trivial zeros on the critical line.

The Dirichlet beta function

The explorer implements the Dirichlet beta function \( \beta(s) \) using its series representation:

\beta(s) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^s}

Similar to the eta function, this series converges for \( \operatorname{Re}(s) > 0 \).

The Gamma function

The Gamma function \( \Gamma(z) \) is implemented using the Lanczos approximation (\( g=7, N=9 \)):

\Gamma(z) \approx \sqrt{2\pi} (z+g-0.5)^{z-0.5} e^{-(z+g-0.5)} A_g(z-1) \] where \( A_g(z) \) is the partial fraction expansion: \[ A_g(z) = p_0 + \sum_{n=1}^8 \frac{p_n}{z+n}

using the following coefficients:

\( p_0 = 0.99999999999980993 \)
\( p_1 = 676.5203681218851 \)
\( p_2 = -1259.1392167224028 \)
\( p_3 = 771.32342877765313 \)
\( p_4 = -176.61502916214059 \)
\( p_5 = 12.507343278686905 \)
\( p_6 = -0.13857109526572012 \)
\( p_7 = 9.9843695780195716 \times 10^{-6} \)
\( p_8 = 1.5056327351493116 \times 10^{-7} \)

Numerical Singularities: Regions where the function evaluation produces NaN or infinite values are rendered as a dark terrain covered with green grid-like squares reminiscent of The Matrix . These regions typically arise near overflow conditions, or severe floating-point instability in the GPU shaders. Rather than hiding such failures, the visualization exposes them directly as part of the numerical structure of the computation.

The Mandelbrot function

The explorer visualizes the Mandelbrot function by computing the sequence \( z_{n+1} = z_n^2 + c \) recursively, starting at \( z_0 = 0 \), where \( c \) represents the position in the complex plane. The iterations parameter controls the maximum depth of the recursion.

Using the Zoom Factor: The `Zoom Factor` controls the scale of the visible region in the complex plane. Larger zoom values progressively magnify smaller portions of the function, revealing increasingly intricate structures and fine details that are not visible at larger scales. As the zoom increases, the explorer rescales the domain coordinates before evaluating the function. This effectively allows navigation through smaller regions of the complex plane while preserving the same rendering pipeline. In the Mandelbrot visualization, deep zooms expose self-similar formations, filamentary boundaries, and highly complex recursive structures characteristic of fractal geometry.