CaleTZ is a Typst package for visualizing Calabi-Yau manifolds using CeTZ 3D drawing primitives.

It generates high-quality, parameterizable 3D surfaces representing slices of Calabi-Yau spaces. This package is ideal for physics and mathematics visualizations, allowing you to explore these complex geometries directly within your Typst documents.

Calabi-Yau Projection n=5 Cross-section View Topological Structure

CaleTZ visualizes a 3D projection of a complex 1-dimensional sub-manifold (an algebraic curve) related to the Fermat Quintic Calabi-Yau threefold.

The surface is defined by the locus of points satisfying the Fermat equation in $\mathbb{C}^2$:

To visualize this 4-dimensional object ($z_1, z_2 \in \mathbb{C}$) in 3D Euclidean space, we use a parameterization based on hyperbolic functions and perform a dimensional reduction.

Let $w = a + ib$ be a complex parameter. We define the coordinates using the hyperbolic identity $\cosh^2 w - \sinh^2 w = 1$:

where:

• $n$ is the degree of the manifold (typically $n=5$ for the quintic).
• $k_1, k_2 \in {0, 1, \dots, n-1}$ determine the Riemann sheet branches.
• $a \in [0, \pi/2]$ and $b \in [-\pi/2, \pi/2]$ are the domain parameters.

Since we cannot view 4 dimensions directly, we project the coordinates into $\mathbb{R}^3$. The visualization uses the standard mapping:

Here, $\alpha$ is a mixing angle that determines how the imaginary components are projected onto the vertical $z$-axis, revealing the hidden dimensions of the manifold.

#import "@preview/caletz:0.1.0": calabi-yau
#set page(width: auto, height: auto, margin: 1cm)
// Render a Calabi-Yau projection with degree 3
#calabi-yau(
 power: 3,
 angle: 0.4,
 subdivisions: 20,
 scale-factor: 4.0
)

This project is distributed under the MIT License. See LICENSE for details.