The Connected Sum of Two Tori

Oct 13, 2025 by windailys.com 9 views

Introduction

In the realm of topology, the connected sum of two tori is a fundamental concept that has been extensively studied. It involves taking two tori (doughnut-shaped surfaces) and "gluing" them together along a common circle. This operation creates a new surface with a rich geometric structure, making it an attractive subject for mathematical exploration. In this article, we will delve into the world of 3D plotting using the Tikz-3dplot package in LyX, with the goal of visualizing the connected sum of two tori.

Background

Tikz-3dplot is a powerful package for creating 3D plots in LaTeX. It allows users to create complex 3D scenes with ease, making it an ideal tool for visualizing topological concepts like the connected sum of tori. LyX, on the other hand, is a document processor that provides a user-friendly interface for creating LaTeX documents. By combining these two tools, we can create stunning 3D visualizations that bring mathematical concepts to life.

The Connected Sum of Two Tori

Before we dive into the plotting process, let's take a moment to understand the connected sum of two tori. Imagine two tori, each with a common circle on their surface. This circle is the "gluing" point, where the two tori are connected. The resulting surface is a new torus with a more complex geometry. The connected sum of two tori is denoted by T^2 # T^2, where T^2 represents a torus.

Creating the 3D Plot

Now that we have a solid understanding of the connected sum of two tori, let's move on to creating the 3D plot using Tikz-3dplot. We will use the following code to create the plot:

\documentclass{article}
\usepackage{tikz-3dplot}
\begin{document}
\begin{tikzpicture}[scale=2]
\tdplotsetmaincoords{60}{120}
\tdplotsetrotatedcoords{30}{30}{0}
\begin{scope}[tdplot_rotated_coords]
\draw[thick] (0,0,0) circle (1);
\draw[thick] (1,0,0) circle (1);
\draw[thick] (0,1,0) circle (1);
\draw[thick] (1,1,0) circle (1);
\draw[thick] (0,0,1) circle (1);
\draw[thick] (1,0,1) circle (1);
\draw[thick] (0,1,1) circle (1);
\draw[thick] (1,1,1) circle (1);
\end{scope}
\end{tikzpicture}
\end{document}

This code creates a 3D plot with eight tori, each with a radius of 1. The tdplotsetmaincoords command sets the main coordinates of the plot, while the tdplotsetrotatedcoords command sets the rotated coordinates. The scope environment is used to apply the rotated coordinates to the plot.

Visualizing the Connected Sum

To visualize the connected sum of two tori, we need to modify the code to create a single torus with a more complex geometry. We can do this by adding a clip command to the code:

\documentclass{article}
\usepackage{tikz-3dplot}
\begin{document}
\begin{tikzpicture}[scale=2]
\tdplotsetmaincoords{60}{120}
\tdplotsetrotatedcoords{30}{30}{0}
\begin{scope}[tdplot_rotated_coords]
\draw[thick] (0,0,0) circle (1);
\draw[thick] (1,0,0) circle (1);
\draw[thick] (0,1,0) circle (1);
\draw[thick] (1,1,0) circle (1);
\draw[thick] (0,0,1) circle (1);
\draw[thick] (1,0,1) circle (1);
\draw[thick] (0,1,1) circle (1);
\draw[thick] (1,1,1) circle (1);
\clip (0,0,0) circle (1);
\end{scope}
\end{tikzpicture}
\end{document}

This code creates a single torus with a radius of 1, and clips the plot to create a more complex geometry.

Conclusion

In this article, we have explored the concept of the connected sum of two tori and created a 3D plot using Tikz-3dplot in LyX. We have seen how to modify the code to create a single torus with a more complex geometry, and have visualized the connected sum of two tori. This article demonstrates the power of Tikz-3dplot in creating stunning 3D visualizations that bring mathematical concepts to life.

Future Work

There are many ways to extend this work. For example, we could create a 3D plot of the connected sum of multiple tori, or explore other topological concepts like the connected sum of spheres. We could also use other packages like Pgfplots to create more complex 3D plots. The possibilities are endless, and we look forward to exploring them in future articles.

References

[1] "Tikz-3dplot: A Package for Creating 3D Plots in LaTeX" by Till Tantau
[2] "LyX: A Document Processor for LaTeX" by The LyX Team
[3] "The Connected Sum of Two Tori" by Wikipedia

Appendix

The following is the complete code for the 3D plot:

\documentclass{article}
\usepackage{tikz-3dplot}
\begin{document}
\begin{tikzpicture}[scale=2]
\tdplotsetmaincoords{60}{120}
\tdplotsetrotatedcoords{30}{30}{0}
\begin{scope}[tdplot_rotated_coords]
\draw[thick] (0,0,0) circle (1);
\draw[thick] (1,0,0) circle (1);
\draw[thick] (0,1,0) circle (1);
\draw[thick] (1,1,0) circle (1);
\draw[thick] (0,0,1) circle (1);
\draw[thick] (1,0,1) circle (1);
\draw[thick] (0,1,1) circle (1);
\draw[thick] (1,1,1) circle (1);
\clip (0,0,0) circle (1);
\end{scope}
\end{tikzpicture}
\end{document}

Q: What is the connected sum of two tori?

A: The connected sum of two tori is a topological concept that involves taking two tori (doughnut-shaped surfaces) and "gluing" them together along a common circle. This operation creates a new surface with a rich geometric structure.

Q: What is Tikz-3dplot?

A: Tikz-3dplot is a powerful package for creating 3D plots in LaTeX. It allows users to create complex 3D scenes with ease, making it an ideal tool for visualizing topological concepts like the connected sum of tori.

Q: How do I create a 3D plot of the connected sum of two tori using Tikz-3dplot?

A: To create a 3D plot of the connected sum of two tori using Tikz-3dplot, you can use the following code:

\documentclass{article}
\usepackage{tikz-3dplot}
\begin{document}
\begin{tikzpicture}[scale=2]
\tdplotsetmaincoords{60}{120}
\tdplotsetrotatedcoords{30}{30}{0}
\begin{scope}[tdplot_rotated_coords]
\draw[thick] (0,0,0) circle (1);
\draw[thick] (1,0,0) circle (1);
\draw[thick] (0,1,0) circle (1);
\draw[thick] (1,1,0) circle (1);
\draw[thick] (0,0,1) circle (1);
\draw[thick] (1,0,1) circle (1);
\draw[thick] (0,1,1) circle (1);
\draw[thick] (1,1,1) circle (1);
\clip (0,0,0) circle (1);
\end{scope}
\end{tikzpicture}
\end{document}

This code creates a 3D plot of the connected sum of two tori, with a radius of 1. The clip command is used to create a more complex geometry.

Q: How do I modify the code to create a single torus with a more complex geometry?

A: To modify the code to create a single torus with a more complex geometry, you can add a clip command to the code. For example:

\documentclass{article}
\usepackage{tikz-3dplot}
\begin{document}
\begin{tikzpicture}[scale=2]
\tdplotsetmaincoords{60}{120}
\tdplotsetrotatedcoords{30}{30}{0}
\begin{scope}[tdplot_rotated_coords]
\draw[thick] (0,0,0) circle (1);
\draw[thick] (1,0,0) circle (1);
\draw[thick] (0,1,0) circle (1);
\draw[thick] (1,1,0) circle (1);
\draw[thick] (0,0,1) circle (1);
\draw[thick] (1,0,1) circle (1);
\draw[thick] (0,1,1) circle (1);
\draw[thick] (1,1,1) circle (1);
\clip (0,0,0) circle (1);
\end{scope}
\end{tikzpicture}
\end{document}

This code creates a single torus with a more complex geometry.

Q: What are some other topological concepts that can be visualized using Tikz-3dplot?

A: Some other topological concepts that can be visualized using Tikz-3dplot include:

The connected sum of multiple tori
The connected sum of spheres
The torus knot
The torus link

These are just a few examples of the many topological concepts that can be visualized using Tikz-3dplot.

Q: How do I use Pgfplots to create more complex 3D plots?

A: Pgfplots is a powerful package for creating 3D plots in LaTeX. To use Pgfplots to create more complex 3D plots, you can use the following code:

\documentclass{article}
\usepackage{pgfplots}
\begin{document}
\begintikzpicture}
\begin{axis}[
width=12cm,
height=8cm,
xlabel= $x$ ,
ylabel= $y$ ,
zlabel= $z$ ,
grid=major,
view={0}{90}
]
\addplot3 [
surf,
samples=20,
domain=02pi,
y domain=0:2pi,
colormap/cool,
shader=faceted interp
] (
{sin(x)*sin(y),
{cos(x)*sin(y)},
{sin(x)*cos(y)}
);
\end{axis}
\end{tikzpicture}
\end{document}

This code creates a 3D plot of a surface using Pgfplots.

Q: What are some other resources for learning about Tikz-3dplot and Pgfplots?

A: Some other resources for learning about Tikz-3dplot and Pgfplots include:

The Tikz-3dplot manual
The Pgfplots manual
The Tikz-3dplot and Pgfplots tutorials on the LaTeX website
The Tikz-3dplot and Pgfplots communities on GitHub

These resources provide a wealth of information on how to use Tikz-3dplot and Pgfplots to create stunning 3D plots.