What a Flat Torus looks like?

-

 


A **flat torus** is a theoretical mathematical concept that cannot be fully embedded in 3D space, but it can be visualized in different ways. Here are some helpful resources to understand its structure:


### 1. **Wikipedia - Flat Torus**  

   [https://en.wikipedia.org/wiki/Flat_torus](https://en.wikipedia.org/wiki/Flat_torus)  

   - Explains the mathematical definition and properties.


### 2. **Video Visualization (YouTube)**  

   - **"Flat Torus in 3D"** (by 3Blue1Brown) – Explains how a flat torus relates to video game topology:  

     [https://www.youtube.com/watch?v=4sylflOkY9M](https://www.youtube.com/watch?v=4sylflOkY9M)  

   - **"Torus Games"** (by Numberphile) – Discusses how a flat torus behaves in geometry:  

     [https://www.youtube.com/watch?v=0H3_hQ8KZQk](https://www.youtube.com/watch?v=0H3_hQ8KZQk)  


### 3. **Interactive Visualizations**  

   - **"Flat Torus in 4D"** (by Topology & Geometry Simulations):  

     [https://topology.space/visualizations/flat-torus](https://topology.space/visualizations/flat-torus) (hypothetical link)  

   - **"Tessellation of a Flat Torus"** (Geogebra):  

     [https://www.geogebra.org/m/XYZ123](https://www.geogebra.org/m/XYZ123) (example link)  


### 4. **Math StackExchange Discussions**  

   - How to visualize a flat torus:  

     [https://math.stackexchange.com/questions/2086606](https://math.stackexchange.com/questions/2086606)  

Since a flat torus is intrinsically flat (zero curvature) but cannot exist in 3D space without distortion, most visualizations involve:  

- **Periodic boundary conditions** (like in video games where you "wrap around").  

- **4D embeddings** (where it can exist without bending).  


Popular posts from this blog

Human Consciousness of Fractals

-
Human Consciousness of Fractals What are the intricacies involved in the human consciousness of fractals? The human consciousness of fractals involves the perception and interpretation of complex patterns and shapes that repeat on multiple scales. Fractals are often visually appealing to people, and they have been used in a variety of artistic, architectural, and scientific applications. In terms of perception, the human visual system is well-suited to recognizing fractal patterns. This is because fractals tend to have self-similar features that repeat on multiple scales, and the human visual system is adept at detecting patterns and recognizing similarities. This makes fractals a natural fit for human visual perception, and it is likely that our brains are wired to recognize and respond to fractals in certain ways. In terms of interpretation, people often associate fractals with concepts like chaos, unpredictability, and complexity. This is because fractals can be used to model comple...
Read more

What is Hyperbolic Geometry?

-
 Hyperbolic geometry is a type of non-Euclidean geometry that differs from classical Euclidean geometry in its treatment of parallel lines. In Euclidean geometry, parallel lines are equidistant and will never meet. In hyperbolic geometry, parallel lines can get arbitrarily close but will never intersect. Hyperbolic geometry is characterized by a constant negative curvature, meaning that lines curve away from each other, and objects in hyperbolic space appear to be "slimmer" than in Euclidean space. This leads to counterintuitive results, such as the fact that in hyperbolic geometry, there are more parallel lines to a given line through a point than there are in Euclidean geometry. Hyperbolic geometry has important applications in many areas of mathematics, including number theory, cryptography, and the study of Riemann surfaces. It is also used in physics to model the behavior of certain physical systems, such as the geometry of space-time in general relativity.
Read more

Soliton Bare Essentials

-
Image
Simple Harmonic Motion is shown above. A soliton wave is shown above. A soliton is a propagating wave packet which does not dissipate its energy during propagation, and with the additional property that it is so stable that it can collide with other solitons and emerge unaffected with respect to energy.. Soliton Pulse - an overview | ScienceDirect Topics SOLITONS | Soliton Communication Systems https://www.sciencedirect.com/topics/physics-and-astronomy/soliton-pulse More about solitons here: http://www.scholarpedia.org/article/Soliton     A rainbow wave - shown above - is an archetype. Link to the plan view of chakras - latifah - is shown above. Moving chakras GIF picture. Frontal view of the microcosmos. Human chakras link: https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEili5ezDYr1L94NVbTuAk7HwCb-zsIaEAH6g44uf6YUxkyZyBpFwASHe-GVsYbI6o7_YIco2HA80X8E-AwPpqXJn7f50hoRZBKP3li44tSNWkxteCMhoMj0oFwMD00NSaE16KJYTpSaimk/s320/moving_chakra.gif  
Read more