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

Using concepts in semiotics in as great detail as possible, how to measure the DNA molecule for its aesthetic value and the extent that principles of aesthetics play a part before its formation?

-
Using concepts in semiotics in as great detail as possible, how to measure the DNA molecule for its aesthetic value and the extent that principles of aesthetics play a part before its formation? Measuring the aesthetic value of a DNA molecule and determining the extent to which principles of aesthetics play a part before its formation involves applying concepts from semiotics, which is the study of signs and symbols. While semiotics is traditionally concerned with the analysis of human communication, we can adapt its principles to explore the aesthetics of the DNA molecule. However, it is important to note that assessing the aesthetic value of a DNA molecule is highly subjective and speculative, as aesthetic judgments are typically rooted in human perception and cultural contexts. Nevertheless, I will provide a conceptual framework that can be used for this analysis. Signs and Symbols: Semiotics recognizes signs as entities that convey meaning. In the context of DNA, we can consider th...
Read more

Video with Transcript: Fibonacci Numbers hidden in the Mandelbrot Set - Numberphile

-
Image
0:00 So, today I'm gonna tell you about how the Fibonacci sequence appears in the Mandelbrot set. 0:04 Hopefully, your mind will be blown by the end of that sentence. 0:06 So the Fibonacci sequence, the rule is that you take the previous two numbers, 0:10 and you add them together to get the next. Right? So we started with 1 1. Their sum is 2. 0:15 The next number will be the sum of 1 & 2, which is 3. 0:19 The sum of 2 & 3 is 5. The sum of 3 & 5 is 8. I'll do one more. 0:24 And then, you can continue on. 0:26 Brady: "Easy." 0:27 Easy, right? And this occurs everywhere that has interesting connections to things in nature, and all of that, 0:31 but I just want to show you where it appears on the Mandelbrot set. 0:35 So, slightly less easy. So, the Mandelbrot set is a special object inside of the complex plane. So the plane of complex numbers. And 0:43 the way you cook this thing up is by 0:46 considering a certain type of dynamical system. So if you give me ...
Read more

Islamic Aesthetics - Prof. Nasr

-
Islamic Aesthetics -  Seyyed Hossein Nasr  Work:  Islamic art and spirituality - Islamic art and spirituality : Nasr, Seyyed Hossein : Free Download, Borrow, and Streaming : Internet Archive Introduction   Aesthetics, as a branch of philosophy dealing with beauty and art, was not treated as a separate subject by Islamic philosophers (falāsifa or ḥukamā). Unlike Western philosophers such as Hegel or Croce, Islamic thinkers addressed aesthetics indirectly. To understand Islamic aesthetics, one must examine passages from Peripatetic (mashshā’ī) philosophers, the Illuminationist (ishrāqī) school, and Sufi writings, as well as oral traditions transmitted by masters of Islamic arts.   Peripatetic Philosophers and Aesthetics    Islamic Peripatetic philosophers primarily discussed aesthetics in relation to poetry, often commenting on Aristotle’s Poetics. Key figures include:   - Al-Fārābī: Viewed poetry as "imaginative syllogistic proof by ...
Read more