Featured Research

from universities, journals, and other organizations

Mathematics: First-ever image of a flat torus in 3-D

Date:
April 25, 2012
Source:
CNRS (Délégation Paris Michel-Ange)
Summary:
Just as a terrestrial globe cannot be flattened without distorting the distances, it seemed impossible to visualize abstract mathematical objects called flat tori in ordinary three-dimensional space. However, a team of mathematicians and computer scientists has succeeded in constructing and visually representing an image of a flat torus in three-dimensional space. This is a smooth fractal, halfway between fractals and ordinary surfaces.

Image showing the isometric embedding of a square flat torus in 3D space, seen from the outside (above) and from the inside (below). Different oscillation waves, called corrugations, can be distinguished.Together, the corrugations form an object that resembles a fractal and has a rough appearance.
Credit: © Borrelli, Jabrane, Lazarus, Thibert

Just as a terrestrial globe cannot be flattened without distorting the distances, it seemed impossible to visualize abstract mathematical objects called flat tori in ordinary three-dimensional space. However, a team of mathematicians and computer scientists[1] has succeeded in constructing and visually representing an image of a flat torus in three-dimensional space. This is a smooth fractal, halfway between fractals and ordinary surfaces.

The results are published in The Proceedings of the National Academy of Sciences.

In the 1950s, Nicolaas Kuiper and the Nobel laureate John Nash demonstrated the existence of a representation of an abstract mathematical object called flat torus, without being able to visualize it. Since then, constructing a representation of this surface has remained a challenge that has finally been met by scientists in Lyon and Grenoble. On the basis of the Convex Integration Theory developed by Mikhail Gromov in the 1970s, the researchers used the corrugation technique (oscillations). This reputedly abstract mathematical method helps to determine atypical solutions to partial differential equations. This enabled the scientists to obtain images of a flat torus in 3D for the first time. Halfway between fractals and ordinary surfaces, these images show a smooth fractal.

These findings open up new avenues in applied mathematics, especially in the visualization of the differential equations encountered in physics and biology. The astounding properties of smooth fractals could also play a central role in the analysis of the geometry of shapes.

[1] The team brings together four researchers from Institut Camille Jordan (CNRS/Universités Claude Bernard Lyon 1 and Saint-Etienne/Ecole Centrale de Lyon/INSA de Lyon), GIPSA-lab (CNRS/Grenoble-INP/ Universités Joseph Fourier and Stendhal-Grenoble 3) and Laboratoire Jean Kuntzmann (CNRS/Universités Joseph Fourier and Pierre Mendès France/Grenoble-INP/INRIA).


Story Source:

The above story is based on materials provided by CNRS (Délégation Paris Michel-Ange). Note: Materials may be edited for content and length.


Journal Reference:

  1. V. Borrelli, S. Jabrane, F. Lazarus, B. Thibert. Flat tori in three-dimensional space and convex integration. Proceedings of the National Academy of Sciences, 2012; DOI: 10.1073/pnas.1118478109

Cite This Page:

CNRS (Délégation Paris Michel-Ange). "Mathematics: First-ever image of a flat torus in 3-D." ScienceDaily. ScienceDaily, 25 April 2012. <www.sciencedaily.com/releases/2012/04/120425094348.htm>.
CNRS (Délégation Paris Michel-Ange). (2012, April 25). Mathematics: First-ever image of a flat torus in 3-D. ScienceDaily. Retrieved October 22, 2014 from www.sciencedaily.com/releases/2012/04/120425094348.htm
CNRS (Délégation Paris Michel-Ange). "Mathematics: First-ever image of a flat torus in 3-D." ScienceDaily. www.sciencedaily.com/releases/2012/04/120425094348.htm (accessed October 22, 2014).

Share This



More Computers & Math News

Wednesday, October 22, 2014

Featured Research

from universities, journals, and other organizations


Featured Videos

from AP, Reuters, AFP, and other news services

Thanks, Marty McFly! Hoverboards Could Be Coming In 2015

Thanks, Marty McFly! Hoverboards Could Be Coming In 2015

Newsy (Oct. 21, 2014) — If you've ever watched "Back to the Future Part II" and wanted to get your hands on a hoverboard, well, you might soon be in luck. Video provided by Newsy
Powered by NewsLook.com
Robots to Fly Planes Where Humans Can't

Robots to Fly Planes Where Humans Can't

Reuters - Innovations Video Online (Oct. 21, 2014) — Researchers in South Korea are developing a robotic pilot that could potentially replace humans in the cockpit. Unlike drones and autopilot programs which are configured for specific aircraft, the robots' humanoid design will allow it to fly any type of plane with no additional sensors. Ben Gruber reports. Video provided by Reuters
Powered by NewsLook.com
Japanese Scientists Unveil Floating 3D Projection

Japanese Scientists Unveil Floating 3D Projection

Reuters - Innovations Video Online (Oct. 20, 2014) — Scientists in Tokyo have demonstrated what they say is the world's first 3D projection that floats in mid air. A laser that fires a pulse up to a thousand times a second superheats molecules in the air, creating a spark which can be guided to certain points in the air to shape what the human eye perceives as an image. Matthew Stock reports. Video provided by Reuters
Powered by NewsLook.com
Apple Enters Mobile Payment Business

Apple Enters Mobile Payment Business

AP (Oct. 20, 2014) — Apple is making a strategic bet with the launch of Apple Pay, the mobile pay service aimed at turning your iPhone into your wallet. (Oct. 20) Video provided by AP
Powered by NewsLook.com

Search ScienceDaily

Number of stories in archives: 140,361

Find with keyword(s):
 
Enter a keyword or phrase to search ScienceDaily for related topics and research stories.

Save/Print:
Share:  

Breaking News:

Strange & Offbeat Stories

 

Space & Time

Matter & Energy

Computers & Math

In Other News

... from NewsDaily.com

Science News

Health News

Environment News

Technology News



Save/Print:
Share:  

Free Subscriptions


Get the latest science news with ScienceDaily's free email newsletters, updated daily and weekly. Or view hourly updated newsfeeds in your RSS reader:

Get Social & Mobile


Keep up to date with the latest news from ScienceDaily via social networks and mobile apps:

Have Feedback?


Tell us what you think of ScienceDaily -- we welcome both positive and negative comments. Have any problems using the site? Questions?
Mobile iPhone Android Web
Follow Facebook Twitter Google+
Subscribe RSS Feeds Email Newsletters
Latest Headlines Health & Medicine Mind & Brain Space & Time Matter & Energy Computers & Math Plants & Animals Earth & Climate Fossils & Ruins