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 September 17, 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 September 17, 2014).

Share This



More Computers & Math News

Wednesday, September 17, 2014

Featured Research

from universities, journals, and other organizations


Featured Videos

from AP, Reuters, AFP, and other news services

2K Drafts Face-Mapping Tech for New Game

2K Drafts Face-Mapping Tech for New Game

AP (Sep. 17, 2014) — "NBA 2K15" is angling for a slam dunk with an innovative new way to put players in the game. Gamers will be able to digitally graft lifelike 3D renditions of their faces onto virtual players using the PlayStation 4 and Xbox One cameras. (Sept. 17) Video provided by AP
Powered by NewsLook.com
FBI Finishes $1 Billion Facial Recognition System

FBI Finishes $1 Billion Facial Recognition System

Newsy (Sep. 15, 2014) — The FBI announced it plans to make its Next Generation Identification System available to law enforcement, but some privacy advocates are worried. Video provided by Newsy
Powered by NewsLook.com
A+ for Apple iPhone Pre-Sales

A+ for Apple iPhone Pre-Sales

Reuters - Business Video Online (Sep. 15, 2014) — Apple says it received a record 4 million first-day pre-orders for its new iPhone 6 and iPhone 6 Plus, pushing delivery dates into October. Bobbi Rebell reports. Video provided by Reuters
Powered by NewsLook.com
Microsoft to Buy 'Minecraft' Maker for $2.5B

Microsoft to Buy 'Minecraft' Maker for $2.5B

AP (Sep. 15, 2014) — Microsoft will acquire the maker of the long-running hit game Minecraft for $2.5 billion as the company continues to invest in its Xbox gaming platform and looks to grab attention on mobile phones. (Sept. 15) 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:
from the past week

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