text-only page produced automatically by LIFT Text Transcoder Skip all navigation and go to page contentSkip top navigation and go to directorate navigationSkip top navigation and go to page navigation
National Science Foundation
Discoveries
design element
Discoveries
Search Discoveries
About Discoveries
Discoveries by Research Area
Arctic & Antarctic
Astronomy & Space
Biology
Chemistry & Materials
Computing
Earth & Environment
Education
Engineering
Mathematics
Nanoscience
People & Society
Physics
 

Email this pagePrint this page
All Images

Discovery
Klein Bottle is a Real Natural in the Zoo of Geometric Shapes

Back to article | Note about images

Image of a two-dimensional representation of the Klein bottle topology.

A two-dimensional representation of a Klein bottle--a shape with no inside or outside, just one continuous surface. A true Klein bottle needs at least four dimensions; in other words, it can't be blown from glass. Two- and three-dimensional representations like this one exist to help us visualize the topology, but they are not completely faithful to the original shape. The surface cannot be built in two- or three-dimensional space without self-intersection, as shown here with the "handle" passing through the side of the surface.

Credit: Thomas Banchoff, Brown University, and Davide Cervone, Union College.


Download the high-resolution JPG version of the image. (47 KB)

Use your mouse to right-click (Mac users may need to Ctrl-click) the link above and choose the option that will save the file or target to your computer.

Photo of Stanford mathematician Gunnar Carlsson.

Stanford mathematician Gunnar Carlsson.

Credit: Stanford University


Download the high-resolution JPG version of the image. (49 KB)

Use your mouse to right-click (Mac users may need to Ctrl-click) the link above and choose the option that will save the file or target to your computer.

Image of a klein bottle with bands removed to show the interior connection.

A two-dimensional representation of the Klein bottle shape with wide bands removed to make it easier to see the interior connection. Two of the colored bands are Mobius strips, but the others are orientable strips with a full twist, each looping through the handle and over the top twice.

Credit: Thomas Banchoff, Brown University, and Davide Cervone, Union College.


Download the high-resolution JPG version of the image. (60 KB)

Use your mouse to right-click (Mac users may need to Ctrl-click) the link above and choose the option that will save the file or target to your computer.



Email this pagePrint this page
Back to Top of page