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 Home National Science Foundation - Mathematical & Physical Sciences (MPS)
Mathematical & Physical Sciences (MPS)
design element
MPS Home
About MPS
Funding Opportunities
Awards
News
Events
Discoveries
Publications
Advisory Committee
Career Opportunities
2013-2014 Distinguished Lecture Series
View MPS Staff
MPS Organizations
Astronomical Sciences (AST)
Chemistry (CHE)
Materials Research (DMR)
Mathematical Sciences (DMS)
Physics (PHY)
Office of Multidisciplinary Activities (OMA)
Proposals and Awards
Proposal and Award Policies and Procedures Guide
  Introduction
Proposal Preparation and Submission
bullet Grant Proposal Guide
  bullet Grants.gov Application Guide
Award and Administration
bullet Award and Administration Guide
Award Conditions
Other Types of Proposals
Merit Review
NSF Outreach
Policy Office
Other Site Features
Special Reports
Research Overviews
Multimedia Gallery
Classroom Resources
NSF-Wide Investments

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