Updated 7/5/01

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Surface Transformations

Often we find that two seemingly different surfaces are actually (at least in some topological sense) equivalent. This section provides animations of some of the more complicated transformations from one form of a surface to another.

Below each animation is a description of the moves used to transform one projection of the surface to the other. (This section relies somewhat upon the Roseman Homma-Nagase moves. If you are unfamiliar with these moves, you should read The Basics.)

For each of the transformations, we rendered the animation with two different textures. Generally, the wood texture is the more stable of the two (the striped texture sometimes moves in an unexpected manner), but for some transformations, the striped texture may be clearer.

Moebius Strip to Cross Cap with a Disk Removed

As mentioned in the Construction section, we can stretch the moebius strip around to make it resemble the cross cap with a single circular disk removed. In this animation, we place the edge of this disk in the plane where the cross cap would normally intersect itself.

 Striped Media Player Quicktime Wooden Media Player Quicktime

We begin by rounding the top of the moebius strip downward, and then round it out a bit more by pulling the bottom corners upward. Then we simply constrict the edge until it all lies in one plane.

Other Klein Bottle Representations

The Klein Bottle can be immersed in 3 space in a number of ways. The following animation shows three of these (and how the surface transforms from one to another). The standard klein bottle is shown first, followed by a more symmetric version, and finally the "pinched torus" klein bottle.

 Front View Back View Striped Media Player Quicktime Wooden Media Player Quicktime Striped Media Player Quicktime Wooden Media Player Quicktime

This animation begins by taking the "handle" of the klein bottle and pulling it upward. For the first half second or so, this is simply stretching the surface. At about the 0.5 second mark though, we are forced to perform a type V move to "untwist" the surface at the klein bottle's "mouth".

The following half second is then just a simple deformation to the symmetric form (which is complete at the 2 second mark). It's useful to note that in this form, the klein bottle has two pinch points and a double curve that resembles a parabola where the surface intersects itself. To transform the surface to the pinched klein bottle, we simply straighten this curve to a line.

Klein Bottle to Cross Caps

The Klein Bottle is known to be equivalent to the connected sum of two cross caps. The following animation shows this transformation. It might be helpful to watch the "Other Klein Bottle Representations" animations first, as the "pinched torus" klein bottle is one of the intermediate steps in this transformation.

 Front View Back View Striped Media Player Quicktime Wooden Media Player Quicktime Striped Media Player Quicktime Wooden Media Player Quicktime

The first three seconds are an abbreviation of the previous "Other Klein Bottle Representations" animation.

The next step is to turn the pinched torus "inside out". You might visualize this by imagining that we made a cut on the back side of the pinched torus, turned it inside out through this cut, and then reglued. (As cutting isn't technically allowed, we actually do this by pushing the surface that was the back of the pinched torus all the way through to the front side.) This inversion process takes 4 seconds (from the 3 second mark to the 7 second mark).

Taking two seconds to flatten out the newly inverted surface, we see what looks like an ellipsoid with a pinch through the middle. In the final two seconds (from 9 to 11) we produce the two joined cross caps by performing a Type IV move on the right hand half of the ellipsoid. (That is, on the right hand half, we pass the upper surface downward through the lower surface.)

Half Klein Bottle to Moebius Strip

As a moebius strip is equivalent to a cross cap with a disk removed (and the connected sum of two cross caps forms the klein bottle), it should be unsurprising that two moebius strips, joined along their edges, form the klein bottle as well.

 Striped Media Player Quicktime Wooden Media Player Quicktime

We show half of the standard klein bottle (cut laterally) initially. Examine the first frame of the animation closely to be sure to convince yourself that this is truely half of a klein bottle. The animation then simply stretches part of the surface upwards until it no longer intersects itself.

Torus with Cross Cap to Klein Bottle with Cross Cap

Unfortunately, the limitations of our software prevented us from animating this transformation. An abbreviated sketch of the process is shown below.

The first picture is a klein bottle connected-summed with a cross cap (1). Using a transformation similar to the one above, we change this to a pinched torus klein bottle (2). Stretching out the surface a bit, we can round it into a more torus-like shape (3). We then perform a Type V move along the line formed by the two double curves (horizontally through the middle of the "torus" in picture 3). This "undoes" the pinch on the left and the half-pinch on the right, but also adds a half-pinch on the right (4). (For a demonstration of a similar move, see the last three seconds of Klein Bottle to Cross Caps.) Stretching the surface a bit, we bring the remaining half-pinch to the outside (5). Finally, we stretch the surface back out into the familiar form of a torus connected-summed with a crosscap (6).