Single-Sided Surfaces ISAMA 2004 CS 39R Single-Sided Surfaces Carlo H. Séquin Title. -- What are these things?? – Let’s look on the Internet … EECS Computer Science Division University of California, Berkeley

Making a Single-Sided Surface ISAMA 2004 Making a Single-Sided Surface You are probably familiar with at least one single sided surface: the Möbius Band. This surface has only one rim in the form of a single closed double loop . . . Twisting a ribbon into a Möbius band

A closed ribbon with a 540°flip. ISAMA 2004 Simple Möbius Bands A single-sided surface with a single edge: A closed ribbon with a 540°flip. This rim can be seen more easily in this static image where it is highlighted in yellow. But Mobius bands come in many different shapes: >>> It can also come in a triply-twisted form – as seen in this recycling symbol. In both cases, the yellow rim circles twice around the center of the figure. Now we can deform this surface so that its rim takes on different shapes . . . A closed ribbon with a 180°flip.

Max Bill’s sculpture of a Möbius band. ISAMA 2004 More Möbius Bands In Max Bill’s sculpture, the rim forms a highly warped 3D space curve. >>> But we can warp the whole surface even more, so that the rim becomes circular, as in the fascinating configuration known as the “Sue-Dan-ese MB”, shown on the right. (By Sue Goodman and Dan Asimov) This is basically a one-sided bottle with a circular rim. Max Bill’s sculpture of a Möbius band. The “Sue-Dan-ese” M.B., a “bottle” with circular rim.

A Möbius Band Transfromation ISAMA 2004 A Möbius Band Transfromation This clarifies how the classical MB can be transformed into the SueDan-ese shape. We start by widening the bottom part by pulling upwards its two sides. This starts to form a basket shape with a twisted handle. Gradually we also widen the handle and un-twist it until the yellow line becomes circular. Widen the bottom of the band by pulling upwards its two sides,  get a Möbius basket, and then a Sudanese Möbius band.

Many Different Möbius Shapes ISAMA 2004 Many Different Möbius Shapes Left-twisting versions shown – can be smoothly transformed into one another Topologically, these are all equivalent: They all are single-sided, They all have ONE rim, They all are of genus ONE. Each shape is chiral: its mirror image differs from the original. Thus, there are many different Moebius shapes, but topologically they are all equivalent! : They all are single-sided . . . Also: They are all chiral: . . . -- Shown here is always the left-twisting version. It is interesting to note, that all these 5 shapes can be smoothly transformed into one another -- with an operation called a regular homotopy.

These are NOT Möbius Bands ! What you may find on the Web under “Möbius band” (1):

These are NOT Möbius Bands ! What you may find on the Web under “Möbius band” (2):

These are NOT Möbius Bands ! ISAMA 2004 These are NOT Möbius Bands ! What you may find on the Web under “Möbius band” (3): This is a Figure-8 Klein bottle. A Klein-bottle results when you fuse two Moebius bands by joining their edges.

ISAMA 2004 TWO Möbius Bands ! Two Möbius bands that eventually get fused together: 2 bands: almost aligned; at 90 degrees against one another; extended half-way to top and bottom; almost fused together at their borders.

Classical “Inverted-Sock” Klein Bottle ISAMA 2004 Classical “Inverted-Sock” Klein Bottle Yes, the Klein Bottle meets these requirements. Most of you probably are familiar with this geometry. Here -- one small ribbon on its surface has been singled out, painted in bright green and orange, and enhanced with surface-normal-arrows to show that it is indeed a single-sided, “non-orientable” Möbius band.

Topological Surface Classification ISAMA 2004 Topological Surface Classification The distinguishing characteristics: Is it two-sided, orientable – or single-sided, non-orientable? Does it have rims? – How many separate closed curves? What is its genus? – How many handles or tunnels? Is it smooth – or does it have singularities (e.g. creases)? Can we make a single-sided surface with NO rims? Mathematicians use just a few parameters to classify all possible surfaces: they ask: . . . As an example, can we make a single-side surface with NO rims?