Mobius bands and tilings

Mark Dow

Three half-twist Möbius bands and tilings

Conceptually these Möbius band (or Möbius strip) patterns are formed by connecting the ends of a rectangle after three half twists. Another common Möbius band has only one half twist. The shape shown here has a rotational symmetry with alternating direction folds that makes it easy to visualize and forms interesting tiles and tilings. It is directly related to the trefoil knot; any cut parallel to an edge results in a trefoil knot. In this sense it can be considered as a set of alligned trefoil knots.
While these patterns evoke the perception of three dimensional shapes, they are in fact two dimensional patterns and each can be constructed by rotations of single diamond elements; the pattern on the diamonds are simple piece-wise linear luminance gradients.

(left click for larger images)

[ To Do: Splitting and trefoil knots]

Mobius triptych

Triple trefoil triptych

Note that while interlinking pairs are possible in three dimensions, this triplet is not -- the bands must interpenetrate near the center.

Tiles and tilings constructed from these patterns. I particularly like this tiling.
Relationship of Möbius band topology to elementary number theory
[ To Do: How they were drawn]
[ To Do: Photos of physical models]
[ To Do: Ceramic or glass tiles?]
[ To Do: Other links]

Tiles (single chirality)	Tilings	Tiles (alternating chirality)	Tilings
mobius_band_1_tile_medium.jpg mobius_band_1_tile_small.jpg mobius_band_1_tile_tiny.jpg	mobius_band_1_tiling_medium mobius_band_1_tiling_small mobius_band_1_tiling_tiny	mobius_band_1_LR_tile_medium.jpg mobius_band_1_LR_tile_small.jpg mobius_band_1_LR_tile_tiny.jpg	mobius_band_1_LR_tiling_medium mobius_band_1_LR_tiling_small mobius_band_1_LR_tiling_tiny
mobius_band_2_tile_medium.jpg mobius_band_2_tile_small.jpg mobius_band_2_tile_tiny.jpg	mobius_band_2_tiling_medium mobius_band_2_tiling_small mobius_band_2_tiling_tiny	mobius_band_2_LR_tile_medium.jpg mobius_band_2_LR_tile_small.jpg mobius_band_2_LR_tile_tiny.jpg	mobius_band_2_LR_tiling_medium mobius_band_2_LR_tiling_small mobius_band_2_LR_tiling_tiny
mobius_band_3_tile_medium.jpg mobius_band_3_tile_small.jpg mobius_band_3_tile_tiny.jpg	mobius_band_3_tiling_medium mobius_band_3_tiling_small mobius_band_3_tiling_tiny	mobius_band_3_LR_tile_medium.jpg mobius_band_3_LR_tile_small.jpg mobius_band_3_LR_tile_tiny.jpg	mobius_band_3_LR_tiling_medium mobius_band_3_LR_tiling_small mobius_band_3_LR_tiling_tiny
mobius_band_4_tile_medium.jpg mobius_band_4_tile_small.jpg mobius_band_4_tile_tiny.jpg	mobius_band_4_tiling_medium mobius_band_4_tiling_small mobius_band_4_tiling_tiny	mobius_band_4_LR_tile_medium.jpg mobius_band_4_LR_tile_small.jpg mobius_band_4_LR_tile_tiny.jpg	mobius_band_4_LR_tiling_medium mobius_band_4_LR_tiling_small mobius_band_4_LR_tiling_tiny
Triple_trefoil_1_tile_medium.jpg Triple_trefoil_1_tile_small.jpg Triple_trefoil_1_tile_tiny.jpg	Triple_trefoil_1_tiling_medium Triple_trefoil_1_tiling_small Triple_trefoil_1_tiling_tiny

Relationship of Möbius band topology to elementary number theory

    [Preface/Rant: The Möbius band's interesting topological properties, particularly its single side / single edge nature and the shapes that are formed when cut in various ways (see Möbius band for examples), are often presented as mystical. This is particularly true in physical demonstrations. While Möbius bands have applications in magic, their association with mysticism is a bit irritating to me personally. While they are not part of most everyday experience, they are quite simple shapes (even young kids understand on a gut level the basic topology of a loop of paper with a twist) and it is this simplicity that gives them such fundamental beauty. And while number theory (relationships between integers) and topology (shapes) are not an explicit part of elementary math cirricula (or any cirriicula), they should be.]

    Without much detail, the topology of the Möbius bands on this page can be described by the number and direction (sign) of folds as you circumnavigate the mobius loop. As you complete one full circle (in the plane of the page), three opposite 1/2 rotations (folds) are encountered. This implies that an ant who travels that path will end up on the opposite side of the band after one orbit of the center:

[ To Do: diagram ]
        Θ_k = (-1)^k, k = 3, where the sign of Θ indicates the side of the sheet relative to the page and k is the number of folds that you've passed through.
        Θ₃ = -1

    After two full orbitss an ant will arrive at the original side of the band near where it started:

[ To Do: diagram ]
        Θ_k = (-1)^k, k = 2*3 = 6
        Θ₆ = 1

    You can check the logic and math for a loop without a twist (not a Möbius band): it requires an even number of folds to project it onto a plane resulting in a return to the same side after one, or any number of circumnavigations:

[ To Do: diagram ]
        Θ_k = (-1)^k,   k modulo 2 = 0 and (n*k) modulo 2 = 0
        Θ₆ = 1

    If the pattern is split down the middle, the shape is the same, but in effect the ant will have to start off center. It will again require two full circlings to arrive at the original point, but this time will not be able to visit the opposite side (it can't "cross the center/cut line", where the cut was made). [ To Do: diagram ]

    Now consider how many times the these ant paths must cross in the plane of the page, keeping track of whether the path passes under or over a different portion of the path. [ To Do: diagram ] This is a metric used in knot theory, a branch of topology, used to decide if two shapes are topologically equivalent (can be smoothly deformed into one another). [ To Do: Show that a cut always results in a trefoil, and a cut in a trefoil always results in a trefoil. ]

    So one way of thinking about this form of Möbius band is that it is a flattened trefoil knot, with an edge glued together (the "cut" is undone). [ To Do: diagram ] Or, equivalently, you can think about trefoil knots as this form of mobius band split down the middle, or any non-intersecting, connected section of the mobius band.

How these Möbius bands can be drawn

I used Paint Shop Pro, version 5.00 to draw the "bands" on this page. I also tried to do it with Adobe Phoshop Elements, and I'm sure it's possible, but I got hopelessly confused.

step 1: Draw a rectangle and fill with linear gradients in the horizontal direction.	step 2: Vertical perspective distortion.
step 3: Cut horizontal bands.	step 4: Adjust color.

[ To Do: Fill out this description, add 120 degree rotations.]

Mark Dow

Three half-twist Möbius bands and tilings

Tiles (single chirality)

Tilings

Tiles (alternating chirality)

Tilings

Relationship of Möbius band topology to elementary number theory

How these Möbius bands can be drawn