Chemistry and Möbius Strips have been banding together for at least several decades now. Could flexagons possibly help visualize the behavior in molecules? Maybe by using VELCRO, magnets or other sorts of means to easily disconnect and reconnect the flexagon at various points, the actual or theoretical processes of bonding and separation could be represented.
To me, of extreme interest are the infinite possibilities derived from making chains of one-half-twist trihexaflexagons. Or one could link other types of flexagons and non-flexagon rings when comfortable manipulation is possible. An opened flexagon band could be inserted in the other flexagon or non-flexagon ring. Then, the ends of the band could be connected to make a chain link.
Chains of any length could be made. These physical theoretical models would help tremendously. The designing of new molecules and compounds along with explaining the structure of existing ones may be simplified.
Following is a picture simulating three linked trihexaflexagons where each trihexaflexagon has the same three faces. The flexagons have been flexed in order to display each different face. The Blue, Orange, and Pink borders indicate the three separate flexagons. Since these are trihexaflexagons with made using only one 1/2 twist, they are easily manipulated.
This demonstration should help you visualize what the
possibilities could be when modeling existing or imagined linked compounds that may have a
Möbius band type of structure.
If you want play around with generating linked
Möbius band type molecules or even just designing your own trihexaflexagons, you can download some Flexagon Blank Templates
A Few References
Topology and Möbius Strip
Möbius Strip Topology
Boch, E. D. "A First Course in Geometric Topology and Differential Geometry. Boston, MA: Birkhauser. This is probably what gave me a much clearer idea of what a paper
Möbius strip would be without any thickness.
Fauvel, J. Flood, R, and Wilson, R. (eds.)(1993). Möbius and His Band. New York, NY: Oxford University Press.
Poppos, T. (1991). More Joy of Mathematics. San Carlos, CA: Wide World Publishing/Tetra.
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