Imperial College London

Henry Rzepa

The Möbius band is an experimental delight. In its original forms, it
came flat-packed as below. The one shown on the left is related to the
international symbol for recycling (if we denote the number of half
twists imparted as *m*, this one has *m*=3). The middle one (*m*=4)
shows a 4-twisted variant, and the one on the right has a 5-twist
(*m*=5). These all come from Möbius' original sketches, found amongst
his belongings when he died. In this post they will form the basis for
some experiments in molecular chirality.

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[![](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2011/02/higher_twist-1.jpg "higher_twist-1"){.size-full
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Flat-packed Möbius bands
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These Möbius bands are all chiral, which means they cannot be
superimposed upon their mirror image. We may in fact give the two forms
labels, **M** and **P** (similar to the [left and right handed helical
forms for
DNA](http://www.ch.ic.ac.uk/rzepa/blog/?p=3326){target="_blank"} noted
in a previous post). Armed with a selection of these rings, I list below
some experiments in paper cutting that you could try. I will use the
notation *M*~*m*~ or *P*~*m*~ to denote the handedness and twistedness
of each strip and **+** to denote glueing.

1.  Build two strips, each *m*=1 and glue them together orthogonally,
    then cut down the middle of each strip with a pair of scissors. The
    process is already illustrated with lots of nice photos (using
    pink-coloured strips) [on this
    blog](http://threesixty360.wordpress.com/2008/02/14/valentines-day-math/){target="_blank"}
    and the outcome below is transcluded here from the original post. If
    the strips are flat-packed first, then they will fit into an
    envelope, and all that would be needed for the recipient to complete
    the valentine is the preceding instructions (plus some glue and
    scissors)!  
    ![](http://threesixty360.files.wordpress.com/2008/02/mobius-heart-5.jpg?w=350){decoding="async"}
2.  My purpose here is to take this basic experiment and to suggest
    variations, using the following variables; the chirality ***M*** or
    ***P***, the number of twists ***m*** in each, and the total number
    of strips used ***N***. As is noted in the original instructions, a
    valentine is only produced if one *M~1~* and one *P~1~* strip is so
    joined (*N*=2), what chemists would call a heterochiral *pair*. What
    happens when a homochiral pair is used instead? The chemical term is
    that we end up with diastereomers, in other words
    *M~1[^+^]{style="font-style: normal;"}~P~1~* and
    *M~1[^+^]{style="font-style: normal;"}~M~1~* have a diastereomeric
    relationship, and *P~1[^+^]{style="font-style: normal;"}~P~1~* and
    *M~1[^+^]{style="font-style: normal;"}~M~1~* would have an
    enantiomeric relationship (as of course do *M* and *P* themselves).
3.  One repeats experiments **1** and **2** using heterocoloured strips
    rather than the same colour as above. How are the two colours
    distributed?
4.  Experiments **1-3** are then repeated for *M~2~* and *P~2~*
5.  Experiment **4** is then repeated for *M~3~* and *P~3~*
6.  One can now move on to ***N***\>2. For ***N***=3, one might
    construct a [isotactic
    polymer](http://www.ch.ic.ac.uk/rzepa/blog/?p=2300){target="_blank"}
    P*~1~*+P*~1~*+P*~1~* or a [heterotactic
    polymer](http://www.ch.ic.ac.uk/rzepa/blog/?p=2300){target="_blank"}
    *P~1~*+*M~1~*+*P~1~*.
7.  The cutting down the centre of each strip does not have direct
    chemical analogy you might think, but in fact if you relate the cut
    to the node in a *p*-atomic orbital, one can quickly move into
    Möbius conjugation and aromaticity. One might ask whether any of the
    preceding experiments might relate to the [molecular
    trefoil](http://www.ch.ic.ac.uk/rzepa/blog/?p=2084){target="_blank"}
    I described in another post? Or these [lemniscular
    octaphyrins](http://www.ch.ic.ac.uk/rzepa/blog/?p=399){target="_blank"}?
8.  There are other variations. Thus ***N***=2 dimers constructed with
    dissimilar twists as P*~1~*+P*~3~* or perhaps even P*~1~*+M*~2~.* I
    will not try to list all the permutations.
9.  If your head is not yet swimming, consider a tetramer ***N***=4, but
    now complete a second supercycle by joining the first band to the
    fourth.
10. And as a final flourish, is it possible to give the supercycle
    described in experiment **9** a twist before you join the ends
    together? Would it matter if that twist were ***M*** or ***P***?
    Would it matter if ***N*** were even or odd?  The chemical analogy
    here of course is to cyclic (and supercoiled) RNA molecules, which
    are increasingly implicated in the transition from pre-biotic to
    post-biotic chemistry.

I have to confess I have not tried the majority of the above experiments
myself! If anyone does and gets anything interesting, do tell. What I
hope I have illustrated here is how these simple experiments in
twisting, gluing and cutting simple strips of paper may actually tell us
something about molecules and their polymers and perhaps life itself.

::: {.wp_orcid_field}
[https://orcid.org/0000-0002-8635-8390](https://orcid.org/0000-0002-8635-8390){target="_blank"
rel="author"}
:::


The Möbius band is an experimental delight. In its original forms, it came flat-packed as below.

Messaggi di Rogue Scholar

Valentine chemistry