The Square Knot and the Granny Knot An Analogy for Diastereomers Dirk Tavernier State University of Gent, Krijgslaan 281 (S4bis), B-9000 Gent, Belgium The following statement is easily rationalized.
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For a c h i d ahied. " ."ioinine either of it4 two enantiomornhs ~
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(e.g.,A-) witheach of the twornanriornorphs ofanorher rhirni object ( e g , H* and H-I gcncrates t w o anlsnrnetne rr.r., nerther suprrposable nor enantiumorphic forms A'H* nnd A T - .
Choose a plane through A+and B+;reflect B+across that plane to produce B-. The distances between any point on A+ and any two mirror-related points (one from B+,the other from B-) will be different. Thus, A+B+and A+B- are anisometric. Stereochemistry The above is the basis of organic stereochemistry, as established by van't Hoff. A tetrahedral carbon bonded to four distinguishable groups constitutes a chiral object; n chiral atoms in a molecule @ve rise to 2 " " chiral diastereomers. These are anisometric species with the same constitution. At ambient temperatures the configuration about a chirat carbon atom is generally maintained, and the 2'" diastereomers can commonly be separated. Additional diastereomeric forms of each of the above defmed 2'" diastereomers may be generated by rotation around chemical bonds. These internal rotations are generally fast, and the diastereomeric forms thus generated cannot be separated at room temperature. Diastereomers due to the presence of two or more chiral carbons in one molecule may be designated by systematic conventions such as the threwerythro convention, the R,S system, and derived terminology (e.g., l,n) or by trivial names (e.g., glucose, galactose).
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Chirality in Everyday Objects The manufacturing industries produce many chiral objects: scissors, nuts, bolts, and automobiles:
two diastereomorphic knots: the square knot and the granny knot (Fig. 2). Chirality and Diastereomerism in Mathematical Knots The figures show mathematical knots, that is, three dimensional nonintersecting closed curves. Physically, an intertwined rope with spliced ends is an appropriate model. The circle is the trivial or "unknotted" knot. The Cloverleaf Knot The simplest knot, called a cloverleaf knot, cannot have less than three crossings of the rope. It may assume, by pulling and pushing of the rope, manifold forms. These forms are generally asymmetric, but there are also three special types, which are distinguished by their symmetry: D3,D2, and C2. A specimen of each is shown in Figure 1. The cloverleaf-like form with Ds symmetry has obviously given its name to the knot. TheD2 form, especially when unspliced, is widely known as the overhand knot. Every form of the cloverleaf knot is chiral. The configuration of rope about the chiral center cannot be inverted by oulline and oushine the roue. This would reauire untvine the knlot and retying it in the opposite confi&ation. c ~ h ; verv first helical turn of the roDe determines the confieuratioi of the knot.)
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The Granny Knot and the Square Knot Tying two overhand knots in a rope in the same configuration about the chiral center generates the granny knot, whereas two overhands of opposite configuration generate the square knot. Chemists would designate the square knot as the meso form of the diasterwmeric pair.
P a i r s of enantiomorphs (pairs of shoes) Enantiomorphs (left-steered cars in nght-keeping trafic, and vice versa) 'Homochiral forms (right-threadedscrews, which vastly outnumber left-threaded ones) Few of the diastereomorphs generated by joining two man-made chiral objects have different names. In fad, the present author is aware of just one example. It concerns I I
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Figure 1. Three specimens of the cloverleaf knot, each of different symmetry. They have the same configuration about the chiral center (left-handedloops).
Figure 2. (a)Two specimens of different symmetry of the granny knot and of the square knot. (b) Threo- and erythre2,3-difluombutane. The threo isomer has a C2 axis in any conformation,whereas the erythro isomer has a plane of symmetry only in the conformation shown. Volume 69 Number 8 August 1992
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Symmetry Elements The granny knot can have rotational symmetry only, but the square knot may also have rotation-reflection symmetry. Figure 2 shows the most symmetrical form of each knot. The knots belong to two different point groups. The granny knot belongs to the Dzgroup with three perpendicular Czaxes. The square knot belongs to the C Zp~u p with a Cz axis, a plane of symmetry, and an inversion center. The granny knot and the square knot may have forms of lower symmetry, Cz and C,, respectively (Fig. 2). Absence of Symmetry Of course there are also forms devoid of symmetry. A comparison with the symmetries of the threo and erythro isomers generated by joining two homomorphic and two enantiomorphic chiral carbon atoms is instructive. Consider 2,3-difluombutane as an example. The general conformation of erythro-2,3-difluorobntanehas no symmetry. However, there is a special eclipsed wnformation that does have a plane of symmetry, and there is a special staggered conformation that has an inversion center. Unlike the square knot, erythro-2,3-diiluombutanecannot have a Cz axis due to the symmetry of the respective building
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blocks: The chiral overhand knot may itself have a Cz axis, whereas the chiral carbon atom is truly asymmetric. More Analogies between Molecules and Knots in Rope Like chemical diastereomers, the granny knot and the square knot have different properties. For instance, the taut granny knot jams, that is, it cannot be untied easily. However, the square knot does not jam, making it the more useful knot. For several years now in this laboratory, we have used the analogy that compares the granny knot and the square knot to the threo and erythm forms. It has been well-received by our students. The diastereomeric pair of the granny knot and the square knot is a macroscopic illustration of the geometric principle underlying van't Hoffs rule. Arope, like a carbon chain, has flexibility. Long-lived chiral elements may be added to the rope by tying a knot, and to the paraffinic chain by effecting a substitution. However, the chiral carbon atom has a b e d position in the chain, whereas the overhand knot may move back and forth along the rope. This similarity, and yet dissimilarity, between knots and chiral atoms help the student to distinguish clearly between configuratio& and conformations.
