Symmetry in Automobile Tires and the Left-Right Problem W. Gregory Jackson University College (NSW),ADFA, Canberra, Australia 2600
Concepts of symmetry can be taught with the aid of a wide range of concrete materials, and in a recent article published in this Journal ( I ) ,automobilewheel trims were an addition to these. Following an interesting encounter with a tire fitting firm, the author discovered that automobile tires are also a fascinating subject (2)for the consideration of symmetry. Indeed they can be an especially convenient tool for discussing symmetry terms used i n chemistry, such as asymmetric, dissymmetric, equivalent, meso, chiral, prochiral and so on, as well as being a useful vehicle for considering symmetry principles of chiral discrimination. All modern automobile tires have at least C, rotational symmetry (n > 1)about the normal axis of the tire. Close inspection reveals that there are additional elements of symmetry in the tread, of various kinds, and overall tires can be classified into one of the following point groups: Dm,, Dnh,D n d , Dn,Cnh,Or Sn. Examples of tires conforming to some of these symmetry patterns are shown in Figure 1, and simplified (stylized) versions appear in Figure 2. Because of the symmetry about the center of the tread, be it a plane or Cz or Sz (= i) axis, tires can be considered as constructed of two halves that are either nonsuperimposable mirror images or identical (neelectinn the side walls of the tires). In some exampies halves that are nonsuperimposable mirror images are ioined so there is a horizontal plane generated; in others 'these mirror-image halves are joined &such a way as to be rotated out-of-phase by a Cz, rotation. Similarly, halves that are identical can be joined in-phase, out-of-phase by precisely C%, and even out-of-phase by a fraction of a turn less than Czn.In practice the value of n varies but is often of the order of 100. The table summarizes the symmetry elements comprising the point groups considered here, and we refer to Figure 2 for the following discussion. The treadless tire has the highest (cylindrical) symmetry, belonging to the point group D., (la). This point symmetry is retained if a central track (lb), or any number of symmetrically spaced tracks (retaining ah), for example lc, are placed about the center of the tread section. Many tires have such tracks, but note that a less symmetrical tread pattern such as that shown in i d (C,,) is not encountered in practice; indeed, there is always some form of left-right symmetry (which excludes C,,, for which C,, is a particular case), but of var-
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ious kinds as considered ahead. A single central track is retained in the diagrams shown in Figure 2, if only to assist in visualizing the symmetry elements. The order (n) of
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Essential Elements of Relevant Point Groups Point Group
Characteristic Elements C n + n C n + m v +ah Cn+nCn+md Cn+ nCn cn + oh C" + m" C"
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Figure 1. Examples of tread patterns in some commercial automobile tires, and their point group symmetry.
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through sloping tread. Finally, it i s noted that any degree of phase-lagging of t h e sloping tread in 5 (to 6 and 7)has no effect on the point symmetry-nCz axes remain, whereas, the point symmetry is systematically reduced (3-8-10) for a corresponding phase-lagging of the Vsplayed tread. The student of advanced chemistry will know that chiral molecules are devoid of any improper rotation axes (S.) which includes a (S1)and i (Sz).Asymmetric molecules have no symmetry at all (C1 point group) and are of course chiral, while many are !issymmetric; that is, they have nly rotational symmetry, be,nging either to the C, point group, or if they have nCz axes perpendicular to C, as well, they belong to the higher symmetry D point group D, (See the table). And so it is with tires. The author could not locate any asymmetric tires (other than ones that were worn non-uniformly!), nor tires with just C. symmetry, but the majority were found to belong to the chiral point group D, (examples, Fig. 1). Herein lies an intriguing problem. All these tires have the property of nonsuperimposable mirror images. This handedness, much like that of a glove (leR- or righbhanded), raises the interesting question as to whether both kinds of tire are available for fitting to automobiles. The author's automobile was recently refitted with new tires, and on close inspection it was noted that the leR and right tires both slanted the same way (Fig. 3A). In the belief that the configuration should have been meso (Figs. 3C or 3D) rather than active, the author returned to the tire dealer. The tires were not "unidirectional" was t h e response. The apparent nondirectionality of the tires was a redherring it transpired, but it did lead to the consideration of a related symmetry problem (see ahead). The tires were of Dloz symmetry (Fig. I), and reversing them on their rims or swapping them IeRright made no difference (because of the CZsymmetry). Less obvious was the fact that the combination left wheel plus tire was different to right wheel plus tire. However, in isolation all four tires were identical and hence swapping made no difference.Asearch was mounted among local tire stocks for a nonsuperimposable mirror image (enantiomeric) tire, but to no avail. All were identical "leRhandedm tires (the tread can be considered to form part of a lefthanded helix, as viewed from either end of the major axis of rotation). Apparently when the first dissymmetric tire mold was made, it had been exactly replicated thereafter. There is a good analogy here to many biologically import-
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Figure 2. Symmetry analysis of tire tread patterns. the symmetry rotational axes along the normal axis of rotation of the tire can be made finite by the introduction of the simplest form of tread, parallel sections as shown in 2. The point symmetry is thus reduced to Dnh.The point symmetry is further reduced if the left and right sides of the tread become out-of-phase,removing the 01,(4, D d ; if this phase-lag is not symmetric, the a" are also removed and the symmetry is lowered to D,, 9. If the tread is oblique, as in 5 (D.), all a are removed. A splayed V-shaped tread retains the ah but the a, and now the nCz are gone, leaving Cd point symmetry, 3. If now the leR and right are symmetrically out-of-phase, this leads to Sn symmetry, 8, while if the phase-lag is less uniform the symmetry is C,, 10. The S. and C, symmetry patterns also can be considered as derived from the D d and D, patterns with the appropriate elements of symmetry removed (or added)
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Figure 4. Clockwise- and anticlockwise-rotating plain tires that can be superimposed (A), and the same tires (now plus axles)that are mirror images and that cannot be superimposed (0).
Figure 3. Two 'an ve' (A, 0) ana two 'meso' (0.C ) a~romobie-pl~st re contgurations, snow ng the mlrror-pane symmetv for the laner and the ack of thls pane symmetry for tne former. ant molecules for which L-amino acids are the classic example. It is of further interest to note that closely similar tires can be found in left- and right-handed forms (Fig. I), but never both forms for the same brand and model tire. Svmmetrv armments are freauentlv used in discussing chiial inte;act$ns and asymm&c .iynthesis and these can he illustrated hv rntating tires. A rotatin~plni~r tire, of itself, does not have a handedness. One c&-be superimposed on the other by the appropriate Czrotation (Fig. 4). In chemical terms it can be said to be prochiral, because the introduction of an axle distinguishes the sides, rendering the rotating tire plus axle combination chiral. Thus, while the tire is rotating, clockwise or anticlockwise, the two configurations are different (enantiomeric). (There will be a leftxight wheel balance from this source of chirality because obviously the left wheel rotates in the opposite sense to the right wheel). Now what if dissymmetric tires are introduced? The orinciole to aoolv is that chiralitv is distinguished only by'chiraiity. This, ehiral tire rotaring in a particular sense should provide a different interaction to its mirror image tire rotating in the same sense (we consider "interaction" ahead). I t is useful for our purposes to consider the action of the tread on a wet road in removing water from under the wheel. In one situation the sloping tread contacts the ground first on the outside, and in the other. it contacts first on the inside. These are not enantiome& situations; in one case water is thrown from under
a
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the tire more to the left andawav from the automobile. and for the other tire water is throw; more to the left under the automobile. It is clear from Firmres .. 3A and 3B that the comhination automobile plus tires does not have a plane of symmetry bisecting the assembly lengthwise, and i t is in the motlon of the automohile, either furward ur backward, that thls lefl-right dilference is manifested. The sense of-rotation has another interesting connotation in as far as so-called "unidirectional" tires are concerned. ks the name implies, such tires are designed to rotate in one sense only, and this sense depends upon which of two ways they are fitted to the wheel. A simple tractor tire illustrates the point (3, Fig. 2). The tire itself has the usual rotational symmetry about its axis and as well a plane of symmetry (crh). It does not have planes of symmetry perpendicular to the plane of the tire (o,), and thus which way the tire goes on the wheel rim is different. In the symmetry jargon-of a chemist, it is a prochiral tire which on fitting becomes chiral. This fact, coupled with a sense of rotation, leads to two distinctly different rotational modes (nonenantiomeric). This can be seen clearly by noting that, on a clockwise rotation the point of the Vtread meets the ground first, whereas for the opposite sense of rotation the back of the Vstrikes the ground first; the former obviously provides the tractor with more bite. The symmetry criterion for strictly unidirectional tires is that they lack plane symmetry perpendicular to the plane of the tire (cr,,). Interestingly, almost all commercial automobile tires are in this category, including our dissymmetric ones. The "conventional" unidirectional tire like the tractor tire involves a front-back bias while in motion. while the dissymmetric tires have a left-right bias. The former can be out on the left or right side. but it must be fixed in a partic;lar way on the r i g , while'the latter should go specifically on the left or right but either way on the rim. Acknowledgment The author is grateful to a reviewer who drew attention to the existence of reference 2 which considers two-dimensional tire tread patterns from the perspective of a mathematician. Interestingly, the same issue (3) depicts the wheels trim patterns considered in reference I . Literature Cited 1. Gsllian, J. A. J Cham. Educ 1990, 67, 549. 2. Montgomery,0. Mathematics k h i n g l U K 1 , Dee. 1984,1617 3. Walter, M. Mothmtics T~oehtngIUKl, Dec. 1984, 11.
