Chapter 10 Chapter 6 Mobius Molecules

Mathematicians confided, The Mobius strip has only one side, If you were to split it in half, You will feel very ridiculous, Because it is still a belt after separation. --anonymous Mathematics can aid in shape design not only on the grandest scale, like a 3.5-story Easter egg, but also on the tiny scale.This chapter tells the story of how David Wolba and his colleagues at the University of Colorado, Boulder, synthesized molecules in strange Mobius strips. The mysterious Mobius strip is a pet of mathematicians.You can make a Mobius strip with a narrow strip of paper, for example, take a strip of paper for an adding machine, twist it halfway, and then connect the two ends of the strip to form a closed loop, which becomes a Mobius strip.

The Mobius strip has only one side and only one side.If you take a paintbrush and paint in the direction of the tape, you will find that when the paintbrush returns to the starting point, it has painted the entire surface of the tape.If you make a kind of magic mark along one side of the tape, you will also immediately believe that the tape has only one side. If you cut the Mobius strip in half along the direction of the tape, sure enough, it's still a strip, as five dozen lines say. In 1858, a scientific society in Paris, France awarded a prize for the best paper in mathematics.In a paper submitted for the competition, mathematician August Ferdinand Mobius of the city of Leipzig, Germany, "discovered" the surface that now bears his name.Mobius only discussed his discoveries from purely mathematical point of view, for example, he did not discuss the possibility of Mobius band molecules existing in nature.

Indeed, Mobius would not have thought of the possibility of molecules such as Mobius belts, because the science of organic chemistry was still in its infancy at that time, and people knew nothing about even the simplest molecular shapes, let alone complex molecules that are mathematically meaningful.At the same time as Mobius' discovery, August Kekuler of the University of Bonn in Germany announced his discovery that carbon atoms could be linked to form long chains, which would become the basis of organic chemistry. Four years ago, Kekule first thought about the carbon chain in fantasy on a stagecoach in London.He recalled: "It was a clear summer night, I took the last stage coach home, sat in the 'roof' seat as usual, through the streets of big cities without pedestrians, in ordinary times, it was full of people. Vibrant city. I fell into fantasy, and seemed to see many atoms dancing before my eyes... I often saw how two smaller atoms united to form even atoms, how 1 larger atom surrounded two smaller ones Atoms; and how the larger atoms hold on to 3 or even 4 smaller atoms, and at the same time, how they all whirl around in a dizzy dance. I also see how the larger atoms form chains... In any case, I am going to spend some of my nights writing down the outlines of these fantasies in my thesis."

Eleven years later, in 1865, Kekule realized that a chain of carbon could rotate around it, forming a ring.And dreams inspired him once again. "I sat writing my textbook, but my work was going nowhere, and my mind wandered. I turned my chair toward the warming fireplace and dozed off. Atoms danced before my eyes again. Smaller atoms cautiously Staying on the substrate. My mental eye, sharpened by this repetitive sight, can now discern larger structures in many forms, arranged in long rows, sometimes more closely stitched together; the whole row twists and turns Moving like a snake. Look! What is that? A snake bit its own tail and whirled mockingly before my eyes like a bolt of lightning, waking me up... That night I deduced the hypothetical in conclusion."

First, Kekulé deduced the structure of benzene, which consists of 6 carbon atoms and 6 hydrogen atoms.Kekulé concluded that 6 carbon atoms form a hexagon with a hydrogen atom attached to each carbon atom. In the 120 years since Kekulé identified the shape of benzene, organic chemists have of course discovered more complex molecular shapes, such as the double helix DNA molecule.But only in recent years have chemists observed molecules shaped like Mobius strips. Mobius molecules are not found in nature, but were synthesized in the laboratory by David Wolba and colleagues.He started by synthesizing molecules shaped like a three-run ladder. (Each rung of the ladder is actually a carbon-carbon double bond, which can be ignored here.) Then bend the ladder around and join the ends so that it actually forms a ring.

One half of the annulus is just a circular strip, while in the other half, when its two ends are joined, the half is twisted, thus forming a Mobius strip. Like the Mobius paper strip, Mobius strip molecules have many mysterious properties.If all three carbon double bonds are broken, the molecule is still a single molecule.The breaking of the carbon double bond is equivalent to dividing the Mobius strip into two halves along the midline of the paper strip.For both molecules and paper tapes, the result is a single tape with twice the original circumference. Chemists have known for a long time that two compounds can have the same molecular formula (that is, a compound composed of the same chemical components in strict proportions), but exist as chemical entities with different properties.This phenomenon can occur if the same chemical components bond to each other in different ways or at different angles.However, two compounds with the same molecular formula, and even the same chemical bond, can be chemically different.How could this be possible?

A subfield of mathematics called topology could explain this phenomenon.It is a mathematical discipline that studies the properties of objects that remain unchanged when they are continuously deformed.Imagine an object made of flexible rubber.Topologists want to know what properties remain the same when objects are pushed and pulled but not punctured or torn apart.The example of the Mobius strip can be used to illustrate this abstract concept.Suppose you have a rubber Mobius strip and you can stretch it in every possible way.No matter how many ways you try to deform it, the resulting shape will always be one-sided.Therefore, the one-sided property is of concern to topologists.When one shape can be continuously deformed into another shape, the two shapes are considered to be equivalent from a topological point of view, so no matter what shape the Mobius strip is stretched into, from the definition of topology That is, they are also equivalent.

Now consider two Mobius strips, one made of a rubber band twisted in one direction, and the other made of a rubber band twisted in the opposite direction. Topologically, are these two Mobius strips equivalent?They are not equivalent.Neither can morph into another shape.If you look at one of these two bands in a mirror, then you will see that its reflection is much like the other band; the bands are mirror images of each other. Here I must pause to issue a denial to avoid malicious letters from mathematicians.Mathematicians are a bunch of weirdos, and topologists don't confine themselves to three-dimensional space.But in the four-dimensional space, they can prove that the Mobius strips in the mirror can transform into each other.However, I will still insist on restricting our discussion to three dimensions, since the shape of molecules, the main object of our investigation, is always observed in three dimensions.So I would reiterate that in 3D the mirrored Mobius strips are topologically distinct.

The key to why two chemical compounds with the same composition and the same chemical bonds have completely different entities is that there may be completely different mirror images from a topological point of view. Because the right and left hands are both well-known mirror images, it is customary for people to refer to objects that are opposite to their mirror images as left-handed or right-handed.Which of a pair of mirror images is called an image is a matter of habit.This is just like there is no absolute position on the right side of the street, it depends on the direction you walk.Two kinds of Mobius belts have been called right-handed and left-handed Mobius belts, but there is no need to worry about which is right-handed and which is left-handed.Molecules also exist in right-handed and left-handed forms, which are called chirality, borrowed from the Greek word for "hand (Cheir)".

Both right-handed and left-handed Mobius strips are examples of mirror-image shapes that are topologically distinct but have equivalent mirror-image shapes.Now take a simple figure as an example. A circle is the mirror image of itself. Obviously, from the topological point of view, the circle is equivalent to itself. Another example is the letter R.If the figure R is made of soft rubber, it can be transformed into its mirror image by means of topological deformation. However, the molecules are not made of soft rubber, and physical constraints prevent them from deforming in any way.Still, the R-shaped molecule can transform into its mirror image without bending—indeed, without bending at all.This time, if you put the letter R and its mirror image in hard plastic on the table, you can turn one into the other just by picking it up and flipping it over.

This kind of transformation is called rigid transformation because the object always maintains its rigidity. Many organic molecules are rigid chiral molecules: it is quite different in rigidity from its mirror image.The human body clearly prefers chiral molecules of a certain chirality.For example, most proteins are composed of L-amino acids and dextrose.When chiral molecules are synthesized in the human body, only chiral molecules with the desired chirality can be produced. But when chiral molecules, such as drugs, are synthesized in the laboratory using non-biological methods, the result is a 50-50 mixture of right-handed and left-handed molecules.When a patient takes a drug, it is difficult to get rid of molecules that are not in the desired form, so they are given a mixture.In general, molecules in undesired forms are biologically inert and simply pass through the body without any effect.Still sometimes harmful. In the early 1960s, it happened to pregnant women Events of taking Zaridomide drug.The right-handed molecule in the drug has the required sedative properties, while the left-handed molecule can cause deformities in newborns. Stephen Mason, a professor of chemistry at the Royal College of London, in an article published in the British weekly "New Scientist", noticed that of the 486 synthetically produced chiral drugs included in the income standard drug handbook, only 88 were composed of the required chiral molecules of.The remaining 398 are all half-and-half mixtures.Mason concluded: "They are all used in a specific environment (the human body), and a certain chirality will be favored. But what will the effect be?" When an organic chemist analyzes a new molecule, the first thing he does is try to determine whether the molecule is rigid, chiral, that is, distinct in rigidity from its mirror image.Topology can be used here.Topologically, if a molecule is different in nature from its mirror image, then they are also different in rigidity, since a rigid transformation can only be one of many transformations that can be done topologically.Also take R and its mirror image discussed above as an example.In morphing from one to the other, an intermediate shape is obtained, which is symmetrical in that its left half is the mirror image of its right half. Topologists know that if a shape can be deformed into something with reflection symmetry, then that shape can itself be deformed into its mirror image.This means that if a chemist can make a molecule acquire a shape with reflection symmetry, he can remove the chirality of the molecule. This insight often proves useful.Wolba had already synthesized Mobius strips of molecules from three-level ladders, and he invited me to directly observe a similar synthesis from two-level ladders.Is the resulting shape chiral?As shown in the figure below, it is not chiral since it can transform into a shape with reflection symmetry. Unfortunately, this explanation does not seem to work for tertiary Mobius molecules.After many thought experiments, Wolba speculated that it seemed impossible for it to deform into a shape with reflection symmetry.If the deformation already exhibits reflection symmetry, then he concludes that the third-order Mobius shape can be deformed into its mirror image.However, is this backstory correct?Does any deformation that fails to show reflection symmetry mean that the molecule itself cannot be deformed into its mirror image? The problem is that the answer is too easy.Wolba asked me to consider two rubber gloves, one for the right hand and one for the left. Gloves are obviously mirror images, but topologically, are they equivalent?Of course, the gloves are not equivalent in rigidity, because if we flipped one of the two gloves to get a mirror image like we flipped the letter R, that wouldn't work.However, if we turn either glove inside out, then we can make the gloves equivalent. (The topologist thus finds himself in the peculiar position of not being able to recognize the glove as being either right-handed or left-handed.) At no step in the process of turning the glove inside out does the glove have reflection symmetry. We might be able to conclude that the glove is a counterexample: a shape that is topologically equivalent to its mirror image but does not have reflection symmetry during its deformation.This conclusion may be wrong.It's just that we didn't deform the glove enough.If we pull the glove hard, we can, at least in theory, deform the glove into the shape of a disk, and then the glove has reflection symmetry (reflection symmetry along any diameter). The point of the above discussion is that some of Wolba's studies in chemistry have raised an important question for topologists: if it is impossible for a certain shape to have reflection symmetry during deformation, can it be concluded that, from topological Apparently, the shape itself is not equivalent to its mirror image?This is a fundamental question, but no one seems to have raised it in the mathematical literature. The whole question ties into an important philosophical question: Do new concepts in the physical sciences often inspire new concepts in mathematics?Or vice versa?In other words, which came first, the physical sciences, or mathematics?Many philosophers have encountered this question, and it is the same as the well-known question about which came first, the chicken and the egg, and the answer seems unsatisfactory. In both cases the conclusion one draws seems not to be an irrefutable proof, but a purposeful experiment.Some domineering mathematicians, following in Plato's footsteps, asserted that their discipline was divorced from the reality of physics.They argue that numbers exist even when there are no objects to count.Less stubborn mathematicians admit that science and mathematics are closely related, but they insist that mathematics come first.As evidence, they point to group theory, a branch of mathematics born in the 1830s that has absolutely no use in physics and has only recently been used by particle physicists to study Discovered set of subatomic particles. Physicists, however, believed their discipline came first, and believed that history was on their side.For example, Isaac Newton created the famous branch of mathematics, calculus, because he needed a mathematical tool at that time to analyze extremely small space and time intervals.And I think that the only just conclusion is that both mathematics and science complement each other, although such a judgment is neither inspiring nor informative.The story of the Mobius strip is a good example of the intricate and mutually reinforcing relationship between mathematics and physical science. The Mobius strip proposed in the 1858 essay competition, which only created pure mathematics, has now been developed in chemistry and has been skillfully used by chemists, raising many problems for pure mathematicians. You can take comfort in the fact that the Mobius strip is not only at the service of chemists, but also of industrialists. B． F．Goodrich has obtained the patent right of Mobius conveyor belt.In normal conveyor belts, there is more wear and tear on one side of the belt.In the Mobius conveyor belt, the stress can be distributed to "both sides", which can double its service life.