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Let us step back and consider another interesting analogy that can be established between the double Möbius strip, its principle of unity through perpetual duality, and the DNA (Watson, Crick and Wilkins): the model of the double helix is composed of two serpent-like intertwining spirals, representing a biological reflection of the archetypal idea of time as a spiral, creating a reunion of the linear and the cyclical aspects of time as a perpetual flow. [...]
How a closed, circular bilateral ribbon can be transformed, without cutting it or creasing it, into a unilateral ribbon (or "mobius strip")?
If we were sitting face-to-face I could show you how to manipulate the bilateral ribbon so as to solve the puzzle. But it is difficult to describe that process in words, or even with a diagram. So I will do the next best thing. I will show you, by working backwards from the solution to the problem, how a unilateral ribbon (a "mobius strip") can be DISASSEMBLED so as to create a closed circular bilateral ribbon. This will literally put the answer in the palm of your hand. All you will have to do is apply the process in reverse order!
First you must make a simple (first-order) mobius strip. You'll have to once more get out a pair of scissors and a sheet of paper. This time, however, draw a rectangle that is 10 inches long and 1/4 inch wide. It is important that you use these measurements, as you will momentarily see. Now, holding one end of the ribbon stationary, twist it 180 degrees clockwise along is central axis, bring the ends of the ribbon together, and scotch tape it. If you do this correctly, you should wind up with a mobius strip -- a figure like the bottom one in the picture above. Now make a small hole in the mobius strip somewhere along its central axis (the line, in the picture, that travels lenghtwise up the middle). Cut along the central axis all the way around the strip. What kind of figure do you have now, after making the very last snip? Unless you've made a mistake with the scissors, you should have another closed circular ribbon in you hand -- although the 'circle' that it makes will be bigger (twice as big) and the ribbon will thinner (half as thin). And -- low and behold -- it is a BILATERAL ribbon. It should indeed be exactly like the one had in hand when trying to solve the puzzle if you previously constructed accordingly -- by twisting a strip 720 degrees. This is the figure that you were supposed to 'transform, without cutting it or creasing it, into a unilateral ribbon'! Only now you see that it IS possible - and precisely HOW it is possible. The solution, in other words, involves wrapping the bilateral ribbon around on itself, EDGE TO EDGE, as you have just seen!
How what happens when one splits unilateral ribbons is different from what happens when one splits bilateral ribbons? And also how there is a hidden similarity? One way in which DNA -- which, as everyone knows, takes the form of a double-helix (which looks like a twisted ladder) -- can replicate is by splitting the double helix up the middle into two pieces. Cut the rungs of the 'ladder', and the posts fall apart. But if the DNA is in the form of a closed circle, something interesting happens. When it is split down the middle, it will fall -- unlike our BILATERAL ribbon did -- into TWO separate closed circular ribbons.
You can see how that works by making a bilateral ribbon (with one "full" twist of 360 degrees). If you cut it up the middle, the two bilateral pieces it falls into will be linked like the two links of a paper chain -- with one "cross over." Furthermore, as the number of full twists in the bilateral ribbon that you start with increases, the more times the two resulting ribbons will cross over each other. When you split a bilateral ribbon with 7 full twists, you get 7 "crossovers" in the two offspring ribbons -- which will look like the figure above (the small circular figure in the diagram).
When I was playing with this I happened to be see one of Escher's paintings of a mobius
strip -- the one above. It is a "second-order" mobius strip (one with three half-turns, twisted a total of 540 degrees). The curious split up the middle remindeds you of the discussion about the DNA. But the DNA book had not dealt with splitting UNILATERAL ribbons, such as this. Did the same thing happen when one splits a unilateral strip? To find out, make a simple mobius strip (with only one 180 degree half-turn -- like the one you made above, and the one in the top picture). You will be surprised to see that it does not result in two separate closed ribbons, as in the case of split bilateral ribbons, but in one only!
Would the same thing happen if the figure that was being split was a higher order mobius strip (with 3 half turns, or 5 turns, or "n" number of turns -- where "n" is any odd number)? Make a mobius strip with three half
Make mobius strips of even higher orders, and find out that the greater the number of twists in the mobius strip, the more complex will be the knot in the resulting figure, when the mobius strip is bisected! You will begin to wonder, Is there any order to be seen in what happens as one increases the number of twists? And is there a formula that ties together what happens in the two seemingly dissimilar cases -- when unilateral ribbons and bilateral ribbons are split? As the reader who has tried to solve the original puzzle by making such a figure will have realized, these deceptively simple forms are actually rather complex. When the number of twists in the ribbon are increased, and the knots made become more complicated, the sheer complexity of the arrangements tends to defy understanding. But you will be able to come up with a fairly simple rule with respect to splitting, which seems to link the two classes of ribbon -- for at least the first 6 levels of complexity. In the following chart, "parent" means the original ribbon (before slitting up the middle), and "offspring" means the ribbon or ribbons that are created when the parent ribbon is split. The DJ number (short for "DinkelJack"), a constant associated with each case, is calculated by multiplying the respective number in the second column with the respective number in the third column:
What this chart shows is that the "DJ number" is the same for all ribbons whose number of full twists rounds upward to the same integer. And the DJ number seems to increase by increments of 2 as one increases the number of twists in the original ribbon. I don't know if these regularities remain for cases in which the ribbons have full twists greater than 3. In any case, the formula seems to also make some intuitive sense -- for when you cut the unilateral ribbon up the middle you get only one ribbon. But since it is twice as long as each of the two that you get when you cut the bilateral ribbon, there will be twice as many twists in it.
Psycho-Physical Isomorphism
The puzzle demonstrates that one can look at the Escher painting above in a new way. Not only can one see it as a mobius strip (a unilateral ribbon), it can also be viewed as a "supercoiled" bilateral ribbon -- achieved by winding the two-sided ribbon edge to edge with itself! Why is this of significance? Because if ccDNA (which are bilateral ribbons) can be coiled in this way, we'd have an example of a naturally-occuring biological structure which can transform itself from a "one-sided" (i.e., "paradoxical") figure into a "two-sided" ("non-paradoxical") figure by splitting -- and, conversely -- from a "two-sided" figure into a "one-sided" figure, by coiling. Such a structure may thought of as isomorphic with consciousness - which has a similar capacity, by virtue of its liminocentric structure, to be both paradoxical (at its 'extremess) and linear (under ordinary conditions).
A number of theoreticians, in their attempt to understand under what physical conditions consciousness comes into being, have sought to find a physical structure with which it is isomorphic. And the puzzle begins to suggest one way in which the DNA molecule might possibly be viewed as such a structure. Elsewhere in this issue we discuss the work of two men -- Herbert Read and Douglas Hofstadter -- who seek physical structures that demonstrate isomorphism with mental structures. And it is ultimately paradoxical mental structures on which they are focusing attention in their searches. Read points to the mandala, and Hofstadter to a kind of strange looping that takes place in the mental realm. It is Hofstadter's belief that the strange loops into which the mind is capable of twisting itself will "eventually turn out to be at the core of AI [artificial intelligence studies] and the focus of all attempts to understand how human minds work." So when seeking physical structures isomorphic to this one must look, he argues, for physical structures that are somehow themselves "paradoxical." He suggests several places to look --
- in the 'looping back between informational levels' that takes place in DNA;
- in the way in which viral DNA may use a suicidal "trojan horse" strategy to avoid detection and convince its host to attack itself -- a topic that has, ironically, become a major interest since the advent of AIDS, which occured many years after Hofstadter's book was published;
- and in the presence of a 'neural substrate' that would be the physical equivalent of the riddle proposed by Epimenides, the socalled "liar's paradox" -- a sentence which states, "This sentence is false."
To Hostadter's list of proposals one might add the alternative suggested by our puzzle -- bilateral ribbons supercoiled into mobius strips.