An Answer to the Puzzle Presented in 'Losing Face'
Abstract
In Losing Face a simple puzzle was presented.
It is related to issues that impact on our discussion of personality theory (see
About Face Again). In the present paper an answer to the
puzzle is revealed, and some of the interesting consequences of this solution
are discussed. The puzzle helps us to understand, for instance, how the structure of a ccDNA molecule (closed circular DNA molecule, which is a bilateral ribbon) would theoretically permit it to be 'supercoiled' in such a fashion as to constitute a mobius strip (a unilateral ribbon). In the following it is explained why it might be this KIND of physical structure to which those who are in search of biological systems exhibiting an isomorphism with consciousness (as 'liminocentrically' conceived) might turn.
The Puzzle:
The Answer:
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 DIS-ASSEMBLED 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!
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 according
to the instructions in Losing Face - 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. 1
The solution to the puzzle, in other words, involves wrapping the bilateral
ribbon around on itself, EDGE TO EDGE, as you have just seen! When this is done,
and a unilateral ribbon is created, a single line is created by the abutting
edges, and a circle is formed. And this is why the paper
in which the puzzle was presented was subtitled 'A 3-D Puzzle for Any Mathematical Sidewinder Who Has the Stick-to-it-iveness to Remain on the Cutting Edge' - in an attempt
to subliminally hint at the process that one would use to solve the
problem.
I will speak to the possible signifance of this finding in the final section
of this paper. But first I want to discuss 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.
Section Two - DNA Science and Escher
I arrived at the puzzle
Scientists now believe that one end of the ladder can loop back on the other end, forming a closed circle, like in the figure directly below - the ladder's two posts
are shown intertwining.
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. Try it and see. When you
split a bilateral ribbon with 7 full twists, you get 7 'crossovers'
in the two offspring ribbons - which will look like the second figure
to the left (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
Would the same thing happen if the figure that was being split was a higher
order mobius strip (with three half turns, or five turns,
or 'n' number of turns - where 'n' is any odd number)? I made a mobius strip with three half
I made mobius strips of even higher orders, and found out that the greater the number of twists in the mobius strip, the more complex was the knot in the resulting figure,
when the mobius strip was bisected! I began 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 I was 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.
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' (ie, '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). (See The Structure of Consciousness). 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 that I have presented for your consideration 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.' 3 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 -
To Hostadter's list of proposals I add the alternative suggested by our puzzle - bilateral ribbons supercoiled into mobius strips. Footnotes and References 1. Even when you have succeeded, in this way, in proving to yourself that the bilateral
ribbon that you now have in your hand CAN be wrapped into a unilateral ribbon,
it is not necessarily an easy thing to do once you have made the final snip.
You might have to do as I did and, just prior to making the last cut,
draw little numbered lines horizontally across the mobius strip, showing
where to make the edges meet once the bisection is completed and the figure
falls apart.
2. The book is by Maxim D. Frank-Kamenetskii, a biotechnology professor 'best
known for his contribution in the field of DNA topology, supercoiling,
and unusual structures' - (Unraveling DNA - the Most Important Molecule
of Life, 1997, Perseus Books, Reading Mass)
3. Douglas R. Hofstadter, Godel, Escher, Bach: an Eternal Golden
Braid, 1979, (New York: Vintage Books), page 714.
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DNA has a double-helix shape that looks like a twisted ladder (left). 
A Trefoil - 
A more complex knot