The Biography of a Dangerous Idea

[ya]

http://www.metacafe.com/watch/331665/video

The Möbius strip is a mathematical construction demonstrating an evolution of a two-dimensional plane into a three-dimensional space; by merging the inner with the outer surface, it creates a single continuously curved surface. It allows returning to the point of departure after having completed a tour by following a path along its surface. This paradox can be explained by the fact that even though the strip has only one side, each point corresponds to two sides of its surface. Another interesting situation appears when a band with joined ends is cut in half lengthwise until getting back to the starting point: a single band twice as long as the original is produced if its ends have been rotated for 180°, but rotating its ends for 360° forms two interlocking rings. Yet the Möbius strip is far more than just a mathematical abstraction.
     
Symbolically, its two-dimensional projection forming the figure eight represents infinity and cycles, but can also be found in many natural phenomena related to fluid dynamics and the analemma. The latter is known as a representation of the virtual path of the sun projected to the surface of the Earth. It reveals the dynamics of sunlight as a source of our vision and an instrument of construction of our space-time perception. Representing temporality, the cyclical nature of processes and eternity, it is no wonder that the twisted ring is an archetype, a symbol of infinity, present both in alchemistic iconography as the serpent biting its tail, named the ouroboros and in contemporary consumer society as an icon of recycling.

Eight as a Symbol. The figure eight, also used as the symbol of infinity in mathematics, to graph relations between the numbers 5 and 4, 6 and 3, 7 and 2, 8 and 1 in an asymmetrical way. Recent studies of analemma which also mimics the figure eight show that its asymmetry is a result of the sun being projected to the curved surface of the Earth, but purging this deformation produces a symmetrical figure eight. Numbers between one and eight are paired as twins in multiplication tables: in this way, number nine is the sum of numbers 5 and 4, 6 and 3, 7 and 2, 8 and 1; presented in the sigma code (the sum of digits of a number), the pairs of numbers will always sum up to 9 even if both of the twins have been multiplied. These relations can be graphed in the form of a figure eight or a Möbius strip: when such a graphed numeric orbit flows through the inflection point (number 9) in a clockwise sense, it subsequently reverses itself into its own twin partner, continuing the flow through an anticlockwise orbit. Quite unlike the stationary circles, energy is released into the numbers so that they spin, one out of the other ... the bending and twisting in and out of separate energies, the big and the small, connected by a continuous movement through the eye at the centre of the storm of numbers."



Symbolically, the eternal revolving is traditionally symbolised by the ouroboros, representing a continuous circle of creation. The circumference completes the centre to suggest the idea of God. Being a symbol of the manifestation and cycles, the ouroboros represents unity, self-nourishment, union of matter and spirit; in hermetic tradition, it symbolises the union of Isis and Osiris - the Female and the Male principle represented also by two intertwining serpents related to the Sun and the Moon; as such, it has also been extensively used in Alberti's architecture. It symbolises a dialectics of life and death, the dynamics of circle, infinite movement, universal animation and is therefore extremely interesting as a subject of research in architectural animation. The ouroboros is a creator of time, duration and life and continuously returns to itself. The alchemists' Big Whole is a cosmic spirit, a symbol of eternity and cyclic time, also used by as a symbol of Divinity. An outstanding parallel can be drawn to the Zen tradition, based on the dynamic sphere of the two opposite principles in a perpetual interaction, the Yin and the Yang. 

INTRODUCING THE DOUBLE MÖBIUS STRIP. A continuous topological entity results if the simple band with a continuous (yet still a composite) surface is duplicated and consequently transformed by applying a negative scale (x = -1, y = -1). This inversion results in a closed, continuous structure that we called the Double Möbius Strip. The same process can be applied to a simple twisted band describing a volume. The fractal geometry of the double Möbius strip becomes evident by identifying its characteristics as applied to architectural space. If the two intertwining bands of the double Möbius strip represent the structure of a wall and a floor, the two architectural articulations can become mutually entangled and exchanging while following a path along the two surfaces. Starting on a horizontal surface representing the floor results in moving along the surface of the wall become floor after having completed a tour and vice versa. It is also interesting to notice that entering the structure from a particular band always results in revolving to the starting point of the identical band. If the double Möbius structure is transparent, its users can walk along the paths of both strips representing floor-become walls-become floor and never interfere with the users starting from a different entrance placed at the other strip as the surfaces never intersect.


     
These interesting characteristics do not only indicate the non-Euclidean nature of the double Möbius strip, thus making it an ideal polygon for formal and conceptual research in post-Cartesian architecture, they also refer to a possibility of reversing and even uniting archetypal binary notions of surface/volume, space/time, inside/outside, matter/media etc. 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 (discovered by 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. The idea of unity through continuity also appears as the most important characteristic trait of structure in Lacan's lecture on Structure & Reality. If the previous Gestalt notion of good form has been related to its function of joining and producing the "unifying unity", Lacan on the other hand reconsiders good form as a "countable unity" one, two, three etc., as an integer that can be used for counting with the formula (n + 1). In this way, the question of "one more" actually becomes the key to the genesis of numbers and has been applied in our attempt to generate the double Möbius strip.
     

[terajoules]

Awesome thread!

[apeiron]

As we read here, the Möbius strip is a loop with a single twist, such that surface α becomes surface β when followed around; inside becomes outside, outside inside. A Kleinian bottle, on the other hand, may be theoretically constructed (for the attempt to make one would tear the surfaces of the material used) by placing a left-handed Möbius strip on a right-oriented one, superimposed point for point.



Just as the two-dimensional Möbius strip demonstrates the melding of two surfaces into one, so the Kleinian bottle describes this same property, but with an added dimension, the "forth" dimension. The interplay of surfaces α and β illustrates well the interrelationality of apparent dualities but the reality of the hole of the bottle disrupts the formness of the bottle; it introduces discontinuity and engages the ontological dimension of human being; that is not just another framework for reflection but a dimension that entails the prereflective depths of Being. The act of incorporating the hole engages us in an active process of making whole the threefold continuities of subject, object and space. By consciously embracing such a paradox a person may realize concretely their own incipient wholeness.

[C A P T C H A]

     
[...] 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 (discovered by 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



turns, like the one in Escher's painting above. And I cut it up the middle. What I discovered was that not only did the resulting figure remain (as in the case of the first-order mobius strip) a single closed circle, but now it also had a knot in it - a socalled 'trefoil' knot (named this way because it is a closed loop that crosses itself three times)!

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.

[C A P T C H A]

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.

[V e r a]

[...] 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 double-helix of DNA replicates by untwisting and separating its two strands, then each strand links with free available amino acids to form an exact duplicate of itself, creating a new double helix. While the linking between the bases along the helical strands, adenine, thymine, cytosine and guanine (A,T,C,G), is key-in-lock, forming AT, CG, TA or GC pairs, the overall resulting strands are exact duplicates of the original--mirror images. DNA strands are not complementary opposites; there isn't a male strand and a female strand or even a right strand and a left strand. The DNA molecule reproduces by reflection, by forming a mirror image of itself. DNA replicates, so to speak, "homosexually"

[gestalt]

     
These interesting characteristics do not only indicate the non-Euclidean nature of the double Möbius strip, thus making it an ideal polygon for formal and conceptual research in post-Cartesian architecture, they also refer to a possibility of reversing and even uniting archetypal binary notions of surface/volume, space/time, inside/outside, matter/media etc. 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 (discovered by Watson, Crick and Wilkins): the model of the double helix is composed of two serpent-like intertwining spirals

Your mentioning of the binary notion and DNA reminded me of a very interesting discovery I read about some time ago:

Calculating that 0 + 0 = 0, 1 + 0 = 1, and 0 + 1 = 1 is normally no big deal. When the calculations are done in the lab using DNA molecules, however, these elementary manipulations look considerably more interesting. Researchers at the Mount Sinai School of Medicine in New York City reported some years back that they have developed an algorithm that permits the use of single-stranded DNA reactions to add binary numbers. More impressively, they had the experimental evidence to back their scheme. Since 1994, when computer scientist Leonard M. Adleman of the University of Southern California first demonstrated the feasibility of a molecular approach to solving mathematical problems, researchers focused on finding ways to link mathematics and biochemistry to perform different kinds of computations. Their long-term hope is that DNA-based computers will eventually prove superior in speed, memory capacity, and energy efficiency over electronic computers for solving certain kinds of problems. Most research efforts have tried to take advantage of the enormous number of DNA molecules that can be packed into a small volume. Adleman, for example, solved a combinatorial problem by generating all the possible combinations as different strands of DNA, then searching for, isolating, and identifying the one strand representing the correct solution.

In contrast, Mount Sinai's Frank Guarnieri and Carter Bancroft have concentrated on developing a DNA-based addition algorithm, which demands only that the correct output be produced in response to specific inputs. Consequently, the addition operation requires a quite different model for the use of DNA in computing than that used previously for search procedures. A single strand of DNA consists of a chain of simpler molecules called bases, which protrude from a sugar-phosphate backbone. The four varieties of bases are known as adenine (A), thymine (T), guanine (G), and cytosine (C). Any strand of DNA will adhere tightly to its complementary strand, in which T substitutes for A, G for C, and vice versa. For example, a single-stranded DNA segment consisting of the base sequence TAGCC will stick to a section of another strand made up of the complementary sequence ATCGG. The links between pairs of bases are responsible for binding together two strands to form the characteristic double helix of a DNA molecule.

The researchers first assigned 3-base units to letters of the alphabet, numerals, and punctuation marks.



Adding binary numbers, represented as strings of 1s and 0s, requires keeping track of the position of each digit and of any "carries" that come up when 1 is added to 1 to give the result 10. For example, adding 11 to 01 means starting with the digits farthest to the right of each number: 1 + 1 = 10, so 0 goes in the first place from the right, and 1 is carried over to the next column. When the carried digit is added to the two digits in the second position from the right (1 + 1 + 0), the result is 10, with 0 in the second position from the right and 1 in the third position to give the final answer 100.

  1 1
+ 0 1
-------
1 0 0

Converting this procedure into manipulations of DNA molecules demands the use of DNA sequences that not only represent strings of 0s and 1s but also allow for carries and the extension of DNA strands to represent the answers. In their DNA addition algorithm, Guarnieri and Bancroft use special sequences that encode the number in a given position (0 or 1) and its position from the right. For example, the first digit in the first position is given by two DNA strands, each consisting of a short sequence representing a "position transfer operator" (which carries information to the adjacent position), a short sequence representing the value of the digit (0 or 1), and a short sequence representing a "position operator." In their Science paper, Guarnieri, Bancroft, and Makiko Fliss supply DNA representations of all possible two-digit binary integers (00, 01, 10, 11), which can then be added in pairs. Adding such a pair involves four steps, in which the appropriate complementary sequences link up and strands are successively extended to make new, longer strands, finally yielding the correct output.

The researchers term this set of steps a horizontal chain reaction. Input DNA sequences serve as successive templates for constructing an extended result strand. Like a tape recording, the final strand encodes the outcomes of successive operations, yielding the digits of the answer in the correct order. The growing strand is also an active participant in the addition algorithm because the output strand for each operation (reaction) serves as the operator (primer) for the succeeding operation. Thus, the resulting DNA strand serves both as an operator that transfers information during the addition algorithm and as a tape that records the outcome of the algorithm. What they've done with the horizontal chain reaction is to start getting DNA molecules to communicate with each other. To test their algorithm in the lab, the team combined in a test tube the DNA strands representing the two numbers to be added, along with the chemicals needed for the strand extension reactions. In this way, they successfully determined the sums 0 + 0, 0 + 1, 1 + 0, and 1 + 1 in the form of DNA strands of the appropriate molecular size. The necessary biochemical procedures took about 1 or 2 days of lab work for each calculation.

[know]

Your mentioning of the binary notion and DNA reminded me of a very interesting discovery I read about some time ago:

Calculating that 0 + 0 = 0, 1 + 0 = 1, and 0 + 1 = 1 is normally no big deal. When the calculations are done in the lab using DNA molecules, however, these elementary manipulations look considerably more interesting. Researchers at the Mount Sinai School of Medicine in New York City reported some years back that they have developed an algorithm that permits the use of single-stranded DNA reactions to add binary numbers. More impressively, they had the experimental evidence to back their scheme. Since 1994, when computer scientist Leonard M. Adleman of the University of Southern California first demonstrated the feasibility of a molecular approach to solving mathematical problems, researchers focused on finding ways to link mathematics and biochemistry to perform different kinds of computations. Their long-term hope is that DNA-based computers will eventually prove superior in speed, memory capacity, and energy efficiency over electronic computers for solving certain kinds of problems. Most research efforts have tried to take advantage of the enormous number of DNA molecules that can be packed into a small volume. Adleman, for example, solved a combinatorial problem by generating all the possible combinations as different strands of DNA, then searching for, isolating, and identifying the one strand representing the correct solution.

In contrast, Mount Sinai's Frank Guarnieri and Carter Bancroft have concentrated on developing a DNA-based addition algorithm, which demands only that the correct output be produced in response to specific inputs. Consequently, the addition operation requires a quite different model for the use of DNA in computing than that used previously for search procedures. A single strand of DNA consists of a chain of simpler molecules called bases, which protrude from a sugar-phosphate backbone. The four varieties of bases are known as adenine (A), thymine (T), guanine (G), and cytosine (C). Any strand of DNA will adhere tightly to its complementary strand, in which T substitutes for A, G for C, and vice versa. For example, a single-stranded DNA segment consisting of the base sequence TAGCC will stick to a section of another strand made up of the complementary sequence ATCGG. The links between pairs of bases are responsible for binding together two strands to form the characteristic double helix of a DNA molecule.

The researchers first assigned 3-base units to letters of the alphabet, numerals, and punctuation marks.



Adding binary numbers, represented as strings of 1s and 0s, requires keeping track of the position of each digit and of any "carries" that come up when 1 is added to 1 to give the result 10. For example, adding 11 to 01 means starting with the digits farthest to the right of each number: 1 + 1 = 10, so 0 goes in the first place from the right, and 1 is carried over to the next column. When the carried digit is added to the two digits in the second position from the right (1 + 1 + 0), the result is 10, with 0 in the second position from the right and 1 in the third position to give the final answer 100.

  1 1
+ 0 1
-------
1 0 0

Converting this procedure into manipulations of DNA molecules demands the use of DNA sequences that not only represent strings of 0s and 1s but also allow for carries and the extension of DNA strands to represent the answers. In their DNA addition algorithm, Guarnieri and Bancroft use special sequences that encode the number in a given position (0 or 1) and its position from the right. For example, the first digit in the first position is given by two DNA strands, each consisting of a short sequence representing a "position transfer operator" (which carries information to the adjacent position), a short sequence representing the value of the digit (0 or 1), and a short sequence representing a "position operator." In their Science paper, Guarnieri, Bancroft, and Makiko Fliss supply DNA representations of all possible two-digit binary integers (00, 01, 10, 11), which can then be added in pairs. Adding such a pair involves four steps, in which the appropriate complementary sequences link up and strands are successively extended to make new, longer strands, finally yielding the correct output.

The researchers term this set of steps a horizontal chain reaction. Input DNA sequences serve as successive templates for constructing an extended result strand. Like a tape recording, the final strand encodes the outcomes of successive operations, yielding the digits of the answer in the correct order. The growing strand is also an active participant in the addition algorithm because the output strand for each operation (reaction) serves as the operator (primer) for the succeeding operation. Thus, the resulting DNA strand serves both as an operator that transfers information during the addition algorithm and as a tape that records the outcome of the algorithm. What they've done with the horizontal chain reaction is to start getting DNA molecules to communicate with each other. To test their algorithm in the lab, the team combined in a test tube the DNA strands representing the two numbers to be added, along with the chemicals needed for the strand extension reactions. In this way, they successfully determined the sums 0 + 0, 0 + 1, 1 + 0, and 1 + 1 in the form of DNA strands of the appropriate molecular size. The necessary biochemical procedures took about 1 or 2 days of lab work for each calculation.

I read that the blueprint stored in DNA, an organism's genome, is, in effect, the program that describes how an organism builds itself and functions throughout its life. This information is subdivided into many discrete packages of instructions (genes). Each gene is typically associated with a particular function or trait (such as the instructions for producing the hemoglobin molecule used by red blood cells). An organism's DNA program is not read in its entirety from start to finish, but is broken down into many smaller units, each of which can be accessed as needed. An igene, like a gene, is a set of computer instructions that can be incorporated into other, more complex programs. Just as the gene for hemoglobin doesn't describe how to build an entire blood cell, an igene that describes how to calculate the sine of a number is a component that deals with a small part of a larger task. This modular approach for packaging instructions allows us to create symbols that are shorthand for an otherwise complicated set of instructions, and to combine these symbols to describe complex processes in shorthand form. The key difference between an igene and a gene is the igene contains computation instructions whereas a gene describes how to build a protein that is used by an organism.



When an alien civilization first encounters our hypothetical radio message (or we detect a similar message ourselves), all they will see is an apparently endless stream of binary numbers. At first, the message will appear to be hopelessly jumbled and devoid of meaning. If it is not formatted so that it contains clues about its structure, the party on the receiving end may never figure out how it is organized, and will never be able to get beyond receiving the signal. The first step on the path to comprehension is to figure out how the series of numbers is organized into the equivalent of words, groups of words, groups of groups of words, and so on. A good metaphor to use is the general format of the information encoded in DNA. Learning the format of DNA is not the same thing as learning the meaning of the instructions encoded in DNA. The first step is to figure out how the information encoded in a genome is broken down into smaller subunits of information. Knowing how genetic information is organized doesn't mean that we understand what every gene does, but merely that we know how to parse this data into the equivalent of words and sentences.

DNA does this by using special sequences of base pairs, the genetic equivalent of letters, to denote the end of a sequence of instructions. We will do something similar with our series of binary numbers by using special sequences of binary digits to play the role of parentheses. We'll use these parentheses to bracket other numbers and to create groups of numbers, and groups of groups of numbers, as in the following example:

(((1001)(1010)) ((1000)(1010)) ((1001)(0101))
  ((1000)(1100)) ((1001)(0101)))
 
Which can be read in decimal form as:

(((9)(10)) (( 8 )(10)) ((9)(5)) (( 8 )(12)) ((9)(5)))

This example, thanks to the parentheses, is easy to break down into words and groups of words, and groups of groups of words. While this doesn't tell us anything about what this statement means, it does tell us where to start in breaking the message down for analysis. Now compare the above example with the following statement:

10011010100010101001010110001 10010010101

This is merely the first example without the parentheses. Imagine that you had received a message like this, except that instead of a few digits, you were looking at millions or billions of digits. Where does one word end and the next begin? How could you tell whether words are combined to form groups or groups of groups? Without a clue about how the message is organized into subunits, it would be very difficult to interpret. This type of structure tells the receiver how to parse the message, or break it down into its basic units.

[know]

So, how can we describe the idea of a parenthesis to an alien? Biology can teach us a lot about how to pack an immense amount of information into a small package. The entire human genome, in effect the program required to build a human being, consists of about 3 billion DNA base pairs, or roughly 6 billion bits of information. This is roughly equivalent to the amount of information stored on a single CD-ROM. What this demonstrates is that nature can condense everything it takes to build a human into this space, whereas Microsoft can barely manage to squeeze its suite of Office software into the same real estate. We can learn a great deal from nature's economy of words.

The DNA molecule encodes the information needed to build most life forms on Earth. The DNA molecule uses different combinations of base pairs to represent different amino acids, which are used to assemble more complex molecules, called proteins. Special combinations of base pairs represent the start of a series of instructions to build a protein. The DNA molecule encodes information using four different molecules: A (adenine), T (thymine), C (cytosine) and G (guanine). These 4 molecules, when found in groups of 3 (or triplets), form basic genetic words. These words can represent an amino acid (a building block for proteins), or they can represent a special "stop" word to mark the end of a sequence (or a word at the end of a sentence). It's helpful to think of DNA encoding as we would an alphabet, just like the English alphabet. Forget, for a moment, about the fact that we're dealing with chemicals here. Think of each base pair as a letter: A for adenine, T for thymine, C for cytosine, and G for guanine.

By itself, a single letter means nothing. In order for them to mean something, these letters must be combined to form words. The information in DNA is organized in triplets, or sets of three base pairs. These triplets are analogous to words in English. For example, the sequence GAC is the word that represents aspartic acid. The format of DNA is more rigid than English. Every word in DNA is built from three base pairs (letters), no more, no less. These triplets are also referred to as codons. Since DNA uses three letters (molecules) to represent each word, and the letters have four possible values, DNA can encode a total of 64 different states in each three-character (letter) word. This means that DNA could, in theory, encode a maximum of 64 different words. In practice, DNA encodes a total of 22 amino acids. This is because several different combinations can code for a single amino acid. (This appears to be an error-correction mechanism, so a random error in transcribing one letter in a DNA word will not necessarily produce the wrong amino acid.)

These words can, in turn, be combined to form groups of words. The German language offers a good analogy. Many German words are formed by combining several words to create a single compound word. In the same way, several DNA words can be strung together to describe a protein that is formed using several amino acids. For example, combining the words arm, band, and uhr forms the word Armbanduhr. This example translates in English to wristwatch, or literally, "arm + band + clock."

A typical DNA instruction can be read in a format such as the following:

STOP : Alanine + Aspartic Acid + Glycine + Alanine : STOP

This instruction can be read as: "Build a protein by combining alanine, aspartic acid, glycine, and alanine." It would be coded in DNA as:

TAA GCC GAT GGA GCC TAA

The Stop instruction is important because an organism would otherwise produce infinitely long, tangled blobs of amino acids, instead of useful proteins that perform a specific function such as transporting oxygen in blood (i.e., hemoglobin). We're going to do something similar with our endless stream of binary digits by creating two special series of "start" and "stop" instructions. One series, 111000111000111000, will always indicate the start of a word. Another series, 101000101000101000, will always indicate the end of a word. So, whenever we see the sequence of digits 111000111000111000, we see an open parenthesis, "(" and whenever we see the sequence of digits 101000101000101000, it can be interpreted as a close parenthesis, ")"

NOTE: The number 111000111000111000 doesn't have an inherent meaning. This number was chosen at random to use as an example throughout the book. In a real system, the sender could use any string of digits as a delimiter to separate symbols and groups of symbols. Parentheses can also be described by assigning special states to the transmitted signal itself. If, for example, the message is embedded in a pulsed laser beacon, the sender could use special colors to describe "(" and ")" symbols (e.g., red=0, orange=1, yellow="open parenthesis," green="close parenthesis").

At first, the recipient of the message will see nothing more than a series of binary numbers with no apparent beginning or end. However, upon closer inspection, there will be certain sequences of digits that recur throughout the message. One of the first things our recipient will do is to start analyzing the series of digits to look for order or repeating patterns in the message. The open parenthesis and close parenthesis sequences will appear repeatedly throughout the message. The recipient will most likely look for repeating patterns in the message by analyzing the frequency with which different combinations of digits appear. This type of analysis, although it is requires a lot of computation, is fairly easy to do.

The trend that this frequency analysis will reveal is that these two series of digits occur repeatedly throughout the message. They also occur in a predictable order. An open parenthesis symbol will be followed by some data, and then by a close parenthesis symbol. The open and close parentheses symbols will also be encountered in equal numbers throughout the message as a whole. This, by itself, does not reveal the meaning of the open parenthesis and close parenthesis sequences, but their use throughout the message is a strong indication that these sequences are important to deciphering the message. The recipient will also know that, if the message contains useful information, it will most likely be organized into smaller units and subunits of information. So, the first thing the recipient will want to do is figure out how to parse the message, or to break it down into those smaller blocks of data. Once the recipient discovers that the "(" and ")" sequences appear throughout the message, and sees that they almost always occur in pairs, it should be fairly easy to figure out that they are being used to bracket information--to define the start and end of a word or group of words. They will most likely try many different approaches before finding the right solution. When the recipient figures out that the "(" symbol equals 111000111000111000 and the ")" symbol equals 101000101000101000, the basic structure of the message will be revealed. For example:

11100011100011100011100011100 01110001001101000101000101000 11100011100011100011001100101 00010100010100011100011100011 10000001101000101000101000111 00011100011100001010110100010 10001010001110001110001110001 10011001010001010001010001010 00101000101000

Although we can see a repeating pattern in this sequence of digits, it is difficult to see how this message is organized. What we'll do now is to replace the sequence 111000111000111000 with a "(" symbol to denote the start of a word. We'll replace the sequence 101000101000101000 with the symbol ")" to denote the end of a word. When we perform the translation, the series of digits above is reduced to:

((1001)(11001100)(0001)(010101)(11001100))

Now the structure to this message is revealed. We can see relatively short words bracketed by "(" and ")" symbols, and can also see groups of words bracketed by "(" and ")" symbols to create expressions. While this doesn't tell the recipient anything about what the words mean, it does reveals the basic structure of the message. Once the recipient can parse the message into individual words and groups of words, they can then set about the task of determining what these numeric words mean. The basic trick we're using here is to introduce an obvious, repeating pattern into an otherwise unintelligible message. In effect, what we're doing is repeating the following message over and over:

{useful information starts here} 10101011110011011010101011111 1
01010101010000001100110100110 1 {useful information ends here} {useful information starts here} 1011111100000111 {useful information ends here}

This approach allows the recipient to discern the basic structure of the message using only simple statistical analysis tools. This approach may also be easy to decipher because it mimics the way information is coded in biological systems, and therefore may look familiar to our distant recipient (assuming biological information is encoded in something similar to DNA on other planets, which it may not be.) Once these special symbols are known, a simple computer program could be employed to perform a search and replace operation, much like a word processor does. At this point, the goal is not to translate the message itself, but to figure out how it is organized into symbols, groups of symbols, groups of groups of symbols, and so on. Next, we'll look at how we can use this system to create a vocabulary of symbols that we can use to build a progressively more and more sophisticated message.

[l n]

I read that the blueprint stored in DNA, an organism's genome, is, in effect, the program that describes how an organism builds itself and functions throughout its life. This information is subdivided into many discrete packages of instructions (genes). Each gene is typically associated with a particular function or trait (such as the instructions for producing the hemoglobin molecule used by red blood cells). An organism's DNA program is not read in its entirety from start to finish, but is broken down into many smaller units, each of which can be accessed as needed. An igene, like a gene, is a set of computer instructions that can be incorporated into other, more complex programs. Just as the gene for hemoglobin doesn't describe how to build an entire blood cell, an igene that describes how to calculate the sine of a number is a component that deals with a small part of a larger task. This modular approach for packaging instructions allows us to create symbols that are shorthand for an otherwise complicated set of instructions, and to combine these symbols to describe complex processes in shorthand form. The key difference between an igene and a gene is the igene contains computation instructions whereas a gene describes how to build a protein that is used by an organism.



When an alien civilization first encounters our hypothetical radio message (or we detect a similar message ourselves), all they will see is an apparently endless stream of binary numbers. At first, the message will appear to be hopelessly jumbled and devoid of meaning. If it is not formatted so that it contains clues about its structure, the party on the receiving end may never figure out how it is organized, and will never be able to get beyond receiving the signal. The first step on the path to comprehension is to figure out how the series of numbers is organized into the equivalent of words, groups of words, groups of groups of words, and so on. A good metaphor to use is the general format of the information encoded in DNA. Learning the format of DNA is not the same thing as learning the meaning of the instructions encoded in DNA. The first step is to figure out how the information encoded in a genome is broken down into smaller subunits of information. Knowing how genetic information is organized doesn't mean that we understand what every gene does, but merely that we know how to parse this data into the equivalent of words and sentences.

know, you're so funny!