Mathematics Of The Fifth Prophet
On the New York State Lottery pages you can find that the chances of winning the jackpot are 1:175,711,536. Since you choose five numbers from 1 to 56, and a mega-ball number from 1 to 46, they arrive at this figure using mathematical combinations. The order of the first five numbers you choose does not matter, so you are choosing 5 numbers out of 56, or (56 5) = 56! / (51! * 5!) , AND 1 number out of 46, or (46 1) = 46! / (45! * 1!) .
This couldnít be happening. He felt split into pieces. One copy of Sam was inside the ping pong ball with their sixth number, like Da Vinciís Vitruvian man with outstretched arms and legs inside the circle, or on one of those amusement rides to spin you around and make you dizzy. One copy of Sam sat on the couch next to his hysterical wife. Sam and the sixth number, the megaball, bounced and rolled, rolled and bounced, and yet he wasnít dizzy at all. Then he was falling, falling down the chute and approaching the already chosen numbers at a great speed. When he hit, there was an explosion, like the Big Bang at the creation of the universe. Annie had knocked over the lamp, and Yolanda was screaming, "And the megaball number IS ..."
Annie was yelling, "Oh my God! Oh my God! Oh ... my ... God!!" Sam tried to breathe. Yolandaís review of the winning numbers disappeared before he could check them, but Sam knew there wasnít any doubt. In his head, Einsteinís face appeared and said, "God doesnít play dice with the universe." Samís face appeared inside Einsteinís and responded, "Maybe God decided to get back into the game!"
However, recall that Sam's numbers were called out in the same order in which he had submitted them to the clerk. That increases the odds against winning to 1:56*55*54*53*52*46 = 1:21,085,384,320.
Catastrophe Theory is a mathematical theory of sudden change. It was created by the French mathematician Rene Thom in the early 1960's.
It was mentioned as part of Ari Bosronov's lecture (pg 121?) and by Abu BinBob as being related to the caustic waves in the swimming pools (pg 130-131?). However, it is explained in more detail in Dr. Jennifer Beseret's introductory lecture on the topic of Catastrophe Theory in the section of the same name beginning on pg 137. The claim is that catastrophe theory is involved in the sudden change from one level of complexity to the next.
"So lets get started with some mathematics you may not be familiar with, Catastrophe Theory. The French mathematician, Ren Thom, created the theory in the 1960's. There are other people involved, Hassler Whitney, for example, but this is not a history of mathematics course so we wonít discuss who did what any further."
"Isnít that the theory of dynamic systems. I thought that was called Chaos Theory?" one of them said, ...
Jen responded ..., "You are confused. Chaos and Catastrophe may sound alike to you, and both fields of mathematics are theories about dynamic systems, but Chaos deals with self-similarity and fractal geometry. Catastrophe is topological. You are familiar with Topology of course?"
"Thom worked out the relationship of topological singularities to the maxima and minima of calculus. He could see how the former would Ďunfoldí into arrangements of the latter, imposing a structure on them. To know the arrangement of the max and min of a dynamic process would be to know its qualitative behavior. How many topologically different structures were possible? For a very wide range of processes, only seven stable unfoldings, the seven Ďelementary catastrophesí are possible. These unfoldings are called catastrophes because each of them has regions where a dynamic system can jump suddenly from one state to another, although the factors controlling the process change continuously." She paused for breath, ...
"Each of the seven catastrophes represents a pattern of behavior determined only by the number of control factors, not by their nature or by the interior mechanisms that connect them to the systems behavior."
..."These catastrophes model a wide variety of processes ...," she said, and began to display the graphs, "... caustics ... engineering ... phase transitions, as in explosions ..."
"Boom!" said a boy in the middle, ...
She continued her list of applications of catastrophe theory, "... the nervous system of an animal or a human being can convey an infinite variety of messages. The transmission of every nerve impulse depends on appropriate levels of just a few substances, especially sodium and potassium, inside and outside each nerve cell. Key features such as these can serve as the control factors in models based on catastrophe theory ..."
"So the models are multidimensional?" Sarah asked, bringing the crowd back from the brink.
"Yes, exactly," Jen responded showing relief, "Your recognition of that truth was a sudden change in your mental state, a good catastrophe." Jen was waiting for the bad catastrophe.
"Letís examine this in more detail." Jen returned to the first graph. "We can use the cusp catastrophe graph to model examples of territorial behavior, aggression, and group formation to illustrate observed patterns of behavior and their changes from one mode into another."
..."Will this be on the exam?" "No, it wonít be on the exam, look at the list," Jen said sharply, ...
"This is the least interesting ..." " ... of the elementary catastrophes. Because itís simple!" she snarled, thinking, "Simple, just like you!" But she still had control of herself, if not the class. So she tried to continue, "Obviously, only a few things can happen."
"The system can be balanced at the point of inflection," she said raising her voice over the increasing cross conversations. "Boom" boy started trying to balance his pen on the tip of his finger. Jen grabbed for it. "Oooh!" from the class. "The system can be essentially unstable, having no minimum," Jen said a bit louder. "Or the system can move to a state of minimal potential, if one can exist," she said more quietly, as if giving up.
"Consider a rubber band ..." The fat kid in the second row, woke up and took the wide rubber band from around his overstuffed notebook. Then the rubber bands started flying. The nasty comments started flying.
Finally, she had lost control of the class. Everything in the classroom was in motion. Desks were being pushed back as some students started to walk out. Others were throwing papers and shooting rubber bands like a junior high school class. Someone had unplugged his earphones from his laptop and the unfiltered music spilled out into the classroom. There was only one student not moving. It was the girl with the green ribbon and pony tail, Sarah. Suddenly, Jen stopped yelling at the class and looked directly at Sarah and smiled. Then she raised the projection screen from in front of the blackboard and pointed to what was written on it. "Homework #1: Describe what just happened. State the control variables, and draw the appropriate catastrophe graph. Due Thursday."
I don't have the tools to show you the pure mathematics of catastrophe theory on this page at this moment. It involves drawing three dimensional graphs. In the future I will update this page. However, the mathematics is embedded in the reality around you, and I can supply an example of one of the elementary catastrophes and its associated graph which should make you understand that this theory describes your universe.
Imagine standing in a large room. On the floor of the room is a large rug. Perhaps it is a hyperrug, woven by members of FoM from Hotan. The pattern is intricate, but the hyperthreads form a golden grid that shines through the material. You want to move the rug to the other side of the room, so you grab hold of it in the middle of one of the long sides, and try to pull it across the floor. But the rug is heavier than you thought and all you manage to do before stopping for breath is to fold your edge onto itself a bit. If you look at your edge of the rug, the fold forms a backwards "S", with your hands holding the top in the air. The other three edges are still flat on the floor, although they may have moved some.
As you stand there trying to catch your breath, look at the pattern on the rug. In the center is a large red circle enclosing a green leaf pattern. Outside the circle, the pattern is a brown and yellow image of rocks and pebbles. When the rug was flat on the floor, you placed your pet mouse down on the rug so that she could get her daily exercise. Round and round, lap after lap she ran, following the red circle. But now, as she makes one more circuit (counter-clockwise) she falls off the edge of the fold you have made. She stops and looks at you reproachfully for such a catastrophe.
Now to get more mathematical. Assume that the far left corner of the rug is the origin of a three-dimensional coordinate system. Along the far edge of the rug you can measure from the origin and let it be the positive x-axis. The left edge of the rug is the y-axis, with positive values increasing as the approach your edge. The third axis, the z-axis, measures the height of the rug above the floor. The red circular path on the surface of the rug could now be described by a formula, but we won't get that detailed here. Most of the rug still has z-value of 0. The path of your pet mouse changes gradually, or smoothly, step by step, even as it climbs slightly up hill onto the folded part of the rug. Then, suddenly, it changes "discontinuously" from a positive value of z at the edge of the fold, to a 0 value of z back on the floor. This event is the "catastrophe" or discontinuous change that the theory is interested in.
Catastrophe theory is especially interesting when used as applied mathematics. Instead of describing your pet mouse running around the red circle on the hyperrug, the same model can be used to describe the physical events of boiling and condensation. The x-axis and the y-axis in general represent control factors, while the z-axis represents behavior. Let the x-axis represent pressure, and the y-axis represent temperature. The z-axis represents the behavior, the density of water. On the right side the water is liquid. On the left side, the water is gas. Choose a point, A, in the liquid region, and a point, D, in the gaseous region. You can draw a path from A to D that avoids the fold and gradually increases in z-value. You can also follow a path from D to A that drops off the fold before continuing along the bottom surface to A.
Woodcock, Alexander, and Monte Davis. Catastrophe Theory: The Revolutionary New Way of Understanding How Things Change. New York: E. P. Dutton, 1978. ISBN: 0-525-07812-6
Chaos Theory is a mathematical theory of change and its dependence on initial conditions. It studies systems which evolve over time, dynamic systems. Of special interest are those systems that are sensitive to their initial conditions. Although these systems are totally dependent on their initial conditions, their behavior often appears to be random and chaotic. However, they are completely deterministic.
Chaotic behavior is also observed in natural systems, such as the weather.
Mathematical Dimensionality and Fractals
Fractal geometry is the study of self-similarity, which is represented by iterative or recursive equations. The term "fractal" was first used by Benoit Mandelbrot in 1975.
"Politics, religion, philosophy, science, mathematics, economics, war and peace, and dozens of other subjects that, in the past, had been poured into his mind and into his soul, were swirling and twirling around in there like some fractal design from Chaos theory." (pg 3)
Most people have seen some images of fractal designs even if they don't know the mathematics behind them. Those of us who are more familiar with the term really can "see them" everywhere.
"Actually he was searching for patterns everywhere. Nature was full of symmetry and fractals and tilings and other interesting mathematics." (pg 72)
"Sam sat down on the steps of his back stoop under the American flag and caught his breath. Some leaves had already fallen and were covering the herringbone pavers with fractals. " (pg 449)
I have given the reader a literary introduction to the subject in the section entitled "Fractals." Describing the images in words, without images and without resorting to pure mathematics, was a challenge. Hopefully, I can add the images on this webpage, but first, here is some of the section at issue without the entire story line.
... Like a bearded ĎSimon Saysí master, he instructed them to, "Look at your watch. Look to the left. Look to the right. Look up at the ceiling. Look down at the floor. Look straight ahead. Look behind you. Look at your watch." When they finished and were loosened up a bit, he continued, "Thatís about all any of you see of the universe, Samís (x, y, z, t), and unless someone has trouble with his watch, you can only travel in one direction along the t axis. Are you ready yet?" he snapped at Silvio.
Silvio was quite nervous speaking in front of this group of people, ... he couldnít stall any longer. "Iím ready, Ari."
Silvio put up his title slide, "Fractal Geometry and Hyperspace." He began nervously to repeat an obviously memorized talk, "Consider a line segment of unit length. Like this one, in figure (a). Then, in figure (b), we subdivide it into three equal parts. Next, in figure (c), we use the middle subsegment as the base of an equilateral triangle, and erase the base. You now have ... in figure (d), a polygonal line, a polyline, consisting of four segments, each with a length of 1/3. So the total length of the polyline is ... 4/3."
... "Repeat the subdivision process on each of these four segments. You now have a polyline with 16 subsegments, each of length 1/9. It looks like a little star rising from some surface, with two little bumps on either side." Sam knew Ari would skewer Silvio for that description.
"Let the diagrams speak for themselves. Leave the poetry out!" Sam heard Ariís voice.
Silvio said, "Letís start over. Go back to the beginning ... and consider the original unit length line segment. Take three of these and form an equilateral triangle. Subdivide each of the sides of the triangle, as we did before with the line segment. You should have ...", and he switched slides again, "... a six pointed star, with twelve edges. Repeat the process of subdivision, ..." ... "... and you get a much more complicated looking object. Like a snowflake. If we repeat the process indefinitely, we get what mathematicians call a Koch snowflake." ... "If you look at the flat version, it sort of looks like clouds on the horizon or vegetation."
Silvio took a deep breath. ... "One interesting property of this curve is derived from considering the length of the curve. Let the function, L( ), represent the length of the curve. Then L(1) = 1, means that the length of the curve with no subdivisions is 1 unit. L(1/3) = (1/3) * 4, calculates the length of the curve with a single subdivision. In general, L(1/3) = (1/3)* 1/ (1/3)D ... D is the dimension of the curve, and in this case it is 1.2618... . That is, the curve is of fractional dimension! It is also infinite in length, continuous, and nowhere smooth, so it canít be drawn. But it exists! And if you think of the snowflake version, we have just described an infinitely long, folded string, fitting inside a finite circle of area Pi/3."
"Now shrink the original radius ... you have a string inside every point! A very complicated universe indeed!" said the Asian man. Ari turned the lights back up, and Silvio, sweating slightly, packed up the overhead projector and collected his slides.
... (pgs 114-117)
Dr. Abu BinBob has his apprentices, scattered around the world, trying to decipher the alien message. Obviously, they must be competent in fractal geometry to do so.
""Dad turned him on to Escher and fractal designs, now all he sends relates to fractals," Julie said." (pg 128)
Just then Abu returned from his elsewhere, and showed he was at least partly here. "They use addition modulo two, and a pseudo random-number generator based on a numerical feedback loop just as fractals are based on a geometric feedback loop." And then he was gone again, gazing at the honey he was dripping onto a biscuit. The rest of them looked at each other and laughed. (pg 129)
Dr. Jennifer Beseret's lecture is about Catastrophe Theory, and I don't need to include more description of fractals and Chaos Theory at this point. Assume you were required to have taken that course before you signed up for her class.
"Jen responded politely but firmly, "You are confused. Chaos and Catastrophe may sound alike to you, and both fields of mathematics are theories about dynamic systems, but Chaos deals with self-similarity and fractal geometry. Catastrophe is topological. You are familiar with Topology of course?" she moved to stand directly in front of him as she asked. " (pg 139)
Dr. Ari Bosronov, however, never holds back on any mathematics.
"Fractals and hypercubes were flying all over the board, developing into chaos theory and catastrophe theory." (pg 121)
After all the genetic engineering, I thought it only appropriate to place some real-life fractal plants into my landscape. Fractals are often used to depict foliage in computer graphics applications, so I bio-engineered them for planting by Oyun in Hotan near the Golden Jade.
"The fractalies plants were rippling in the wind all down the right side of the road. They were twice as tall as when they were planted last year, so Jenís experiment must have worked. They had genetically modified one of the desert bushes to improve its ability to absorb what little moisture there was in this climate, and to help the roots get some nutrients out of the sand. These were a gray-blue, but the ones out by the two smaller hostels in the mountains were pinkish." (pg 153)
"In the morning Oyun had the weavers, the guards, the farmers, the bike riders, and the others lined up next to the fractalies hedge, along the dirt road sloping down from the Golden Jade to the clinic door. " (pg 369)
"The old Golden Jade was visible in the distance, greener than before, as were the farms up river due to Jenís genetically engineered drought resistant trees and fractalie shrubs. " (pg 431)
I only mention it once, but I claim that the efficient generation of energy by the hyperthreads requires the proper orientation of those threads with respect to each other. What better patterns to use than fractals?
"In the end, it was all about patterns and mathematics. Single sheets of hyperfabric began with simple grids of hyperthreads, then evolved into space filling curves, fractals, and chaos theory. That was easy." (pg 406)
Wolfram, Stephen. A New Kind Of Science. Champaign: Wolfram Media Inc., 2002. ISBN: 1-57955-008-8
As Sam develops the fundamental theorems of the Family of Man, while sitting on his back porch, he ponders time and space.
Theorem III. Your good deeds and your evil deeds last forever.
"What is time? If time is part of this physical universe, and time and the universe seem to have a beginning, then what is time embedded in beyond this universe? Something bigger, of course, in order to be an embedding. That Ďlarger timeí could be called, forever, or eternity, but Iíll refer to it as hypertime."
Hypertime! Sam wished he had a better word to use than "hyper."
"Everything you do is permanent in time/space/hypertime. Heaven and Hell may exist in hyperspace, and thus here on earth as well. They may exist now. Judgement Day is always here."
"If you wish to express it poetically, then recite this from William Blakeís, ĎAuguries of Innocenceí:
To see a world in a grain of sand
And a heaven in a wild flower
Hold infinity in the palm of your hand
And eternity in an hour
Q.E.D." (pg 63-64)
Once this was posted on the web, it was noticed by several key characters. Just like you have noticed it!
He started searching the CST site to see if any new papers were mentioned, and saw nothing relevant but three of his own more recent journal articles, "Geodesics in Hyperspace", and the other two. (pg 70)
It didnít take long to read the Family of Man pages. Ulysses had seen enough genocide in the Balkans and Darfur to agree with the major premise, but he was an atheist and no mathematician, so the part about God in hyperspace just sounded nutty to him. (pg 77)
To the Catholics, since souls exist all over the place both here and in what you are calling Ďhyperspaceí, go ahead and pray or talk to your saints. (pg 106) ... Burt
So what is hyperspace and hypertime, you ask? If hyperspace is where the action is, what the universe looks like, it has to be explained to the reader. In the section "Multiple dimensions and hyperspace" I try to do that.
Multiple dimensions and hyperspace
When Ari got back to his office, Sam was standing in the hallway. ... but Samís mathematical skills were more in the form of coming up with wild and crazy ideas that sometimes showed insight, but often were just weird. Today, though, Ari had a surprise for Sam.
... Then he heard Ari say, "OK, Sam letís hear your spaghetti theory to get this thing started."
"Youíre kidding!" Sam said nervously.
"For real. Itís inspirational!" Ari said handing him some pens, and sat down next to the Asian man.
"Well, you asked for it," Sam said and went to the front of the room. He began, "If a function space, F, has its elements, f, with all the properties of a point in Cartesian space, and if we can extend a finite dimensional Cartesian space with points (x, y, z, t) to an infinite dimensional Cartesian space with points (x, y, z, t, ...), then some part of the function, f, corresponds to the x coordinate of the point, some part to the y coordinate, some part to the z and some to the t coordinate of the point. There is also a correspondence between the rest of the function, f, and the rest of the coordinates."
He relaxed a bit and continued, "In other words, one end of the function is in our knowable, seeable, physical universe and the rest of it is somewhere else. And this is true for every point in physical space! That opens up a lot of possibilities. Sort of like putting a whole lot of angels onto the head of a pin."
... "Remember, a mathematical point, (x,y,z,t) has no diameter. A mathematical line, from point P1 to point P2, has no thickness. You learned that in high school geometry class. Any yet, we picture a function, such as a sine wave, like a piece of spaghetti. A very thin piece of spaghetti. And a point in space is smaller than a speck of pepper, smaller that a molecule of water, smaller than an atom of carbon."
Sam turned and was able to write at least one formula on the board, "The speed of light, c, in Einsteinís formula, e = mc2, is one of the supposed constants of the universe. Nothing can travel faster than the speed of light through space. Thatís why science fiction writers invented hyperspace, to get from planet Earth to planet X in a reasonable length of time."
Sam turned around again, "This new mathematics Ari is exploring gives us our hyperspace. One end of our spaghetti function is placed at our pepper point on a plate of magic pasta in Fermiís Frank House in Peoria, and the rest of it lies in hyperspace. Perhaps ( ..., a, b, c, t2, ... ) lies on another plate of magic pasta on planet X. So how can we fit through such a small keyhole? Take a pack of uncooked spaghetti functions from a circular set of points (x2 + y2 = r2) on Earth and find the other end."
"Ari makes fun of my description, but it canít be too far off if he wants me to tell it to you," Sam said, heading for a seat. (pgs ...-114)
The next section also refers to Hyperspace, but its emphasis is more on fractals.
Silvio put up his title slide, "Fractal Geometry and Hyperspace." ... (pg 115)
And then again, Dr. Lalu mentions it in his physics lecture.
"We believe that time, energy, and space are dynamic aspects of a larger multiverse of hyperspace. (pg 120)
But, Dr. Bosronov is the expert in the theory of hyperspace.
Fractals and hypercubes were flying all over the board, developing into chaos theory and catastrophe theory. (pg 121)
Dr. Bosronovís recent discoveries have described a location for souls, embedded in hyperspace. (pg 147)
After the announcement of Dr. Bosronovís Unified Theory, pictures of Ari with Einsteinís hair started appearing in all different variations. The media was full of discussions of imagined associated breakthroughs in science such as time travel, hyperspace jumps, invisibility, and so on.(pg 150)
Vu Dhu Yen turned some of the theoretical mathematics into art.
That was part of her inspiration for the scene, but it was also influenced by the discussions of hyperspace she had had with Oyun after dinner that evening. Yenís artists would copy the images into other media, and the originals would be sent off to Paris for Jacquesí art gallery. (pg 164)
Dr. BinBob was able to discover the hyperspace coordinates of some astronomical objects in the "alien message", but that was as far as he could go.
One of Abuís theories as to the meaning of the message was that it was an SOS from an alien space traveler, that he had accidentally intercepted. But there was nothing other than the repeating of the message, and the hyperspace coordinates, to support that. (pg 173)
It was the consultation with Ari that led to the next breakthrough. Together they determined that the optical effects in Abuís pools were caused by interference or combination with what Ari was hypothesizing as "hyperlight." Looking beyond the basic three or four dimensional reality, they discovered the hyper-coordinates, but they kept this information within the Family. (pg 173-174)
Computational Geometry is the study of algorithms for solving geometric problems. The name was first used in a 1975 paper by Dr. Michael Ian Shamos, although the field grew out of problems from computer-aided design and manufacturing (CAD/CAM), computer graphics, geographic information systems (GIS), robotics, and other disciplines. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry.
Although computational geometry is not mentioned specifically in The Fifth Prophet, geometry is. The solution of the genetic problems studied by Dr. Jennifer Beseret would implicitly involve geometric algorithms. Computational geometry today is closely associated with computational biology, and its use in modeling the structures of DNA and genes and protein molecules would need to be extended in order for the breakthroughs of Dr. Beseret to actually occur. Dr. Abu BinBob's work in decrypting the alien language encoded in the caustic light patterns would require complex visibility algorithms.
Goodman, Jacob E., and Joseph O'Rourke. Handbook of Discrete and Computational Geometry. New York: CRC Press, 1997. ISBN: 0-8493-8524-5
Preparata, Franco P., and Michael Ian Shamos. Computational Geometry: An Introduction. New York: Springer-Verlag, 1985. ISBN: 3-540-96131-3
Computability, Complexity, and Language
Turing Theory is named after the British mathematician Alan Turing, who in 1936 developed one of many abstract mathematical models that describe the basis of how computers work. It is part of a field of study called Computation Theory, Complexity Theory, Automata Theory, or the Theory of Formal Languages. One of the goals of these studies is to determine what can and what can not be "computed" by a machine.
"Turing machines are basic abstract symbol-manipulating devices which, despite their simplicity, can be adapted to simulate the logic of any computer algorithm. A Turing machine that is able to simulate any other Turing machine is called a Universal Turing machine (UTM, or simply a universal machine). A more mathematically-oriented definition with a similar "universal" nature was introduced by Alonzo Church, whose work on lambda calculus intertwined with Turing's in a formal theory of computation known as the Church-Turing thesis. The thesis states that Turing machines indeed capture the informal notion of effective method in logic and mathematics, and provide a precise definition of an algorithm or 'mechanical procedure'." [Wikipedia]
Many years ago, while I was working on my thesis, I attended a seminar given by Barry Cohen, a Ph. D. candidate at Stony Brook University. It was entitled, "Computational Biology and Turing Machines." Although I can't remember many details of the talk, and I can't decipher the notes that I took that day, I clearly remember the shock and excitement I felt at the clear connections Mr. Cohen presented between the genetic structure of man and the mathematical components of Turing Machines. Since the mathematics forms the foundations of computing and algorithms, and the genetics forms the foundations of the human biology, it is not a big leap to imagine interesting science fiction consequences.
A quick "google" of the key words led to the following, more recent, and contradictory point of view.
Professor Denis Noble Are organisms Turing Machines? Similarities and differences between genetic and computer code.
The idea of DNA as a computer program was invented by Jacob and Monod in the 1960s when valve computers were fed by code on paper tape. Applied to living organisms, the paper tape became DNA, the machine obeying the instructions became the rest of the organism. The idea was that a 'genetic program' was to be found on the DNA 'tape'. An organism could therefore be regarded as a Turing machine. Our knowledge today of the complexity of molecular genetic mechanisms, and of the extensive control that the organism and environment exert via epigenetic and other processes, leads to a very different analogy. Organisms are 'interaction machines' not 'Turing machines'. This opens the way to a radical re-assessment of the central dogmas of biology (Noble, 2006, 2008). http://www.allhands.org.uk/2008/programme/denisnoble.cfm
One use of Turing machines and languages is Dr. BinBob's work on translating the alien message.
Abu BinBob then started them off in a different direction, "I hate to introduce science fiction into this, but there is an interesting overlap in the research areas of myself, Ari, and Dr. Beseret. We are all interested in the theoretical aspects of language. Mathematics, computer algorithms, genetic code, Turing machines, animal communications, alien languages, and so on. Perhaps our work will allow us to improve existing translation devices so that they are actually practical to us all." (pg 101)
A second use of Turing machines was Dr. Beseret's recognition that what she was seeing in her research was the code for a simple counter.
Damn! It was like trying to remember somebodyís name when they came up to you with "Hi, Jen! Long time no see." You know you know it. You know you should be able to say it. But it isnít until later, when you arenít trying so hard, that it jumps up out of nowhere. That was what had just happened to Jen. One of the simplest subroutines in Turing theory was a counter. It was so simple that it was only a single statement in most higher level languages. Increment the counter. Thatís what those things were in the alien DNA, counters! It was like Abu finding the coordinates in the message. (pg 357)
Cohen, Daniel I. A. Introduction to Computer Theory. New York: John Wiley & Sons, Inc., 1986. ISBN: 0-471-80271-9
Davis, Martin D., and Elaine J. Weyuker. Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science. San Diego, Academic Press, 1983. ISBN: 0-12-206380-5
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