by Dr. A. R. Davis

Mathematics, Physics, Computer Science, etc.

These pages describe the "hard science" behind my "science fiction." In my earlier books I spent considerable time and effort gradually leading the reader from the real world into my own creations. When I wrote about the "reality field" I threw everyone into the mysterious globular cluster without warning, and sprinkled the science in later as needed. The boundary between my fiction and your reality is not simply some new technological breakthrough, it is an exploration of the regions science has yet to study. Exactly who is this "observer" science postulates in its formulae? Where do they live? These pages paint a picture of the path to the observer's door.

- Mathematical
- Probability - the lottery ticket
- Catastrophe Theory - folding the universe for FTL
- Chaos Theory
- Mathematical Dimensionality and Fractals
- Computer Science
- Computational Geometry
- Computability, Complexity, and Language
- Genetics and Biological
- DNA

- Astonomy and Cosmological
- SETI
- Gliese
- Alien Prince of Peace

- Physics
- String Theory
- Hyperspace

- Genealogy
- Physics
- Time Travel
- Parallel universes
- Schroedinger's Cat
- Butterfly Effect
- Heisenberg Uncertainty Principle

- Genetics and Biology
- Multiminds

- Astronomy - Alpha Janus and Beta Janus and the planet Al Theris
- Mathematics
- Zeno's Lemma
- Parallel Algorithms

**Chaos Theory**

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.

Wikipedia: http://en.wikipedia.org/wiki/Chaos_theory

Hyperspace ...

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 ‘larger time’ could be called, forever, or eternity, but I’ll 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 Blake’s, ‘Auguries of Innocence’:

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 hourQ.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 didn’t 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 ‘hyperspace’, 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 Sam’s 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 let’s hear your spaghetti theory to get this thing started."

"You’re kidding!" Sam said nervously.

"For real. It’s 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 Einstein’s formula, e = mc

^{2}, is one of the supposed constants of the universe. Nothing can travel faster than the speed of light through space. That’s 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 Fermi’s 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 (x

^{2}+ y^{2}= r^{2}) on Earth and find the other end.""Ari makes fun of my description, but it can’t 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. Bosronov’s recent discoveries have described a location for souls, embedded in hyperspace. (pg 147)

After the announcement of Dr. Bosronov’s Unified Theory, pictures of Ari with Einstein’s 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. Yen’s artists would copy the images into other media, and the originals would be sent off to Paris for Jacques’ 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 Abu’s 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 Abu’s 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**

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 somebody’s 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 isn’t until later, when you aren’t 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. That’s 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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