Android Mind

The physics of everyday miracles

2011-07-30T21:05:00.000-07:00

Are there really miracles?

We are accustomed to hear --and repeat-- that the laws of physics are unalterable through time, and that everyday there is a miracle --visible or not-- around us.

All everyday activities are supposed to be ruled by immutable laws. The force of gravity is the most representative and easily recognized of all the physical laws.

In opposition to that, the miracles are supposed to be a break of some physical law.

If there is an deadly car accident but nobody dies, people say its a miracle, but if somebody die in the same accident, its because "such is life", or that "its the will of God".

However, the awesome miracles and the immutable physical laws are the same thing. It is hard to digest, but every time somebody comes alive out of a deadly accident is because of the same physical laws that provoked the accident.

Everything that happens around us has some physical explanation. If some action is unexplainable then it is just that: "not yet explained".

A book that explains many everyday physical phenomena.

Physics for entertainment, by Jakov Perelman is a simple book easy to read and follow. It answers hundreds of questions like:

Is it possible to make fire with ice?
What's the secret of painting people and drawing faces that seem to follow us everywhere we move?
Is it possible to make soap bubbles that last ... for years?

Click this link to download this free EBook now!

What are ordered pairs?

2011-06-29T20:17:00.000-07:00

An ordered pair is the intuitive idea that objects can be flipped in different positions in such a way that the order in which we take them make different entities.

This "definition" may sound a little abstract, but a few examples should bring the idea comprehensible.

When we think about the basic Cartesian coordinate system of two axes, we immediately think of two "real number lines" intersecting at 90 degrees.

The figure above shows an example of how we intuitively use ordered pairs when we plot graphs of real functions.

In this example, The function is any abstract one-one (1-1) rule Y = f (x). When the variable x on the X-axis assumes or takes the value a, then the function f assigns the value b on the Y-axis to that choice x= a of on the X-axis.

Hence, we are necessarily and intuitively talking about the ordered pair (a, b). This entity (a, b) is an ordered pair because the function f explicitly and uniquely assigns the value b to the unique value a.

The notion of ordered pair is not limited to the usage of the real numbers. We can choose the second entry of the ordered pair to be an imaginary number. In that case, the Y-axis is no longer a real-numbers axis, but an imaginary numbers axis. In that case the ordered pair is simply a complex number.

The ordered pairs are very useful when we deal with transformations, specially transformations of plane figures.

For example, the following transformation, made up of two parametric equations:

transforms a circular area of the plane into a dome in space.

To dramatize the results, a picture of a cat is shown before this transformation, and after the parametric equations are applied to the cat's photo.

In this example we are implicitly using triplets, that is, ordered pairs of three entries, like (x, y, z). The first two entries of the triplet are for the locations of the points of the cat's photo, and the the third entry of the triplet is for the amount of "deformation" applied to each point of the photo.

Transformations and ordered pairs are very interesting subjects, because they are not so abstract after all.

Interested in more examples of transformations as in the example above? Then download this free E-Book: The Golden E-Book of Graphs of Mathematical Functions: A selection of some beautiful mathematical surfaces from the domain of the real and the transcomplex numbers system.

Interested in an in-depth development of the foundation of the complex numbers from the standpoint of the ordered pairs? Then download this free E-Book: Foundations of Transcomplex Numbers: An extension of the complex number system to four dimensions.

Or simply remember this link: http://4DLab.info as a source of free E-Books and critical articles about mathematics and mathematics.

Is the search for perpetual motion an utopia?

2011-06-13T09:06:00.000-07:00

The pursuit for perpetual motion is, maybe, as old as the invention of the wheel. Everybody wants to save energy, specially human energy.

People want to be creative, no doubt about this, but people also hate to do the same thing over and over again. Therefore, inventing a machine that could create workforce with virtually a minimum of input energy is the perfect artifact to substitute the human sweat.

The earliest recorded intent to create a non-stop rotating device dates back to the thirteenth century. The sketch was made by Wilars de Honecort, a French architect.

Wilars wrote about this machine the following:

Many a time have skillful workmen tried to contrive a wheel that shall
turn of itself: here is a way to make such a one, by means of an uneven
number of mallets, or by quicksilver.

How was it supposed to work? By what logic was this wheel supposed to keep turning and turning indefinitely?

Perpetual Motion, by Percy Verance is the book that explains it all. Perpetual Motion, the EBook recently edited by Datum is a source book that not only collects all the historic efforts and contrivances to create ever-spinning wheels, but in addition to that, in it we can find the explanations why the machines, or sketches, can't work.

Download the free EBook: Perpetual Motion.

About the sketch shown above he says:

Seven mallets, or arms, each loaded with a heavy weight at the end, are jointed at equal distances to the circumference of the wheel, so that those which happen to have their joints below the diameter of the wheel will hang freely down, but if the wheel be turned round by hand or otherwise, the weights of those which are on the ascending side will,
in succession, rest on its circumference, and will, in that position, be
carried over the highest part of the wheel and downwards on the descending side, until the arms that bear them are brought into a vertical
position and a little beyond it, and then the weight will fall suddenly
over and rest on the opposite position on the circumference of
the wheel, until its further descent enables it to dangle freely as before.

According to modern physics, a truly perpetual motion machine can never be accomplished because it would violate the basic laws (or principles) of thermodynamics.

The most basic of the thermodynamic principles states that:

Energy can neither be created nor destroyed. It can only change from one state to another.

In every machinery, in every kind of work, some heat is lost or dissipated, and since heat is a form of energy, the energy output is always less than the energy input, thus, we cannot retrofit the output energy as an input feed, because in the next cycle we are feeding less energy than in the first cycle. No matter how well are machine parts lubricated, every moving part generates friction, thus, everything that moves is a potential energy loser.

However, some people still claim having invented some kind of perpetual motion machinery --and believe it or not-- in some rare cases, they obtain invention patents for their claims. In the EBook Perpetual Motion the author includes some old cases where patents were granted for those esoteric devices, even when they could not withstand real-world tests.

Registering an invention is not an easy process. It takes specialized attorneys or lawyers. It takes making specialized drawings, and it takes clearly defining your claims. The invention must be new; you must claim and sustain and prove that your patent claims patent are genuinely new. The hardest part is answering the Examiner's comments to your claims. Those people will respond to your invention with a lot of other claims from similar inventors, and you have to prove that your idea is different and your claims are unique.

To conduct some preliminary search about existing patents you can use the excellent Google patent search. There you can read and download millions of patents applications, their drawings, and their claims; be it space vehicles, toys, lamps, dolls, car seats, stoves ---or a soap dispensing method as shown in the accompanying illustration--- or whatever invention you can think of.

Every society needs good and ingenious inventors. But the process of inventing is not achieved by merely drawing ideas on paper. Those ideas must be in accord with the laws of physics that keep the world running. Ignoring those laws and ignoring those principles is a loss of time, a loss of money, and creativity lost.

EBooks about spinning tops and gyroscopes

2010-04-29T19:29:00.000-07:00

There are many peculiar things about spinning tops and gyroscopes: they are essentially the same kind of scientific toy. They come in many sizes and colors; they are even the kind of toy you can make at home.

Almost every child/teen from almost every part of the world have played with a top. Long hours of healthy fun and everlasting moments; they are wonderful for making friends. Tops are good, cheap and non aggressive toys.

On the other hand, since gyroscopes are humble artefacts --like the pendulum-- it is easy to overlook the many interesting --and sometimes transcendental-- things that we can make with them.

Gyroscopes are refined toys with many scientific applications. Everywhere stability is needed, a spinning gyroscope is hidden somewhere. Aircraft instruments, marine navigation, ballistic missiles, submarines.

For example, Léon Foucault (1819-1868), by just using a simple pendulum he constructed was able to demonstrate that the Earth spins around itself. His experiment was so shockingly simple and transcendental that swinging pendulums are common in museums and physics buildings in many universities and colleges.

Datum is giving away a new book edition of this science classic in gyroscopes: Spinning Tops by Professor John Perry (1850-1920). The book comes with many illustrations for many experiments that can be made at home or at school.

There is a 1871 short novel by Edward Bulwer-Lytton (1803-1873) entitled The Coming Race, very famous for introducing a civilization living underground that conquered an unknown form of super energy they called the Vril that could be used either for good or evil. (The Coming Race is also available at Datum for free download). But what's the connection between a book about gyroscopes, Spinninf Tops, and a novel about people living deep in enormous cavers, The Coming Race?

Well, none is the sequel of the other, however, Prof. John Perry mentions in his Spinning Tops that no matter the great power of the Vril-Ya people in the underground, they failed to discover that the Earth rotates. According to him, even if you live underground, it is possible to discover the Earth's rotation just by using a spinning top, or to put things more academi, by using a gyroscope. You don't need to see the Sun in order to prove the Earth's rotation.

Are there coordinates for the fourth dimension?

2010-03-31T18:37:00.000-07:00

The interest in the fourth dimension is ever increasing.We all keep asking: Can there really exist a fourth dimension? In what direction should we look to find it? Why there are so many interpretations of it?

Is the spiritual fourth dimension the same as the physicists interpretation? What is doing the mathematics about it?

Well, to begin with, we all agree that WE ALL LIVE in three dimensions. That's a good start, but we do not all agree which one is THE FIRST DIMENSION. We don't know which dimension is the SECOND DIMENSION; so, how can we all agree which dimension should be the FOURTH DIMENSION?

To study the fourth dimension from the geometrical or algebraic standpoint of view we should associate dimensions with coordinates in a spatial hyperspace. This approach leads us to ask: Are there coordinates for the fourth dimension?

In the article The coordinates of the fourth dimension I use the figure of a little dinning table to ask which one of the four legs of the table we should say is the THE FIRST LEG, which one of the four legs is the FOURTH LEG.

Take the challenge, read the article, and be the first one to answer the question and fill the blanks: In any table, the first leg is determined by ...!

The myth of the hollow Earth: some myths never die

2010-02-28T16:21:00.000-08:00

Yes, we see it everyday; some myths never die.

People need to believe in something. We need fairies, unicorns, angels, demons, mermaids, dragons, goddesses, winged horses, gargoyles, gryphons, etc.

Same thing happens in science: no matter the advances in astronomy people keep searching and reading their daily horoscope. If this makes them self-assured and happy, its OK. Why destroy their illusions? Maybe myths are an inherent need deep within us.

However, the myth of the hollow Earth is another type of myth, because it is showdown, a confrontation between physics and traditional beliefs. But physics, or geography, or geology do not work this way. Solid hypotheses and experimentation to confirm the experiments are needed.

There are many variations of the hollow Earth idea:

The Edmund Halley's idea that the Earth's inside is made up of concentric hollow spheres and four poles.For Halley, the spheres' diameters are in proportion to some planet orbits.
The Cleves Symmes's idea that the Earth is made up of concentric --not as Halley postulated-- but with a strong will to go himself into the Earth's core.
The Cyrus Reed Teed's concept that the Universe was inside the Earth's crust cradled 'in the hands of God."
The William Reed dream-concept that the Earth's interior is warm and with volcanoes as in the outside.

But believe it or not, some of those ideas still survive. Some people are make plans to travel to the North Pole to take a look at the entrance to the interior of the Earth.

See the complete article and more in: What do you know about the hollow Earth and the hollow Moon?

The return of the fear of comets

2010-01-31T16:51:00.000-08:00

Comets are cyclic phenomena. We know they move in elliptical orbits; a fact discovered by Johann Baptist Cysat (1587-1657). The completion of a cycle (a period) is different among them. Some comets fade away, and some of them never return. There are comets with period of less than 30 years, other may take longer than 200 years to return.

Comets do not come to visit or frighten us: the simply orbit around the Sun. They approach the center of our solar system, make a turn, and disappear into the dark Cosmos possibly to never return.

Among the familiar comets is the well-known Halley's Comet; the most recent apparition was in 1986, although very faint for the naked eye. Halley's Comet is expected to return by the year 2061.

But with every return of a comet, specially if it is expected to be a great comet, the fear for them returns.

David Berg (1919-1994) was a cult preacher that induced great fear for the comet Kohoutek of 1986 that --according to him-- was supposed to cause great disaster and to signal a colossal event.

Here are some of his warnings:

ACCORDING TO OUR OWN CALCULATIONS 1986 SHOULD BE ABOUT THE TIME OF THE FINAL TAKEOVER OF ONE WORLD GOVERNMENT BY A WORLD DICTATOR known in the Bible as the “Antichrist,” and the beginning of his reign of terror!

THE MOST BRILLIANT COMET OF THE PAST CENTURY WAS THE GREAT COMET OF 1882 which appeared and disappeared within about the same year of the deaths of two of the most influential men in all modern history, and upon whose teaching both the faith and fate of a Godless world now hangs: Charles Darwin and Karl Marx! Both sealed the doom of Churchianity and Capitalism and the end of an age, if not the world!

Read a complete article about this malevolent induced fear here.

Can the infinite be randomized?

2009-12-30T20:35:00.000-08:00

Is the infinite always the same? Is the infinite tomorrow the same as is today? How many infinites are there? Only one? How about if there are infinite infinites? How about if all of them are not the same, one to another?

How about if one of those infinites is not linear, without any kind of order, but random? Why not?

We have a naive and simple idea of the infinite. In the best of the cases we think of the infinite as the unending series 1, 2, 3, .... In the worst scenario we think of the infinite as the quantity of the grains of sand in all the deserts and beaches in the Earth.

It was Archimedes of Syracuse, more than two millennia into the past, who proved that it was impossible that the grains of sand were infinite because he was able to give a fair estimate of the grains sand that could placed in a sphere the size of the orbit of Saturn.

He knew that it was impossible to count grain by grain all the deserts and beaches. So it is impossible to prove that the grains of sand are finite by enumerating them one by one.

So, his approach was to establish an upper limit to the amount of grains that can hold the planet Earth. This is an indirect proof that the sand cannot be infinite.

Download the free EBook The Sand Reckoner, the book where he developed his proof.

But it we have an infinite book in our hands, how big can it be? Infinite in size? Infinite in weight?. Infinite in volume?

Have you ever heard of The Book of Sand? It’s a short story about an infinite book with no beginning and no ending. But this book has a finite amount of pages; how come?

The infinite is incredible and surprising! What is going to limit the limitless?

Read the article Three unexpected behaviors of the infinite and see three unforeseen aspects of the infinite.

Why is it so difficult to reason about the prime numbers?

2009-11-30T16:13:00.000-08:00

In the history of mathematics, the prime numbers are ancient as the invention of the natural numbers. To review a little, the natural numbers are the simple positive integers 1, 2, 3, ... --a series without end.

In this series of the natural numbers, when we are given some number, it is easy to tell what the next number is; just add 1 to the given number! (This explains why it is funny to ask elementary school kids for the biggest number he/she can think about, and then confront his answer by telling him/her to add 1 to the number he mentioned.)

On the other hand, the prime numbers are not so easy to visualize; their distribution is as if they were randomly distributed.

Let's review for a moment the standard definition of prime number: a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the primes less than 100 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

The number 1 is by definition not a prime number. Note how unusual is the beginning of this series: the first prime is even --the only even number that is prime; all the other following primes are odd. But not every odd number is prime. To the number 3 follows 5; to 5 follows 7, but to 7 does not follows the number 9.

The simple task of finding the prime number that follows another prime impossible. With the prime numbers it happens that given any prime number, there is no way to compute the next prime that follows --or precede-- the given one.

However, the primes are very special in one aspect: every natural number is either a prime or it is the product of some primes. This is a far-reaching assertion, because it surreptitiously states that the natural numbers are in one way or another all made of prime numbers.

Euclid (ca. 300 BC), and Eratosthenes (ca. 200 BC) were the first two Greek mathematicians to work extensively with the properties of the prime numbers. Euclid proved that the primes were infinite (no way to find a last prime), and Eratosthenes devised a method to sieve out the primes from the series of the natural numbers.

Since then, the primes numbers are a constant headache for number theorists, and mathematicians in general.

Historical Quotations About Prime Numbers

However, mathematics is not a private property of mathematicians and philosophers of science. What about the public? They also have something to say. What does the layman have to say about the prime numbers? Do politicians have something to add to this arduous field of mathematics (they always have something to say!)?

Let's see how some imaginary examples of how some famous people would have reasoned about if the number 9 is prime or not.

Christopher Columbus: “3 is prime, 5 is prime, 7 is prime. According to some ancient manuscripts 9 is not a prime number, but beyond the distant horizon of the oceans, in the New World that I am going to discover, there are surely lots of them.”

Dimitri Mendeleev: “3 is a prime, 5 is a prime, and 7 is a prime, but 9 is a noble prime that deserves a separate row in the periodic table of the primes.”

Charlie Chaplin: “3 is a prime, 5 is a prime, 7 is a prime, 9 is the next prime after 8.”

John F. Kennedy: “1 is not a prime number and 9 is not a prime number? Then ask not what the primes can do for you, ask what you can do for the primes.”

Stephen Hawking: “2, 3, 5 and 7 are prime numbers: 9 is not prime, but in the black holes, past beyond the event horizon, anything can happen.”

George W. Bush: “3 is prime, 5 is prime, 7 is prime, and 9 … well, any odd number can be prime as long as it is not 9.”

For more examples about how easily is to be misled when reasoning about the series of the prime numbers, then see the article:

Historical Quotations About Prime Numbers

Is somebody out there? Is it sensible to send messages to other stars?

2009-10-31T18:46:00.000-07:00

Take a look at the quiet night sky. How quiet is it? Don't let the beautiful twinkling stars deceive you! Somebody may be standing and "listening" to us from one of the stars you are looking at.

Should we listen only from them, and at the same time send no message, like --"I am here"-- to them? Isn't there a paradox in this attitude?

Alexander Zaitsev thinks so, and he called it the Paradox of the Great Silence.

Is there a way of detecting messages flying-by over our atmosphere? Do we have the technology to capture every conceivable way of detecting space messages? And more than that, do we have the knowledge and the required wisdom to answer them? Is radio messaging the only way of capturing and communicating our ideas to other stars in our galaxy?

Well, astronomers seem to be short-patience people. They seem to work on a day-by-day basis: the technology that matters is the one we have "today". The first attempt, on 1962, was a three-word message sent to Venus in an experiment from the Evpatoria (Eupatoria) Ukraine, using a deep-space radio-telescope coded using the Morse code pattern.

Later, on 1974, another message was sent from the Arecibo radio-telescope. This one was an elaborate message with information about the the chemical constituents of the organic life here on Earth, and even the double helix structure of the DNA molecule.

Is it fair that a few astronomers reveal the chemical structure of our organic life? Isn't this another case of scientific arrogance, like the one of Sir Arthur Eddington, when he said the he knew exactly how many protons are in the universe?

Then, should we listen only and stop sending messages? This is the Paradox of the Great Silence that stated the Russian astrophysicist.

But the "text-messaging" to the outer space didn't stop there. Then next, on 1999, from Evpatoria --again-- another message was sent, this one in a layout similar to a 23-page book.

What is written (coded) in this message to the stars? You wont' believe your eyes!

But that's not all, almost yesterday, on 2001, was sent another message, this one with the participation of three teens. The Teen-Age-Message (TAM) is --without doubt-- the most sincere and original of all of them.

Why?

Click here to read the full article about those messages!

Books about the fourth dimension

2009-09-30T06:26:00.000-07:00

Now you can download many controversial books from authors out the mainstream. Open your mind to other ideas, to other fields of knowledge.

These are carefully reformatted books for easy and joyful reading, and without password restriction for printing.

Flatland is a book about a journey of a character from a two-dimensional world that visits a one-dimensional kingdom --Lineland-- and that is also visited by a character from the third dimension --Spaceland.

By analogy, since we are three-dimensional, what if we make a similar journey to Flatland, and then to the fourth dimension? Right now, how can we recognize visitors from the fourth dimension?

Selected Papers of Charles Hinton About the Fourth Dimension is a collection of essays from the pen of Charles Hinton, the classic writer that initiated an effort to put order in the chaos of the thinking about other dimensions besides our three-dimensional world.

Readings of the Fourth Dimension Simply Explained is another collection of articles, this time from different authors. Another classic in this subject, sponsored by the well-known magazine Scientific American. Many of the authors were teachers and experts in their field of knowledge.

Another World or the Fourth Dimension is a curious book about the presence of the fourth dimension in the Bible. Is the presence of so many strange experiences narrated in the Bible evidence that the fourth dimension really exist?

The 4-D Doodler is the story of a man that is trapped between the edges of the Spaceland and the fourth dimension. Could this really happen in a future space-time travel? In 2012, or 2100, or 3500...? Is it happening now?

This list of books is not exhaustive. Go to http://4DLab.info for a complete catalog of free books on the fourth dimension, math puzzles, tables, and even the beautifully illustrated: Alice In Wonderland.

Download any of these books now!

The trees that came from the Moon

2009-08-30T15:43:00.000-07:00

In the Apollo 14 mission on January 31, 1971, the astronaut Stuart Allen Roosa took with him about 500 hundred seeds to orbit the Moon with him. The seeds were from five different types of trees: Sycamore, Loblolly Pine, Sweetgum, Redwood, and Douglas Fir.

Upon return to Earth, the seeds that Roosa took with him to the Moon were sent to the Southern Forest Service Station in Mississippi and to the US Forest Service Western Station in California in an attempt germinate them. Surprisingly, nearly all the seeds germinated successfully.

One beautiful example of the Moon Trees is this Sycamore tree in Lincoln State Park, Lincoln City, Indiana.

But are there more trees like this one? And what about the existence of trees in the Moon in the early science fiction literature?

The answer to the above questions are in this more extensive article: What do you know about the Moon Trees?

A free EBook about perspective

2009-07-30T13:52:00.001-07:00

Datum Editorial just released a new EBook for free download.

The Essentials of Perspective is a book about how to draw true-perspective landscapes.

The book, by the late Prof. L. W. Miller, is the perfect acquisition for the starting artist of the pen and ink media.

The book is not a complicated and hard-to-follow textbook in the somewhat-abstract field of projective geometry. Its the opposite: an easy to follow book suitable for high school and college students.

The Essentials of Perspective is a fully illustrated book, with more than one hundred figures.

To drawing or paint a correct perspective we have to take into account the always present illusion of the vanishing point at the far horizon.

The book not only covers the correct depiction of the converging lines in a true perspective, but also shows ho to take the always present shadow of the objects.

Want to unleash the artist within you? Do you want to start drawing simple objects from your surroundings? Don't want to buy expensive art materials? Then, this book and a simple sketchbook (or maybe some loose leafs) is all you need!

Complex numbers: how complex are they?

2009-06-30T18:02:00.000-07:00

The "history" of the integer numbers is a simple one. From the natural numbers 1, 2, 3, ... we move to the positive integers 0, 1, 2, 3, ... then we add the negative integers ... -3, -2, -1, 0, 1, 2, 3, ....

Then we escalate to fractions and decimals and non-terminating decimals (although historically was not in this order) . The ladder continues to the irrational numbers and to the algebraic and transcendental numbers. This is the "world" or universe of the real numbers.

But mathematics is a product of our minds, so this "universe" or field can be further expanded to suit our needs.

The next heaven after the real numbers field are the imaginary numbers; numbers that in combination with the reals make the complex numbers field.

But how complex are the complex numbers? Curiously, they are simple as the "preceding" ones.

The negative numbers haunted the mathematicians and philosophers for many centuries; no wonder the misnomer "negative". Even the number zero took a long time before it was accepted in the kingdom of the mathematics (in Europe, where it was lately accepted.) It was unacceptable to count "backwards".

The imaginary numbers suffered the same fate: no wonder the epithet of "imaginary". The square root of minus 1 was impossible to compute because no number times itself is equal to minus 1.

Take a read at this article: The imaginary numbers are not so imaginary and the complex numbers are not so complex and you will see how easily and beautifully the complex numbers emerge out of the real numbers.

The Deluge revisited ... once more

2009-05-27T16:57:00.000-07:00

The story of the Deluge is a narrative that is mythical for many, however historically true for millions.

The Deluge In The Light Of Modern Science, by William Denton is a critical analysis of this story as it is found in the Christian Bible.

A few extracts of his commentaries are sufficient to grasp Denton's style:

"Noah, his family, and the animals, went in seven days before this time, and left the ark the six hundred and first year of Noah’s life, the second month, and the twenty-seventh day of the month. They were therefore in the ark for one year and seventeen days.What a quantity of hay would be required, the material most easily obtained!"

"An elephant eats four hundred pounds of hay in twenty-four hours. Since there are two species of elephants, the African and the Indian, there must have been four elephants in the ark; and, supposing them to live upon hay, they would require three hundred tons."

"Many animals live upon insects; and this must have been the most difficult part of the provision to procure. There are nineteen species of goatsuckers; and there must have been in the ark two hundred and sixtysix individuals. These birds feed upon flies, moths, beetles, and other insects. What an innumerable multitude must have been provided for the goatsuckers alone! But there are a hundred and thirty-seven species of fly-catchers; and Noah must have had a fly-catcher family of nineteen hundred and eighteen individuals to supply with appropriate food. There are thirty-seven species of bee-eaters; and there must have been five hundred and eighteen of these birds to supply with bees. A very large apiary would be required to supply their needs."

Denton concentrates his analysis of the narrative of the Deluge on the many difficulties Noah had when collecting the animals for his Ark:

"How could the ostriches of Africa, the emus of Australia, and the rheas of South America, get there,–birds that never fly? There are three species of the rhea, or South-American ostrich; and forty-two of these would have a journey of eight thousand miles before them, by the shortest route: but how could they cross the Atlantic?"

However, some questions arise concerning all the water that fell during all those forty days of heavy continuous rain:

"It is as great a difficulty to discover where the water went when the flood was over. We are told that the fountains of the deep and the windows of heaven were stopped, and the rain was restrained. But this could do nothing towards diminishing the water".

Read the whole book! Download it for free now!

Transcomplex Numbers

2009-04-30T08:46:00.000-07:00

Integer numbers, negative numbers, fractions, real numbers, transcendental numbers, irrational numbers, and imaginary numbers are a few of the number types we usually find in mathematics.

Is there no end to this? Is there no "final" type of numbers?

From the standpoint of number fields, all of them can be encompassed into one type called the complex number field.

Wikipedia, in a short background, mentions how the complex numbers emerged:

Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations. The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher.

Complex numbers can also be understood and developed from the standpoint of view of ordered pairs. Today, developing the complex number system from the foundations of set theory and the concept of ordered pairs is possibly the most intuitive approach we have at hand.

For a rigorous development of the complex number system download the free EBook: Foundations Of Transcomplex Numbers: An Extension Of The Complex Number System To Four Dimensions.

This mathematics book is about a way of extending the complex numbers system to four-coordinate variables, maintaining the usual operations attributed to the complex numbers.

Foundations ... is a fully illustrated EBook. See --and Click-- for example, the following figure about how to multiply two ordered pairs:

Complex numbers are usually plotted using the familiar plane Cartesian coordinate system, but transcomplex numbers are four-entry ordered pairs, also called 4-tuples, so they belong to a four-dimensioned space.

In a nutshell, transcomplex numbers are complex numbers whose elements are ordered pairs.

In the following simple illustration, also taken from Foundations ... we can see that out of a four-entry complex number system we can extract four 3-dimensional spaces like "ours".

The chapters of the book are divided as follows:

Ordered Pairs. The whole theory of transcomplex functions is based on the ordered pair concept: from the two-dimension plane up to the four-dimension space.
Complex Numbers. The complex numbers system is derived from the ordered pairs concept.
Transcomplex Numbers. Here starts the extension of the complex numbers into ordered pairs of complex numbers, arriving at the concept of transcomplexs.
The Coordinate System S4. This chapter is devoted to deriving a suitable coordinate system to plot transcomplex functions.
Transcomplex Functions. Functions of complex variables evolve to make space for functions of four-entries ordered pairs.
Transcomplex Surfaces. A radical and totally new perception of surfaces generated by complex variables.
Theorem Proofs. This chapter collects all the proofs of the theorems stated along the book.

Big numbers: can we really understand their meaning?

2009-03-31T16:39:00.000-07:00

I recently wrote an article about the 'unreality' of the Eddington number. The article title is: The Eddington number: a case for scientific arrogance?

Arthur Eddington (1882-1944) was an all-time advocate of the emerging Theory of Relativity since its introduction by Albert Einstein.

The 'Eddington number' is an extremely big number that supposedly represents the exact quantity of proton in the visible universe. This number, sometimes abbreviated as N_Edd, needs 83 digits to describe it fully:

N_Edd = 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296.

The 'problem' with this number is that it is to difficult to 'digest'. How it is possible that with the incipient science methods we have, an the 'unelaborated' artifacts, instruments, and appliances we have to explore our surroundings, can anybody come to tell us the exact count of protons in the universe?

Read the full article for more info and rants about how other scientists and writers have dealt with the problem of big numbers.

Alice in Wonderland

2009-02-26T16:52:00.000-08:00

Is the girl shown here in this oil paint Alice Liddell, the main character of Alice in Wonderland, the all-time hit in the children and adult's literature?

(Picture: One Summer's Day by William Brymner - 1884)

Maybe not, but the picture title and the walking little girl are evocative of the dedication that Lewis Carroll made for Alice in the book he wrote for Alice Liddell, which at this time he titled: Alice's Adventures Under Ground.

In fact, Lewis Carroll wrote the following dedication in the manuscript he handled to Alicia in November of 1864:

A Christmas Gift
to Dear Child
in Memory
of a Summer Day

Now you can download Alice in Wonderland in EBook format from Datum.

This masterpiece of the children's literature is more than a fantastic narrative from the imagination of a prolific writer, photographer, and mathematician. Alice in Wonderland is also the product of an epoch in the search for new horizons in the science.

Previous to Alice in Wonderland, Ludvig Holberg, near a century before, in 1741, wrote an adventure story about a character that goes down a cave to explore the underground world: Niel’s Klim’s Underground Travels. And in 1692, Edmund Halley, a British astronomer and mathematician, put forth the idea of a hollow Earth when he tried to explain the deviations of the magnetic field of the Earth,

Not to mention that Athanasius Kircher, in 1664 published a geological and geographical investigation that culminated with his Mundus Subterraneus (Subterranean World) in which he suggested that the ocean tides were caused by water moving to and from a subterranean ocean.

Almost simultaneously with Alice’s Adventures Under Ground, in France, Jules Verne, in 1864, published his Journey to the Interior of the Earth.

So, the idea of a “habitable” underground was not new to the fiction and fantastic literature writers, and possibly Lewis Carroll was related with some of those published works.

Besides the historical background, Alice's Adventures In Wonderland is a delightful reading, specially if it is fully illustrated as this free Datum edition in EBook format.

Download both books for free --Gratis: Alice In Wonderland and Niels Klim's Adventures Under The Ground. Click on the cover of the EBooks!

Do random numbers really exist?

2009-01-28T18:04:00.000-08:00

While I was collecting information for this month's post, I was also looking for new material for a new free E-Book to compile for my readers. The idea of the so-called random numbers sprung into my mind, so I began to search for this topic.

I found an interesting book review titled: A Million Random Digits with 100,000 Normal Deviates. The book was originally published in 1955 by the RAND Corporation, so the "review" was a little late, but its OK; the author was "reviewing" one of the oldest books in his library. This book can be found and read at Google Book Search.

Doing a deeper search I also found another article and an E-Mail by Mr. Nathan Kennedy complaining to the RAND Corp about their stand that the One Million Random Digits table was of their property and that it cannot be redistributed on the Internet. By a great coincidence, my intention was the same as Mr. Kennedy's, and his intention of putting the table on the Internet was the same as mine. However, and in a great unselfish gesture, Mr. Kennedy generated his own table of random digits and proceeded to place it on the Internet for free as a text file on the same page.

Since he authorized the use of his million digits table, I reformatted the text file as a PDF file, designed a cover page, wrote a small introduction for it, and made an E-Book, to be distributed also for free.

I am not a statistician, so maybe I will never find a practical use for this kind of numerical table, however, random number are of interest for me, and probably for many others, for the degree of strangeness they bear.

Are there really random numbers, or there are random events?

Can we really speak of random "numbers"? Isn't it more appropriate to speak of random "events"? Can we make at least some arithmetic operations with them? Can we add two RNs and still say that the sum is also "random"? Can we multiply them to obtain -without doubt- that the result is also random?

Note that the RNs are not obtained by any formula, or equation, or matrix, or any predefined mathematical operation; they are mainly obtained by algorithms fed by some "physical" events like atmospheric variations, radioactive decay, thermal processes, or the like. Hence, what we are doing is using unpredictable physical events, assign to each "event" a number, and say that this is a RN. Another nonphysical source of random sequences of digits (but sometimes questioned) is by selecting digits or portions of digits of the decimal expression of irrational or transcendental numbers.

But without physical events, can we still generate RNs? There are some rudimentary approaches, but they are mostly mere mathematical curiosities.

The interested reader can find more authoritative articles at Random.org where he/she can obtain instant (real-time) random numbers for lottery, cards, passwords, etc.

Read also another article related with this E-Book at 4DLab.info and find more E-Books to download.

Paradoxes of the infinite

2008-12-30T07:00:00.000-08:00

Can an "object" be finite and infinite at the same time? Contrary to what our intuition dictates, it seems that this duality can arise in mathematics.

We are used to think that the "infinite" is a well defined concept like some of our everyday ideas of "here", "there", etc. But, if that were the case, we would not have so many paradoxes arising from this field of science that Gauss referred to as the "queen of sciences".

Of course, George Cantor did a great contribution fixing the traditional and loose understanding of the infinite, specially introducing a scale of infinitudes when he proved that there are more than one kind of infinite. Since then, many other mathematicians and philosophers had been kept busy untangling some paradoxes that the new scale of the infinite had brought.

Among them, Jorge Luis Borges surfaced and worked literally with great success the problem of random infinite series without a first and a last term.

To learn about Borges' unusual understanding of the infinite follow this series of posts beginning with the post Nobody understand the infinite so well as Borges.

To learn about the unusual paradox of how can an "object" can be finite and infinite at the same time follow this article Top myths about the infinite about Torricelli's trumpet also called the Gabriel's horn.

▀ Is The Book of Sand a book from the fourth dimension?

2008-11-01T07:33:00.000-07:00

This is the fourth article in the series about the popular short story The Book of Sand of Jorge Luis Borges.

I finished the previous article with the following paragraph:

"Despite of this book being infinite, as Borges admits (and transfinite —according to my interpretation) he also suggests that it can be equally finite. Is this possible?"

The goal of the current article is to show that the book Borges bought from a Bible salesman was a mysterious object that was more than infinite in content: it was also a multidimensional and extra-real object, a supernatural book that was capable of existing in several dimensions, and being finite and infinite at the same time.

The Book of Sand is a hyperbook

What are some of the hints that Borges give us that point toward the book extending to other dimensions beyond our 3-dimensioned space? There are two:

At the beginning of the story, Borges mentions his initial intention of starting his story as a geometrical plot. We see this at the very first paragraph where he writes: "The line is made up of an infinite number of points, ... , the hypervolume of an infinite number of volumes". And then proceeds "No, unquestionably this is not —more geometrico— the best way of beginning my story". With this brief introduction he follows the geometrical tradition of Charles Hinton (1853 - 1907) with his books and essays about the fourth dimension like What is the Fourth Dimension, and Scientific Romances, and Edwin Abbott Abbott (1838 - 1926) with his Flatland: a Romance of Many Dimensions —a journey to the 1 and 2-dimensional space and a journey to the fourth dimension. Borges, like them, constructs the dimensioned space as a series or set of infinite points, planes, and volumes, but nowhere else in the story he makes a reference again to the concept of dimensionality. However, and in analogy with those concepts, we will shortly see that he implicitly assumes that a hyperbook can be made up of infinitely many books.

But, what is a hyperbook? Let us first review what is hypercube. Similarly to the line, the plane, and the cube, a hypercube is a cube that is the result of moving or dragging our ordinary cube to the 4th-dimension. But where is the fourth dimension? Although difficult to imagine, the fourth dimension is supposed to be "perpendicular" to our ordinary three dimensions. We (as Borges) arrive at this conclusion by simple induction: the plane is generated by moving a straight line perpendicularly to its length; a solid is obtained by moving a plane perpendicular to its surface. Note that in each case the motion is perpendicular to the dimensions of the previous object.

The hypercube is similarly obtained: by moving a cube perpendicular to the three dimensions it has. Visualizing the hypercube is not an easy task because we cannot imagine where is a dimension that is perpendicular to our daily 3-dimensional world. Note that according with the previous assumptions a plane is a "hyperline", and a cube is a "hyperplane". Furthermore, even a cube can be considered as a "hyperhyperline", but since this concept is little more than meaningless, we can plainly say that the cube is also a "hyperline".

Returning to our hyperbook, let us simplify the shape of the book assuming that it is like a cube (a book can be cubical). Then, a hyperbook is a 4-dimensioned book such that any reduction of its 4-hyperspace to a 3-dimensioned space results in an ordinary book.

Mathematicians not only speak of hyperspaces an hypercubes, but also of hyperspheres, so Borges —who was also related with modern mathematics— simply extended this concept to the books: if mathematicians could conceptualize such hyperobjects, for him was also very easy to conceptualize the hyperbooks.

Attributing to Borges the idea that The Book of Sand is a hyperbook —and not a simple solid— is not a sound reasoning if we cling to the above arguments alone. But there is another passage in his story that reinforces my argument that for him The Book of Sand is not a "mere" infinite book: it is also a book from other dimensions.

Let's quote again the passages to where I'll make reference:

I turned the leaf; it was numbered with eight digits. It also bore a small illustration, like the kind used in dictionaries —an anchor drawn with pen and ink, as if by a schoolboy's clumsy hand.

It was at this point that the stranger said. "Look at the illustration closely. You will never see it again."

Borges noted the place and closed the book, but once he reopened it he never found again the illustration of the anchor.

However, later he found another illustration: a mask. But there was a curiosity among them and he explicitly narrates it for us:

The small illustrations, I verified, came two thousand pages apart.

A book with illustrations every 2000 pages? Take notice that he is not saying that the next illustration is so far: he is saying that all illustrations are so far apart. He is not explicitly referring to those two illustrations, the anchor and the mask; so we can safety assume that he is writing about all the figures of the book.
Isn't this crazy? He says he verified this fact of the illustrations separation, but what type of book of earth has this particularity? Why exactly 2000 pages apart?

Borges used an alphabetical notebook to record the pictures he found:

I set about listing them alphabetically in a notebook, which I was not long in filling up.

The book was somehow full of illustrations because —as he says — it didn't take him too much time to fill the notebook, despite "Never once was an illustration repeated".

The Book of Sand is a dictionary

Remember the quote: "... a small illustration, like the kind used in dictionaries"? From this quote we obtain the second hint: The Book of Sand is a dictionary! This marvelous book —this hyperbook— is a dictionary because:

It never repeats an illustration: every instance of a "book of sand" is just a definition of an object! This is what dictionaries do: they show a single small plate or diagram and then a short or long explanation of what is this object. Every time the book is reopened, a new random definition with its corresponding pictogram appears.
Only a single picture picture per book "instance" appears. Every time he opened the book the whole book is dedicated to the description of the picture he found. Sadly, he didn't understand the accompanying definition definition and prose because "The script was strange to me".

The dictionary was unreadable for Borges because it was written in a foreign language or dialect: "It seems to be a version of the Scriptures in some Indian language, is it not?" he asked to the salesman, but the answer was "No ... I acquired the book in a town out on the plain in exchange for a handful of rupees and a Bible. Its owner did not know how to read. I suspect that he saw the Book of Books as a talisman".

The Book of Sand is a dictionary: what a surprise! No wonder the figures were 2000 pages apart; each picture definition was 2000 pages long so that each "copy" or "instance" of the Book of Sand is dedicated to one particular object or thing. Now we understand why the figures were never repeated: because each "book of sand" consisted of a 2000-page long definition.

The Book of Sand as an object of parallel universes

Is the possibility of multiple instances of the same object in multiple dimensions coexisting together an insane idea? Not at all, because with the triumph of the modern physics the concept of multiverses is just one of the many hard ideas to digest.

But, how and why so many "instances" could occupy the same time-space in such a manner that the book could be held in Borges' hands? The answer is simple: remember that a hypercube is a cube surrounded by cubes in every possible dimension; therefore, in a similar manner, The Book of Sand is a hyperdictionary surrounded by dictionaries in every possible dimensions. In this way, every time Borges opened the book he was opening a dictionary in other dimensions. This is the reason why he could hold a multidimensional infinite dictionary in his hands: he was holding only a 3-dimensional instance of and infinite-dimensional dictionary. All other "copies" or chapters, or "definitions" or "instances" were in very near dimensions: just touching the "real" one, but inaccessible at the same time, as the the figure at right shows.

The notion of parallel universes is not more insane than the notion of a single an unique universe; both extremes are hard to understand. I leave the reader with two simple questions related with the parallel universes idea: What law of physics states that there should be a unique 3-dimensioned space? What law of physics states that there cannot exist more than three dimensions?

Articles in this series
1 - ▀ Nobody understands the infinite so well as Borges
2 - ▀ The Book of Sand of Borges and the Continuum of Cantor
3 - ▀ The Book of Sand is a Transfinite Book
4 - ▀ Is The Book of Sand a book from the fourth dimension?

The 10 top myths about the infinite

2008-10-17T06:18:00.000-07:00

Myth 1: An infinite split by one half is no longer infinite

Let's us take the set of all natural numbers, i.e., the numbers we use to count, like 1, 2, 3, ... We will represent this set by the symbol Z. Each one of the natural numbers is either odd or even; the odd numbers being 1, 3, 5, ... and the even numbers 2, 4, 6, ... Note that the numbers we call even are those divisible by 2. Hence every natural number is either divisible by two or not. Those that are not divisible by 2 are the odd numbers.

Natural numbers = odd numbers + even number

Z = {1, 2, 3, 4, 5, 6, 7, ...} = {1, 3, 5, ...} + {2, 4, 6, ...}

The even numbers are infinite because there is no end to this series. Same with the set of odd numbers: there is no way to find and end to this series. So the infinite set of all natural numbers is the sum of two infinite series; the series of the odd numbers plus the set of the even numbers.

If you take away the infinite set of the even numbers from the infinite set of the natural numbers you are left with an infinitude of odd numbers.

{1, 3, 5, ...} = {1, 2, 3, 4, 5, 6, 7, ...} - {2, 4, 6, ...}

To a similar behavior we are faced if we take away the set of the odd numbers from the set Z.

Therefore, it is not necessarily true that if we split an infinitude in a half, the two parts are no longer infinite.

Myth 2: One infinite added to another infinite is a greater infinite

This one is the opposite of the above myth.

Myth 3: If we increasingly take away infinite elements from an infinite set, eventually, the remaining set is no longer infinite

This is not the same as Myth 1: there we were linearly taking away one integer for each one left.

Suppose that to the set of all natural numbers Z we remove numbers from it using this pattern:

Leave the number 1, but take away the next 2. We are left with {1, 4, 5, 6, ...}

Leave the number 4, but take away the next 5. We are left with {1, 4, 10, 11, ...}

Leave the number 10, but take away the next 11. We are left with {1, 4, 10, 22, ...}

Repeat the pattern over and over again.

Note that with each step we are taking more an more elements away from the original set of the natural numbers. The separation between the remaining integers is wider and wider. If we repeat this process indefinitely, we'll be progressively removing more an more elements. This is far from the first example above where we were removing even or odd numbers only, because in this schema we are removing from both types of numbers.

However, no matter how far we go or how many integers we remove, the remaining set will be always infinite because although the steps are infinite, the elements to be removed are always finite.

Myth 4: There are more fractions than natural numbers

This assertion might appear to be against our intuition because we assume that since every natural number can be expressed as a fraction, like

1 = 1/1,
2= 2/1 = 2/2,
3 = 3/1 = 6/2 = 9/3 ...

we can conclude that there are more ways of expressing fractions than the numbers themselves. However, note that in the pyramidal scheme above, we can count the fractions as follows:

1/1 = is the first
2/1 = is the second, 2/2 is the third
3/1 = is the fourth, 6/2 is the fifth, 9/3 is the sixth,

Hence, no integral fraction can escape our counting scheme. Therefore, the integral fractions are countable which means that there are not more integral fractions than natural numbers.

Myth 5: An infinitude of elements multiplied by another infinitude is always a grater infinitude

Myth 6: Since every fraction can be converted to a decimal then there are as my decimals as fractions

Myth 7: The segment of line from 0 to 1 contains double the points as the segment from 0 to 1/2

Myth 8: The number of grains of sand is infinite.

This is a classic myth. Probably all of us, at some stage of our live, had think that the grains of sands are infinite.

Archimedes is the first documented one to tackle down the needed mathematics to show that it is impossible the for the sand to be infinite. Strictly speaking, what he showed was that we can count how many grains can a universe hold, no matter how big it is.

At his time the observable universe was up to Saturn, so what he did was to compute how many grains can fill a sphere the size of the orbit of Saturn. The mathematics needed to arrive at his conclusion were simple, but ingenuous extensions he devised for the arithmetic of his time was an enormous contribution.

You can download his all-time famous book The Sand Reckoner here.

Myth 9: If a vase is infinitely long, then it must have infinite capacity

This is a beautiful one

Gabriel's Horn

Myth 10: If there were infinite universes out there, in some of them, or at least in one, should be an exact copy of our planet Earth

▀ The Book of Sand is a transfinite book

2008-10-06T18:12:00.000-07:00

In my preceding article The Book of Sand of Borges and the Continuum of Cantor I wrote about the manifest similarities of the appreciations of the concept of infinitude between the literature writer and Argentinian academic Jorge Luis Borges and the German mathematician George Cantor. The article was at the same time a continuation of: Nobody understands the infinite so well as Borges, which is the article number one in this series.

To understand the three articles the reader is encouraged to be related with Borges’ short story The Book of Sand, or read the articles in sequence.

Is The Book of Sand exactly one book, or a book that reshuffles itself every time it is opened?

All the discussion that follows from here to the end of the current article relies on a personal interpretation of the short story The Book of Sand. For me, the book renovates itself every time somebody opens it. Let me quote again the segments of the story that leads to my interpretation. First Borges finds an illustration in the book:

...I turned the leaf; it was numbered with eight digits. It also bore a small illustration, like the kind used in dictionaries —an anchor drawn with pen and ink...

Then the stranger warns him about the infinitude of the book:

It was at this point that the stranger said: "Look at the illustration closely. You'll never see it again".

Borges challenges the vendor by marking the illustration and tries to find it again:

I noted my place and closed the book. At once, I reopened it. Page by page, in vain, I looked for the illustration of the anchor...

the illustration disappeared, or at least he didn't find it. Based on this words is that in my interpretation, the book Borges that bought was not unique in its composition of pages: it was a book that in some mysterious way regenerated itself every time it was reopened. I call every reopening an instance of the book, so each instance is another "book of sand". That explains why in one instance he sees an illustration, and in another instance (another reopening, another self-reshuffling) the illustration was not in its previous place. That is, the illustration of the anchor belonged to and ephemeral instance of all the possible instances of the infinite and incredible Book of Sand.

So for me, the book was infinite in pages and at the same time was an infinitude of books all of them packed into a single one. That multiplicity of infinitudes is the ground upon which we will build and prove the assertion that the Book of Sand is something more than an infinite book: it is a transfinite book.

Relations between two infinite sets

I suggested in the preceding article that Borges’ imaginary book is more than an infinite book: The Book of Sand is a transfinite book. Transfiniteness —a concept introduced to modern math by Cantor— was briefly described in that article. It was also stated in this article that in order to prove that The Book of Sand is a Transfinite book, we must find a function that could establish correspondence between all the possible "books of sand" and the Continuum of the real numbers, and another function that could establish unique correspondence between elements of the Continuum and a corresponding unique "books of sand".

The following drawing, taken from the previous article, will be reused to refresh our quest for mapping between two sets of objects. We will focus on the possible existence of two relations A and B between the two infinite collections of our study. We are heading to prove that there exist such A and B relations, although they need not be necessarily reciprocals.

The point here is the following: we can establish a two-way (1-1) correspondence between every natural number and every even number: simply double every natural number and we'll have even numbers, or simply half any even number an we'll obtain a natural number. In this simple example, the relation of doubling and the relation of halving are mutually reciprocal. In this example, the relation of doubling is (1-1) and the relation of halving is also (1-1).

Let us now take the squaring relation for the natural numbers; in this case for each natural number x there exist one and only one number x². The number 2 when squared produces the number 4, and the number -2 when squared also produces the number 4, but note that squaring never produces two different result at the same time. On the other hand, the “reciprocal” function of the squaring function, the square root function, is not (1-1). The function f(x) = x² is a map that assigns to any number x its unique square x², but its inverse function, the one that assigns the square root to a natural number is not unique, because, as we have seen, for example, 4 has two the square roots: +2, and –2; therefore the square root mapping is not unique. To accomplish some (1-1) mappings we must split this function in two separate (1-1) maps: g(x) = +√(x) and h(x) = –√(x).

So the noteworthy fact about mappings, relations and functions (they are loosely synonymous) is that they produce unique results. What we want is, for example, that if are to compute the area of a square or a rectangle, the result be one and only one unique number. It doesn't matter we are working with finite or infinite sets of objects or numbers; the mapping rules that apply are the always the same: the uniqueness of the result.

With this ultra-brief introduction to mappings, let us begin our task of mapping the set of all possible "books of sand" to the set of all decimals in the number line. To achieve this goal we will divide our arguments in two parts:

Part A is dedicated to prove that there exist an A-relation, where to every "book of sand" we can associate a unique decimal of a chosen subset of the real numbers, and

Part B is dedicated to prove that there exist a B-relation, where to every decimal of a chosen subset of the real numbers we can associate a unique "book of sand".

Part A. For every instance of the Book of Sand there is a unique decimal in the Continuum

To Cantor and his Theory of Sets, the Continuum is the dense and compact set of all real numbers. But there are also many instances in which Continuum are also some special subsets of all decimals such as the decimals between 0.1 and 1.0. In this "little" interval of decimals are the real numbers: 0.1297067, 0.96239..., and so on. In passing we must note that the number 1.0 and the decimal 0.999 ... are equivalent: 1.0 is just another way to write the decimal 0.999 ... This fact is simply demonstrated by adding 1.0 plus 0.999... and dividing by 2 to find which number is between both numbers. Hence, it is of enormous help to work with the Continuum between 0.1 and 1.0 instead of working with all the real numbers together.

To explore the relationship between the the Book of Sand Borges and the Continuum of Cantor suppose that we label by italicized letters as for example s₁, s₂, ... each sequence numbering of our "thought experiment" (see previous article) suggested above. So, among the possibilities of series of pages we have,

s₁ = {... 40514, 999, ... } (The example of pages given by Borges)
s₂ = {... 280, 45, ... } (Any other instance of the book)

Let’s join together all the digits of the pages to make infinitely large numbers, as follows:

n₁ = 40514999 ...
n₂ = 28045 ...

Now we turn those integers into decimals and label the possible decimals by d₁, d₂, ... by simply adding a decimal point to the left, like this:

d₁ = 0.40514999 ...
d₁ = 0.28045 ...

So to the sequences of pages s₁ of the pages ..., 40514, 999, ... we are assigning the decimal d₁ = 0.40514999 ..., and to the sequence s₂ of the pages ..., 280, 45, ... corresponds the decimal d₂ = 0.28045 ...

Have we demonstrated that there is a unique relationship A from the set of all "books of sands" to the decimals as we specified in the diagram? Can we apply a label decimal between 0.1 and 1.0 for every possible sequence numbering of the pages of the Book of Sand by just following the rules specified above?

Unfortunately, no. No matter how convincing the rules may look, those rules are not enough to produce unique mappings. To see why, consider the following different set of pages

t₁ = {... 40, 51, 4999, ... } (Any other instance of the book)
t₂ = {... 2804, 5, ... } (Any other instance of the book).

Note that under the rules above they also produce the same results

g₁ = 0.40514999 ...
g₂ = 0.28045 ...

That is, s₁ ≠ t_1, but d₁ = g₁. Similarly, s₂ ≠ t₂ but d₂ = g₂. Clearly, the rules we are attempting do not produce or generate unique results.

We must seek, then, a special rule to give a series of pages really unique decimal numbers only.

Two important properties of the prime numbers

We can use two fundamental properties of the prime numbers to construct the mapping we need to associate to each Book of Sand a unique decimal. The properties are:

1. For each natural number there is exactly one unique prime number.
2. Each composite natural number can only be decomposed in only one set of prime factors.

What states property number one is that the prime numbers are infinite. As a short review of the prime numbers, the first prime is 2, the second prime is 3, the prime number 20 is 71 and so on. We denote the series of prime numbers by lowercase letter p’s as follows:

p₁ = 2, p₂ = 3, ..., p₂₀ = 71, ...

When we do not have a Table of of Primes at hand, one good resource for finding the n-th prime number is The Nth Prime Page. A prime page by Andrew Booker. For the actual computations below this online server was used.

What states property number two is that the numbers that are not primes, like 4, 6, 9, etc. can only be factored in unique sets of prime numbers; for example, the number 220 = 2 × 2 × 5 × 11, but there is no other way to factor 220, there are no other prime numbers which multiplied give the same result 220.

Using the prime numbers

Returning to the example of the page numbers in the first sequence s₁ let us multiply the prime number in the position 40514 by the prime number in the position 999, and so on. Since now we are dealing with a new map, then we also obtain new numbers n₁ and n₂, as follows:

n₁ = p₄₀₅₄₁ × p₉₉₉ × ... for the sequence s₁

and for the sequence s₂

n₂ = p₂₈₀ × p₄₅ × ....

Now, by property number two of the prime numbers, both numbers n₁ and n₂ are unique and different from each other. That is, under this new assignment, different page sequences generate different prime numbers multiplications.

The steps we now take are directed to obtain unique decimals from those unique prime numbers multiplications.

Since p₄₀₅₄₁ = 487183 and p₉₉₉ = 9707 then n₁ = 487183 × 9707 × ... i.e.

n₁ = 4729085381 × ...

On the other hand, for the number n₂, we have

n₂ = 1811 × 197 × ... = 356767 × ...

given that p₂₈₀ = 1811 and p₄₅ = 197.

But what we’ll have for the sequences t₁ and t₂ in the examples above? Well, since t₁ = {... 40, 51, 4999, ... } and t₂ = {... 2804, 5, ... } etc., then

N₁ = p₄₀ × p₅₁ × p₄₉₉₉ × ..., and

N₂ = p₂₈₀₄ × p₅ × ...

Clearly, n₁ is not equal to N₁ and n₂ is not equal to N₂, even when the digits of the set s₁ are the same digits used in t₁, and the digits of the set s₂ are the same digits used in t₁. This is due to property 2 of the prime numbers above: the multiplication of different primes necessarily produce different results.

Continuing with this part of the proof, let us now denote the digits of the number N₁ by d₁, d₂, ... and the digits of the number N₂ by D₁, D₂, etc. Then

N₁ = d₁ d₂ d₃ ... and N₂ = D₁ D₂ D₃ ...

As an example, for the random number 3735, d₁ = 2, d₂ = 7, d₃ = 3 and d₄ = 5.

Therefore, for the sets s₁ and s₂, we have

The sequence s₁ = {... 40514, 999, ... } can be uniquely mapped to the decimal 0. 587576...., and
the sequence s₂ = {... 280, 45, ... } can be uniquely mapped to the decimal 0.75476....

That is, we have managed to produce a mapping from every instance of the "book of sand" to a uniuqe decimal in the interval 0.1 to 1.0.

Wrapping up the rules of this mapping

How can we assign a unique decimal to a given Book of Sand if there is never a first page, and consequently, never a first factor to find N₁ or N₂?

Borges mentions that the Book of Sand had no first and last page, that’s the reason why we write each sequence of pages beginning and ending with an ellipses. Then how can we deal with the product of the elements of a sequence of numbers that has no beginning and no end? There may exist a page with the number 1 on it, but that doesn’t necessarily constitute being its first page.

Since the multiplication of the numbers is commutative, the order in which we write the pages sequence is irrelevant for the final result. Therefore, we can equally write the elements omitting the first ellipsis as shown below

s₁ = { ... 40514, 999, ... } = { 40514, 999, ... }, and

s₂ = { ...280, 45, ... } = { 280, 45, ... }.

Therefore

s₁ = {40514, 999, ... } -------> 0. 587576...., and

s₂ = {280, 45, ... } -------> 0.75476....

In this way we have managed to associate with each Book of Sand a single and unique decimal in the segment of real numbers from 0.1 to 1.0.

Now we summarize in symbols the function we worked.

Let S_b be any sequence of page numbers of some Book of Sand with elements e₁, e₂, e₃ ... etc. that is, S_b = { e₁, e₂, e₃ ... }. Note that this is equivalent to say that the page numbers are e₁, e₂, e₃ ...

Let N_b be the product of all the primes with positions p_e1, p_e2, p_e3, etc. That is,

N_b = p_e1 × p_e2 × p_e3 × ...

Let D_b be the decimal obtained when we add a decimal point in front of N_b. Suppose the decimal digits of D_b be the digits d₁, d₂, d₃, etc. Then, the final function F that assigns a random Book of Sand S_b to a decimal in the Continuum is:

F (S_b) = 0.d₁ d₂ d₃ ....

In mathematical terms, the set of all "books sand" is called the domain of the function F, and the set of all decimals obtained under F is called the range of the function. In order for a function between a domain and a range to have mathematical significance, it can be one-to-one or many-to-one but not one-to-many.

Under the mapping we are dealing with, any sequence S_b of the Book of Sand can also be called a pre-image or an element of the domain of all the possible copies of the Book of Sand, and its corresponding decimal D_b is also called an image.

Some thoughts about this mapping

Note that with this function we can assign any possible Book of Sand to a unique decimal between 0.1 and 1.0. However, some decimals —in fact, infinitely many decimals— will remain without its copy of a Book of Sand, as for example, the decimal 0.385. Why this? Is it permissible?

If every Book of Sand has infinitely many pages, then under the function F, any book F (S_b) must have an associated number N_b = p_(e1) × p_(e2) × p_(e3) × ... of infinitely many prime factors. For this reason, any ending decimal will NEVER be an instance of the Book of Sand.

However, generating every decimal of the Continuum between 0.1 and 1.0 is not a requisite for the function F (S_b) to be a valid mapping. The requisite is that for every instance of a Book of Sand a different decimal be generated.

Part B. For every nonrational decimal between 0.1 and 1.0 there is a different Book of Sand

We have gone past the first part of the proof, now we must now prove Part B of the chart, that is, that for every nonrational decimal between 0.1 and 1.0, there is a sequence of numbers of pages related in the Book of Sand. Now we are going to demonstrate that for ever nonrepeating decimal in the range 0.1 to 1.0 we can define a function that assigns an instance, that is, a different and unique "book of sand" to that decimal.

Take any decimal between 0.1 and 1.0, for example, 0.52826971068507 ... Let us convert it into a whole number by eliminating the decimal point and then gradually subdivide its digits in groups of 1-digit, 2-digits, 3-digits, and so on like this: [5] [28] [269] [7106] [85907] ....

That to that this decimal 0.52826971068507 ... it corresponds an instance, a "book of sand" with pages 5, 28, 269, 7106, 85907, and so on.

But, what would happen if one or more of the “pages” begin with zero, for example, [5], [28], [028], [0005], ... Which instance would be the associated for the decimal 0.5280280005 ...?

This case warns us that we must improve the correspondence between the decimals and the instances of the Book of Sand. One way to improve the mapping it is to add one or more digits corresponding to the place where they make the partition of pages.

This is the way it will work for the decimal in the example: 5 is the first partition, 28 the second, 028 the third, and so on. We will then have, that to the decimal 0.5280280005 ... now corresponds the book with pages [15], [228], [3028], [40005], ...

This is a solution that guarantees us that no page number begins with a zero.

Now we state in symbols how this map behaves.

Let d_c be a nonrational decimal chosen between 0.1 and 1.0. Let d₁, d₂, d₃, ... be the digits of this decimal. Let n_i be natural numbers made by the digits of this decimal as follows:

n₁ = 10¹ + d₁

n₂ = 10² + d₂ × 10 + d₃

n₃ = 10³ + d₄ × 10² + d₅ × 10 + d₆

Therefore, to the decimal d_c made of the decimals d₁, d₂, d₃, ... corresponds the set made up of the above sequences: S_n ={[n₁], [n₂], [n₃], ...} Each one of the numbers enclosed in brackets corresponds to a page numbered by the number within the brackets.

We can denote this mapping from the decimals to the instances of "the book of sand" by the symbol G(d_c) = S_n.

Recalling the example above, to the decimal 0.5280280005 ... corresponds the book with pages [15], [228], [3028], [40005], ... This mapping between the decimals and the instances of the "books of sand" produces unique numbering for each one of the "books of sand".

Finally, what about the decimals ending in zero or the repeating decimals? What Book of Sand will we assign to the decimal 0.20000.... or to the decimal 0.33333...? Simple, you just don’t take it into account because the Continuum without the decimals ending in zero (the rational decimals) and the Continuum without the repeating decimals (fractional decimals) is still a Continuum. So, if we chose to take as domain the real numbers omitting the rational numbers and omitting the repeating decimals, we still have a Continuum as domain.

The mappings F and G are not mutually reciprocal, that is, the mapping G that assigns a "book of sand" to a given decimal is not the inverse of the mapping F that assigns a decimal to a given "book of sand". If that were the case they would be called one-to-one (1-1) mappings. However, the mappings need not be reciprocals, what is needed is already satisfied: for each element of the domain of each of the functions there corresponds unequivocal images on both mappings.

We have shown the correlation that exists between the Book of Sand and the Continuum of decimals, but the Continuum is more than simply another term for the infinite. The Continuum is the Transfinite, and by Transfinite we understand the infinite that it is not correlated with the infinite we normally use and understand.

That is, the "thought experiment" that we started at the beginning of the second article consisting in enumerating all the possible combinations of pages of the Book of Sand, is not possible to be carried out NEVER for four reasons:

1. We do not have the TIME for all combinations.
2. Even if we have the time, the combinations are just more than infinite.
3. The combinations are TRANSFINITE, so that NEVER can be completed even having an infinite time.
4. The nature of the Continuum and at the nature of the Book of Sand are such that not even passing an infinite number of pages per second and even having an infinite number of seconds we will succeed in all possible combinations of the Book of Sand.

A surprise for Borges and Cantor

Borges knew that the book he had in his hands was infinite; it was “an impossible book”, a “monstrous book” as he describes it. A book with no beginning page and no ending page what else could it be? He though of burning the books for the many nightmares the book had caused him, but “I feared that the burning of an infinite book might likewise prove infinite and suffocate the planet with smoke”.

But we have seen that the book he had in his hands was much more than being and infinite book: it was a Transfinite book. Trying to destroy it will have consequences, no only for the planet but for the whole universe in every possible time and dimension.

On the other hand, Cantor discovered that the decimal numbers in the real line are much more than the simple infinitude of the natural numbers. He discovered that there are infinites that cannot be paired with other infinites in any way.

Cantor created the Continuum and the Transfinite and Borges found an application for them. Maybe, in an unknown space and unfamiliar dimension, they might be talking about the coincidence of the Continuum and the Book of Sand as we are doing it here.

Despite of this book being infinite, as Borges admits (and transfinite —according to my interpretation) he also suggests that it can be equally finite. Is this possible?

This will be seen in the next post.

▀ The Book of Sand of Borges and the Continuum of Cantor

2008-09-28T20:55:00.000-07:00

The previous article

In the previous article entitled Nobody understands the infinite better than Borges I began the presentation of the short story The Book of Sand of the authorship of Jorge Luis Borges. In the story, The Book of Sand is a book that Borges acquired from a seller of books and Bibles that has the particularity that each time you open the book in one page, the page number changes, and the next page number is also changed. The book never had a first and last defined pages, and as if that were not enough, the text and illustrations never appear twice in the same place or on the same page.

The article ended like this:

Imagine that you open that "devilish book" and by some unknown power you can write the sequence of the page numbering as it appears page by page. Now close the book and reopen it again and repeat the process again. You will "finally" obtain all possible orderings of the natural numbers. I cannot show it here, but is possible to prove that your "list" of all possible orderings of the natural numbers is not countable, not even infinitely countable. Thus opening an closing The Book of Sand is an act of delving into the Continuum, an action that possibly Cantor never though of.

In the present article, which we can consider as a continuation of the one above, we will delve into an informal mathematical proof to show that the Book of Sand of Borges is much more than just an infinite book: it is a Transfinite book.

The infinitudes of Cantor

George Cantor conducted an extensive and deep research on the categories of the infinitude; something that was new to the mathematicians of his time. Prior to him it was assumed that all infinitudes were equal.

If we start from the series of the natural numbers 1, 2, 3, ... we say they are infinite; in fact, this series is the infinite series for excellence for its simplicity. But the even numbers series 2, 4, 6, ... it is also infinite, even when they seem to represent the "half" of the natural numbers. Another infinite set of integers is the set of prime numbers, despite the fact that as they progress the "distance" between them is widening. These sets were begun to be called "countably infinite" not because they were exactly "countable" but because they can be related in a one-to-one (1-1) relationship with the natural numbers we use to count.

More surprising is the fact that it can be shown that the fractions are also infinitely countable. Unexpectedly, with the proper arrangement of the elements, all fractions, such as 1/2, 4 /5, 458/245, ... can be infinitely listed, or counted as the first fraction, the second fraction, the third fraction, and so on.

Findings like these led to Cantor to deepen into the concept of "infinity" as we use it daily. Under a lot of opposition and humiliation, Cantor managed to formalize and establish the Theory of Sets, and the arithmetic of the infinites as a strong and indispensable field in the science of mathematics.

One of his sensational findings was demonstrating that there are infinite sets that are higher than the infinite set of natural numbers. There is no way to establish a one-to-one (1-1) correspondence between the natural numbers with those infinites, therefore, we must invent new categories for certain sets. These sets were called by Cantor Transfinite; infinites way beyond the infinity of the natural numbers.

The simplest example of transfinite sets are the set of the real numbers. Real numbers comprise those we commonly call decimal numbers and the infinite decimals that we never can end writing because they are

either "irrationals" as the square root of 2, commonly symbolized as √(2) = 1.414213562373...,
or they are transcendental numbers like the π =3.141592653589...

The discovery was unforeseen because it was not expected that the amount of irrational numbers was "so big" as to need to coin a new term and concept to study them. Note the reader that the contribution of the finite decimal numbers is almost null because the finite decimals can be expressed as fractions and we already mentioned that the fractions are "countable". For example, since 0.500... = 1/2, 0.333... = 1/3, etc. we can take away all those decimals out of the real line and still the remaining reals are transfinite.

This new set of numbers that can not be "counted" with the set of all natural numbers was the one to be known as The Continuum.

The properties of the Continuum are incredible: in the same way as the set of all even numbers is infinite, despite being a subset of the natural numbers, there are also ways to create subsets of real numbers that are equally transfinite as the set of all the real numbers itself. For example, the "small" line segment between the decimal 0.1 and the decimal 1.0 contains a transfinite number of real numbers. To this segment, later, we will give a good use with the Book of Sand of Borges.

The Book of Sand is a Transfinite book

After this brief digression of going deeper into what are the real numbers, in view of the fact that we will need them later, let us now return to the Book of Sand and make a "thought experiment" with it. Recall that the main peculiarity of this enigmatic and esoteric book was that the pages were randomly numerated and that it lacked a fixed first and last page because there were pages popping out of the nothingness at the beginning and at the end of the book.

Now imagine that you open this "devilish book" —as the Bible seller told Borges— and that for some unknown power you are able to write the sequence of the page numbers as they appear page by page. Now close the book and open it again and repeat the same process continuously. If you repeat this without stopping you will "finally" get all the possible combinations of sequences of pages of The Book of Sand.

What relationship exists between all the possible sequences of The Book of Sand of Borges and The Continuum of Cantor? Well, let's see if it is possible to establish unambiguous correspondences between all combinations of pages of The Book of Sand and The Continuum of Cantor.

To show the existence of unique relationships between the two sets requires a demonstration in two separate parts:

That we can produce a function such that for every "book of sand" the function can assign an unequivocal decimal point in the Continuum. This is the relation A in the illustration.
That there is also another function such that for every decimal in the Continuum the function can assign a unequivocal “book of sand”. That is the relation B in the illustration.

Will it be possible to demonstrate the existence of such mappings for the relations A and B as shown in the diagram?

This will be seen in the next post.

What is the shape of a wheel in the fourth dimension?

2008-08-23T19:56:00.000-07:00

In 1909, the renowned magazine Scientific American held a contest where authors were called to submit articles answering the question "What is the fourth dimension?" The magazine received more than two hundreds essays, a respectable quantity for such an obtuse subject at that time.

The judge in charge to select the best articles was Dr. Henry Parker Manning (1859-1956), a mathematics professor of Brown University. Manning was a specialist in non-traditional geometries and algebras like non-Euclidean geometry and quaternions.

One of the rules of the contest was that the articles should not be greater than 2500 words; thus the essays were going be medium sized in length. Another rule was that the essays should be submitted with pseudonyms instead of the true author name. Since each author was writing independently of the others, and from different countries, some repetitions in concepts were inevitable.
Out of the large amount of essays, Dr. Manning edited a book of what he considered the best 22 articles, and wrote an Introduction for them where he exposed his view of some of the articles selected, and even corrected some misconceptions about transformations and manipulations of objects in the fourth dimensions, like turning gloves inside-out. The book was published under the title: The Fourth Dimension Simply Explained.

Reproduced below is the discussion of Manning about what should be a wheel in four-dimensional space.

A wheel of four-dimensional matter, in two dimensions of the shape of a circle and in the other two dimensions very small, would have for axis a flat plate instead of a rod. This axial plate could extend indefinitely in all the directions of its plane without any interference with the wheel. The wheel can slip all around over the axial plate unless held to some position on it, just as with us a wheel may slip along on its axis unless held to some position on it. We may suppose that in a three-space we can see the axial plate and a pair of opposite radii (spokes) of the wheel, appearing to us entirely separate; in this way we can see a two-dimensional hole. Or we can see the entire wheel with a hole through it and an axial rod, cut from the axial plate by our three-space.

Manning included no figures to clarify his ideas, but we can suppose that what he did is that in the same way that a line (an axis) projected into the next dimension would produce a plane, he deduced that an axis holding two wheels, when projected into the next dimension would become a plane. It is not easy to visualize two linked rotating wheels in 4D where their common axis is a plane, but anything about the fourth dimension is not easy.

But Manning goes further and writes:

We can fasten the wheel rigidly to the axial plate so that it will turn with the wheel, the wheel turning in its plane and the plate turning on itself. We may put more than one wheel on an axial plate, putting different wheels at different points on the plate wherever we please. If these wheels are all fastened rigidly to the axial plate we turn them all by turning one. Thus we have a method of constructing machinery in space of four dimensions.

If this is not enough to dazzle your mind, wait until you read this:

The axial plate may itself be a wheel. We may fasten two wheels together at their centers making them absolutely perpendicular to each other. Such a figure can revolve in two ways, the plane of one wheel being the axis plane of the rotation and the plane of the other wheel the rotation plane.

Dr. Manning should be speaking from a strictly mathematical point of view; he cannot be fantasizing about higher dimensions. However in the past article The strange extraterrestrial worlds of Camille Flammarion, in the paragraphs about the controvertible Flammarion's woodcut, I called to the attention to the enigmatic solid wheel that appears at the top of the "woodcut" (the woodcut figure is repeated here). Note how in this woodcut, the two intersecting wheels are drawn like two classic ox cart wheels. Possibly, when the artist carved this --let's call it, cross-wheel, or super-wheel-- he was not thinking about a fourth dimension, he needed not to. What he tried to convey was the idea that beyond the spheres that limit our imagination many things can coexist even when they appeared to be contradictory to our senses. Hence, for this artist, wheels that can move in two directions simultaneously are possible. Manning, speaking without the need to recur to metaphors tells us that this is possible; in a 4-dimensional world.

Adding to his exposition of a 4D-wheel, Manning says:

We might have a spherical wheel; something in three dimensions of the shape of a sphere and its fourth dimension very small. Such a wheel with a one-dimensional hole through it may turn on an axial rod, but its motion is not confined to a definite direction of rotation as is the case with the flat wheel turning in its plane.

Flammarion's woodcut is not the only picture that incorporates a possible 4D-wheel. See that in the next picture there is also the same 4D-wheel element incorporated as part of Ezekiel's vision. In fact, the origin of this idea or metaphor comes from the following verses (Chapter 1 of Ezekiel 15-18 ) of the book of Ezekiel in the Bible:

"As I looked at the living creatures, I saw a wheel on the ground beside each creature with its four faces. This was the appearance and structure of the wheels: They sparkled like topaz, and all four looked alike. Each appeared to be made like a wheel intersecting a wheel. As they moved, they would go in any one of the four directions the creatures faced; the wheels did not change direction as the creatures went. Their rims were high and awesome, and all four rims were full of eyes all around."

Continuing with Manning's Introduction see the following"

A spherical wheel may be used for vehicles. If four dimensional beings lived on a four-dimensional earth; that is, alongside of its three-dimensional boundary, a vehicle with four or more wheels of either kind could be used in traveling over this earth. With a flat wheel he could travel only in a straight line without friction between the wheel and the earth; with a spherical wheel he could travel in any direction in a plane without such friction, but would meet with a slight friction in turning from one plane to another.

Download the free ebook: Readings of The Fourth Dimension Simply Explained.

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We have discussed so far the Introduction that Dr. Henry P. Manning wrote to the book The Fourth Dimension Simply Explained. The edition that Datum is giving for free contains many essays about the fourth dimension that you will surely enjoy. Download it now!

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