Android Mind

The Origins of the Theory of The Hollow Earth

2018-09-04T17:13:00.003-07:00

Edmond Halley (1656-1742), in Philosophical Transactions of Royal Society, raised the idea of a hollow Earth, with a solid sphere in the center. The publication was to explain the magnetic anomalies of the compass readings. Halley suggested that the atmosphere inside the earth was luminous (and possibly inhabited) and that the gas escaped was responsible for the Aurora Borealis.

Edmond Halley with a diagram showing the multiple shells of his hollow Earth theory.

The Hollow Earth Was a Geometrical Necessity

For Halley, the Earth would be a shell of 500-mile thick, with three concentric spheres about the diameter of the planets Venus, Mars, and Mercury. Halley assumed that the spaces between the spheres in his model were representative of the atmospheres of them.

For Halley, the Earth would be a shell of 500-mile thick, with three concentric spheres about the diameter of the planets Venus, Mars, and Mercury. Halley assumed that the spaces between the spheres in his model were representative of the atmospheres of them.

Assuming as a unity the diameter of Earth, Mercury is 0.38 the diameter of Earth, Mars and Venus are 0.53 and 0.95. With these data, we can assume that Halley thought that in the very center of the Earth was the lowest of the spheres, representing Mercury. Then you should follow the sphere of Mars, which in turn should also be hollow to accommodate the Mercury sphere. Finally, the sphere of Venus, very close to the planet's surface, but also hollow to accommodate the field of Mars (which itself was inside that of Mercury). Thus, for Halley, the Earth is not completely empty but occupied by concentric hollow spheres, one inside the other areas [...] rotating at different speeds and oceans [...] filtered inwards" for the provisions that the Creator had taken".

If we recall the geometric model of Kepler, a century earlier, in which he adapted the spheres of the first five orbits of planets within the five Platonic solids, we realize that what Halley did was similar to Kepler's model (1571-1630). Obviously, the Keplerian arrangement was had some solid foundations that are based on actual calculations of planetary orbits. What Kepler did is still valid because the orbits today have the same diameters of yesteryear, but what Halley did is now only a historical curiosity.

However, there is a common denominator between these two dedicated and famous astronomers: in both cases, they tried to find a justification for the existence of planets based on the harmony between the diameters of the planetary orbits. In both cases, the distances and diameters of the planets are not a product of long planetary evolution, but the planets are there because they were put there by the provisions that the Creator had made.

The arrangement of Kepler has a beauty that still one wonders why there is this amazing coincidence. In contrast, in Halley's model, the fact that the spheres corresponding to the planets are placed one inside each other, and the planet Mars is one of them, makes us question why the place Mars, although it is diameter smaller than Earth, its orbit is greater. This takes away the beauty and harmony of Kepler's model and makes us conclude that Halley's model is a model forced one.

The Hollow Earth Was a Geological Necessity

The idea that the Earth is hollow is a long story before Halley made it popular. To review some data, Archelaus ca. 500 (before the Christian era) was talking about how the Earth was "a swamp ... hollow in the center" and by 1616, another person named Balthasar van der Veen had also another theory that incorporated the idea of the hollow Earth. Griffin, in his article: What Curiosity in the Structure: The Hollow Earth in Science expands the list and includes proponents, advocates or hollow Earth mentioning Plato, Aristotle, Lucretius, Seneca, and Dante.

''Earthquakes and volcanoes, the holes in the ground, springs, and wells were sufficient to show that not everything is solid under our feet. The caverns and caves provide direct access to foreign inverted worlds beneath the surface of the Earth, while (in the absence of mechanistic understanding about how they form) fossils and other geological events reasonably suggest that they not only exist but hosts strange creatures", says Griffin.

The Hollow Earth Was a Theological Necessity

And then continues: "The need to reconcile the large number of geographical and geological information with the Mosaic accounts of creation and the Flood was the formulation of theories about the Earth a popular activity at the same time a moral necessity in the intellectual elite the seventeenth century, one that generated a scholarly debate and popular interest at the same time. Two books of that period - in particular, stand out: the Mundus Subterraneus by the Jesuit Athanasius Kircher, published in Amsterdam in 1664 and Sacred Theory of the Earth by cleric Thomas Burnet, in 1681".

It was while gathering data for his research of weather and topographic that Halley met with two difficulties which he described: "not easy to solve". The first was that "there is no magnet that I have heard with more than two opposite poles, while the Earth obviously has four, and maybe more. The second difficulty was that the poles were not fixed on the Earth".

If Halley believed that the earth had more than two poles in his time was because the tabulated data of geomagnetic positions was very limited, incomplete and fragmented, "especially near the poles", says Griffin. So not surprisingly, Halley believed that the Earth had more than two poles. But what most contributed that Halley postulated a hollow Earth was Newton's error in the estimate of the relative densities of the Earth and Moon. "Newton's estimate of a dense Moon provided Halley the key to solving the dilemma with which he lived for eight years". For Halley, the Earth was not only hollow but also had an inner sphere that floated under us.

After pointing out the possible arguments to his theory, Halley displays other arguments. He invokes the rings of Saturn as natural analogy and evidence that nested bodies may share a common center and hold in place by gravity.

''In the days of Halley", Griffin continues, "the question of utility was a significant issue that could not be ignored". That is, he could ask what is the use of a sphere within our hollow Earth and was considered a reasonable question, to which Halley replied: "Why do we think that this prodigious mass of matter will only serve to maintain the surface? Why not best to assume that it has been mandated by the Supreme Wisdom that it can also serve as a surface for the use of living creatures ..."

All this was previously answered Bernard Bovier de Fontenelle, in his book The plurality of worlds where he argued that there must be life on every planet; it is impossible to imagine any other use. Halley, influenced by this thought must have thought that same as above should be like below: if there are spheres outside to sustain life, the ultimate function of an inner sphere should be the same. To the problem of light to these inner worlds, "Halley professes humble ignorance noting that 'there are many ways that we do not know to produce light ..."

To Sir John Leslie is credited with saying that there are two suns, which gave them the names Pluto and Proserpina. Cotton Mather was influenced by these ideas and included them in his book, Christian Philosopher, and Leslie's ideas were the inspiration for Jules Verne's Journey to the Center of the Earth.

Jules Verne's Journey to the Center of the Earth is perhaps one of the best known and disseminated of his prolific career. Journey to the Center of the Earth was published in 1863, but more than a century before, in 1741, Ludvig Holberg, a Danish writer to which Humboldt refers, published a Latin work titled Nicolai Klimii iter subterraneum (Journey to the underworld Nicolai Klimii). Holberg's work seems to be closer in time to the ideas of Halley and Leslie.

Jules Verne's Journey to the Center of the Earth

However, the idea that Leslie suggested that in the center of the Earth there must be two suns, is widespread, but in an intensive search on what was actually proposed by him, or from a reliable source "resulted in that apparently he did not. Alexander von Humboldt, in his book Cosmos: A Sketch of the Physical Description of the Universe, Vol 1, says that Leslie "has cleverly designed the core of the earth full of imponderable matter which has an enormous force of expansion. These arbitrary and adventurous conjectures have led in circles completely unscientific, more fantastic notions. Hollow Earth has been slowly becoming alive with plants and animals, and two small subterranean spinning planets Pluto and Proserpina, were imaginatively supposed to shed light on his tenuous". That is, Leslie apparently, never proposed the Suns attributed to him.

The Hollow Earth Was an Astronomical Necessity

Halley concluded that the Earth must be hollow by purely astronomical reasoning: his reasoning was based on the comparison of relative densities of the Earth and Moon, but based on erroneous calculations of Newton who then corrected them, but Halley continued to maintain his idea of Hollow Earth.

On the other hand, Leslie also came to the conclusion that the Earth must be hollow, but by deductions other than those of Halley. Leslie was based on "the theory of compression of bodies" in which it was assumed that water is incompressible, but Leslie believed that the water he could be.

The common denominator in both theories, Halley's and Leslie's, is that both were based on false premises, but, curiously reached the same conclusion; an outcome even more surprising because it is believed that Leslie did not know the theory of Halley.

According to Griffin, the Suns at the center of the Earth are a must for Leslie because if the earth was hollow, space could be filled simply with air, because air is compressible because it would have ended in disaster. Instead of that substance, "the vast underground cavity should be filled with some highly diffusive, with an amazing elasticity or internal repulsion between molecules ... This leaves only one possibility: ... the only fluid known to possess this characteristic is the Light itself".

This leads us ... the most important and impressive conclusion. The large central concavity is not that dark and depressing abyss that the fanciful poets had painted for us. On the contrary, this spacious internal vault should contain the purest essence of ethereal light in its most concentrated state, glowing with a strong and pervasive effulgence splendor.

Sir John Leslie.

The Largest Library of All Times

2018-09-03T11:27:00.000-07:00

Among Borges' fiction stories there is another one also related to books and infinity titled: La Biblioteca de Babel (The Library of Babel), an incredibly complex and labyrinthic library that has all the books of the universe. This story has been the subject of deep study among many writers especially by the English mathematician William Goldbloom Bloch, professor of mathematics at Wheaton College, in his book: The Unimaginable Mathematics of Borges' Library of Babel published by Oxford University Press.

In The Library of Babel Borges writes:

…each book contains four hundred ten pages, each page forty lines, each line approximately eighty black letters. There are also letters on the front cover of each book; these letters neither indicate nor prefigure what the pages inside will say.

Image credit: Rice+Lipka Architects.

Based in these numbers, Bloch computed that “each book consists of 410×40×80 = 1,312,000 ortho-graphic symbols; that is; we may consider a book as consisting of 1,312,000 slots to be filled with ortho-graphic symbols.”

Borges continues the description of the orthographic symbols used for all the books of this enormous library: “… all books, however different from one another they might be, consist of identical elements: the space, the period, the comma, and the twenty-two letters of the alphabet.” Hence, each book can at most use 25 different characters. Later, he adds: “In the Library, there are no two identical books.”

So, ultimately, there are, says Bloch: about

10^1,834,097

books in the library.

William Bloch made this interesting computation for us when he asked himself: Could our universe possibly contain the Library? “… if the universe consisted of nothing but sand, it would hold at most 10⁹⁰grains of sand”.

Borges' short story The Library of Babel is a good example of how extremely big numbers can fall beyond our comprehension, but nevertheless, they are not close to infinity. Well, no number, no matter how big it is, is ever near to reach infinity.

And then adds, “… suppose that each book is shrunk to the size of a proton, that is, shrunk to about 10^-15 meters across … then a cubic meter of the universe could hold

10¹⁵ × 10¹⁵ × 10¹⁵ = 10⁴⁵

books.”

How does this compare to the size of the universe? This is Bloch's answer to this question:

"It would take

10^1,834,013

universes the size of ours to hold just the books of the Library.”

In other words,

The Library of Babel will extend far beyond millions and millions, and ... millions of parallel universes.

Image credit: Gerd Altmann via Pixabay.

🔳⏺⏺⏺○●

The Curious Infinite of Jorge Luis Borges

2018-09-01T16:59:00.001-07:00

Previous to the Argentine writer Jorge Luis Borges, (1899-1986) many writers, philosophers, and mathematicians have for a long time played and juggle with the language trying to analyze and decipher the infinite.

Jorge Luis Borges, the Argentinian writer, author of the famous short story The Book of Sand.

In The Book of Sand, a translation from the Spanish original El Libro de Arena, Borges shows us an unexpected view of his understanding of the infinitude.

In this short story that bears the same as the title of the collection of short stories where it first appeared, Borges is visited by a Bible salesman that after a brief introduction offers him a special book: "I don't only sell Bibles. I can show you a holy book I came across on the outskirts of Bikaner. It may interest you."

What kind of book can a Bible salesman offer Borges—the writer—that could capture his attention? He was a professor of English literature and librarian with a huge collection of books of his property, so he was accustomed to seeing every kind of book. Thus, what was so interesting in a book that on its spine the title was the simple unexciting and dull phrase: Holy Writ - Bombay?

Borges opened the book at random just to find that the pages were worn, written in an unintelligible tightly printed text with poor typography. But this was not a surprise for him, the true surprise was in the numbering of the pages.

I noticed the one left-hand page bore the number (let us say) 40514

and the facing right-hand page 999.

Was this a misprint? Was it a typographical error? There was only one way of finding the truth: look for the numbering in other pages. And that he did, but

"I turned the leaf; it was numbered with eight digits".

What should we expect after the number 999? The number 1,000, of course, but an eight-digit number is greater than 9,999,999; so the "error" persisted. In fact, no book on the earth is so big, so the first conclusion that comes to our mind is that from the very beginning of the book there was no intention in numbering it correctly.

He told me his book was called the Book of Sand,

because neither the book nor the sand has any beginning or end.

We can reproduce this unordered numbering with any book from a bookcase: choose a book, erase or overwrite all the page numbers and substitute each one with a random one, no matter how big or small.

But there was an additional attribute of The Book of Sand that we cannot do with any book from any bookstand, bookshelf, or library: you see each page only once. This happened to Borges with one of the illustrations of the book:

It was at this point that the stranger said:

"Look at the illustration closely. You will never see it again."

Borges' metaphor of choosing the sand to convey the idea of non-repeating events is perfect. Go to the beach —any sandbox is useful for our example— take a handful of sand, drop all of the grains and keep only one in your hands; the sentence applies perfectly: look at it closely because you will never see it again.

The book was so mysterious and weird that it even lacked the first page, not because there was no number 1 in the first page, but because in some inexplicably, or magical, puzzling and perplexing way there appeared more and more pages between the book cover and the "first page".

The stranger asked me to find the first page.

I laid my left hand on the cover and, trying to put my thumb on the flyleaf,

I opened the book. It was useless.

Every time I tried, a number of pages came between the cover and my thumb.

It was as if they kept growing from the book.

Trying to find the last page of the book was equally frustrating for Borges as baffling can turn to be for us to find the first and the last grain of sand on a beach.

Now find the last page.

Again I failed. In a voice that was not mine I

barely managed to stammer: "This can't be".

A book with no first and last page is nothing more nor less than an infinite book. That was The Book of Sand: an infinite book. Somehow, the book was infinite in pages, but not infinite in weight, nor in volume. The book was not infinitely big, it was an ordinary book, but with the particularity that its pages were constantly appearing and disappearing, new pages substituting existing ones.

The strange salesman that visited Borges was aware of his astonishment with the bizarre book he was holding in his hands: the same thing happened to him—that's the reason why he called it a "devilish book". Thus, ceremoniously he told Borges:

If space is infinite, we may be at any point in space.

If time is infinite, we may be at any point in time.

Borges' concept of infinitude is different

In mathematics, we usually associate the infinite with the sequence of the natural numbers: 1, 2, 3 ... We say that the natural numbers are infinite because they don't ever end. To every number no matter how big it is, we can always add 1 to find a bigger number. Thus, there is no way of reaching a limit, of reaching an end, there is no last natural number. The natural numbers are the best example of the most elementary idea of infinitude. But the sequence of natural numbers is far from Borges' idea of what is the infinite: we already saw that the Book of Sand had no first page; it had no page 1.

But equally important as not having a first page is the fact that the book's page numbering followed no ordered sequence; any number can follow any other number at any moment. The page numbers were random at its purest stage.

If as he says, when he first opened the book he saw the page number 40514 followed by the page number 999, at some other time the same number 40514 may be followed by, let's say, the number 23089.

Can we say then that the book's numbering is just a scrambled number sequence? No, we cannot compare the book's numbering with a scrambled number series because in this case or ordering we always have a first number: the first number we choose for the scrambled sequence. But Borges' book had no first page, so he is not writing about unordered sequences of natural numbers: his metaphor is something beyond that.

Borges' Book of Sand confronts us with a different concept of infinitude: an infinite too far beyond our mental conception, an infinite that avoids any ruling, an infinite that escapes any possible ordering or any possible prediction. To Borges, the infinite is the kingdom where the chaos reign; the infinite is the source of every possible finiteness.

Borges chooses a simple short story to convey his idea of the infinitude because for him the infinite is not only unreachable, but any part of it is also inconceivable. The simple random numbering of a small book like the Holy Writ, that can be held in our hands is enough to take us to the vertigo of the infinitude.

Borges and Cantor: two minds where the infiniteness meet

Georg Cantor, the founder of the Set Theory in mathematics, discovered many other types of infinities. In mathematics there are many manipulations that can be done with the infinite; the infinitude of the natural numbers is just the simplest of them. One of the many breakthroughs in this field came when the mathematician George Cantor (1845-1918) introduced a more complex notion of infinitude with what he called the transfinite numbers which are an infinite class of infinities. For Cantor, there is a ladder of infinitudes, where the infinitude of the natural numbers is just the simplest of all the infinities. For him, this ladder of ever-growing infinity has no end. To this obscure but interesting field of mathematics belongs the abstract field called transfinite arithmetic.

Borges had a literary mindset with deep interests in mathematics; that's the reason why he exposes so excellently—and in a very short story—how the infinite is beyond our comprehension. On the other hand, Cantor was the pure mathematician that worked on the abstract concepts of the theory and the cardinality of sets. From there he discovered that the infinites are infinites in themselves.

The first of the "infinitude of infinites" discovered by Cantor is the one called the Aleph. From there he also introduced the so-called the Continuum. The connection I try to establish between Borges and Cantor is that in The Book of Sand we can begin to understand what is the Continuum without recurring to deep mathematics.

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 and again. You will "finally" obtain all possible orderings of the natural numbers. I cannot show it here, but it 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 Mathematical continuum.

The Sorites Paradox

2018-09-01T15:23:00.000-07:00

The Sorites Paradox also called a Sorites problem, is an argument that arises when a situation is presented in vague terms, or poorly defined circumstances. The name “sorites” derives from the Greek word “soros” meaning “pile” or “heap”.

Curiously, this paradox arose when trying to determine when an amount of grains of sand constitute a “heap of sand” or a “pile of sand” or when there are not enough grains to call it a heap. This paradox is attributed to the Greek philosopher Eubulides of Miletus.

The paradox goes like this: if a heap of sand is reduced by taking single grains of sand one after another, at what exact point does the pile of sand ceases to be considered “a heap”? Removing a single grain of sand at a time does not turn a heap into a non-heap, so the paradox is to find at what point repeating the process of taking away grains one by one turns the heap into a non-heap.

We can also study this paradox by going in an opposite direction; let's start with nothing of sand. Add a single grain of sand: this is obviously not a heap of sand. Adding one more grain is not enough to build a heap of sand; so, what about three grains of sand? How many grains do we need to make a heap?

Obviously, at some amount of grains, we will exclaim: “Now I have a heap of sand!” But if you go backward and extract just one grain of sand out of the heap you just created, can you still make the same exclamation? This is the paradox.

How Do You Imagine the Multiverse?

2018-08-30T14:52:00.000-07:00

Before my formal college background in mathematics, my vision of the infinite was a simple never-ending series of natural numbers without nothing in the way that could stop it.

Later, after getting acquainted with the theorems of Georg Cantor and his revolutionary vision of the transfinite numbers as a stair of many other super-infinities, my mind was furthermore expanded.

As time went on, a little short story titled The Book of Sand came into my hands. I never suspected that there was another twister waiting to squeeze my mind again. The simplicity of this short story by Jorge Luis Borges viewing the infinite as an incomprehensible chaos eternally randomizing and reconfiguring itself changed my mind forever. Borges’ infinity is the opposite of Cantor’s infinity; however, both are shapes of infinity.

Fortunately, the realm of infinity is not a private property of mathematics, nor of literature, nor of any field of knowledge. So, the science of physics—in its modality of cosmology—also raised its voice and said: here I am, I also have something to say about infinity! There are infinite parallel universes too! This universe is part of a multiverse!

But how do you imagine the multiverse? Do you imagine it as in Cantor-infinite style, or do you imagine it as in the Borges’-infinite-style?

The Fourth Dimension According to Charles Hinton

2018-08-23T12:57:00.001-07:00

Surely at some time of your life, you have heard about traveling to the fourth dimension. The are many books—famous books—and many movies—famous movies—around this topic. This means that the topic is of interest to everybody, and not only for a selected group of mathematicians and physicists.

When people are confronted with the subject of higher dimensions, or when asked if they believe in it or not, their answer will vary depending on if their background is more religious than scientific or vice versa.

The fact is that for many people "the fourth dimension" is a place—similar and different to our surrounding three-dimensional world around us. Visualizing it as "place" enable us to enter or exit from it like when going to some kind of theme park. Going to such a place will enlighten and will empower us in such a way that from that experience on to forever we'll be talking about the experience like sacred events that should not be shared with anybody except those "chosen" to experience a trip like ours in the future.

For the privileged beings of this hyperworld, we are seen somewhat as toys that can be manipulated at their will.

In the literature arena of the books, Charles Hinton (1853 - 1907), a British mathematician, occupies the place of one of the initiators among the public the fourth dimension subject. He studied the topic in a reasonable systematic approach, recurring to simple examples and explaining his ideas in nonmathematical terms.

Selected Papers of Charles Hinton
about the fourth dimension

The Selected Papers of Charles Hinton about the fourth dimension is a good compilation of many of his works in a simple nonmathematical language.

Hermann Schubert Probes the Fourth Dimension

2018-08-23T12:51:00.002-07:00

Mathematical Essays and Recreations

Hermann Schubert was a German teacher and textbook author. Mathematical Essays and Recreations is a small collection of articles ranging from the foundations of the number system, the foundations of algebra up to an extensive essay about the fourth dimension.

In the first article: Notion and definition of number, Shubert gives a brief account of how the concept of number evolved within the human mind.

In another of his essays: Monism in arithmetic, the author explores and writes about the elementary rules of algebra and about the importance that mathematical systems can operate with defined rules for any kind of number without making exceptions for some of them. For example, we can define the commutative rule of addition for the natural numbers, as in 2 + 3 = 3 + 2, and the same number should apply if instead, we use natural an imaginary numbers combined, like 2 + 3i = 3i + 2. Following his exposition, we can easily see why the class of complex numbers is such a robust field in mathematics, and how it can be derived from the natural numbers following easy and consistent steps. On the other hand, we can also see why the quaternions—when compared to the complex numbers—lack such popularity and why this class of number is so limited in applications and acceptance.

The third essay in this Datum edition is about the fourth dimension. But contrary to other science authors that rarely touch the fourth dimension from the metaphysical point of view, Schubert is not afraid to confront both doctrines. When dealing with the fourth dimension from the mathematical point of view, Schubert goes carefully from the definition of the point to the definition of dimension to a many-dimensioned space.

The Physics of Everyday Miracles

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

Are there real miracles?

We are accustomed to hearing—and repeat—that the laws of physics are unalterable through time, and that every day 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 a deadly car accident but nobody dies, people say its a miracle, but if somebody dies 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 things. 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

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 a 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?

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 can make different entities.

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

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

Every plane curve is a set of ordered pairs

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 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, especially 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.

A flat image transformed into a 3D-image.

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 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 EBook of Graphs
of Mathematical Functions.

Interested in an in-depth development of the foundation of the complex numbers from the standpoint of the ordered pairs?

Foundations of Transcomplex Numbers.

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, especially 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 the machine that could create a workforce with virtually a minimum of input energy is the perfect artifact to substitute the human sweat.

An early attempt to create a perpetual machine.

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..

Perpetual Motion.

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 sourcebook 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.

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.

Deatil of a patent submission.

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

A spinning top.

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 artifacts—like the pendulum—it is easy to overlook the many interesting—and sometimes transcendental—things that we can make with them.

A small gyroscope.

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.

Spinning Tops.

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 an 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, Spinning Tops, and a novel about people living deep in enormous cavers, The Coming Race?

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 academic, 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?

The legs of a table do not have
any naming order.

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 dining table to ask which one of the four legs of the table we should say is 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 in 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 every day; some myths never die.

A gargoyle.

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

The 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 a 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:

Some people still think that the Earth is hollow.

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 making 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 a period of fewer than 30 years, others 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, especially 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:

Even today, comets are feared.

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 infinities are there? Only one? How about if there are infinite infinities? How about if all of them are not the same, one to another?

How about if one of those infinities 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 on 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 of sand that could be 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 number of grains that can hold the planet Earth. This is an indirect proof that the sand cannot be infinite.

The Sand Reckoner.

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

But what if 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 end. 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.

Statue of Euclid.

Let's review for a moment the standard definition of a 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 follow 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 the 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?

The first coded message sent to outer space.

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, in 1974, another message was sent from the Arecibo radio-telescope. This one was an elaborate message with information about 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 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 won't' believe your eyes!

A single page from a "binary book" sent to space.

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 a doubt—the most sincere and original of all of them.

Why?

Books About the Fourth Dimension

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

Another Word,
or The Fourth Dimension.

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 an order in the chaos of the thinking about other dimensions besides our three-dimensional world.

Some passages from the Bible
can be interpreted as from the fourth dimension.

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?

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.

Some Sycamore trees alive today
are from seeds that traveled to the Moon.

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?

Perspective Drawing

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

Perspective is everywhere.

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

The book, by the late Prof. L. W. Miller, is the perfect acquisition for the 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.

Shadows also show perspective.

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 how 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 are all you need!

Essentials of Perspective.

Drawing shadows of rounded objects is not easy.

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.

Every real number has a specific place on the Number Line.

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 is 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 as 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 later accepted.) It was unacceptable to count "backward".

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 has many interpretations.

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."

The Deluge In The Light Of Modern Science.

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".

Transcomplex Numbers

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

Foundations of Transcomplex Numbers.

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:

Transcomplex numbers are an extension of
the common complex numbers.

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 transcomplex numbers need a 4-dimensional
coordinate system to be represented.

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 pair's 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

Is it possible to count all the protons of the universe?

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, 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

Alice In Wonderland is based on real characters.

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.

Alice's Adventures In Wonderland.

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.

Niels Klim's Journey Under The Ground.

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.