"Mummy's Curse" Legend Won't Die

By Brian Handwerk

Movie mummies are known for two things: fabulous riches and a nasty curse that brings treasure hunters to a bad end. But Hollywood didn't invent the curse concept.

Enlarge

Mummy of King Tutankhamun

Photograph by Kenneth Garrett

The "mummy's curse" first enjoyed a worldwide vogue after the 1922 discovery of King Tutankhamun's tomb in the Valley of the Kings near Luxor, Egypt.

When Howard Carter opened a small hole to peer inside the tomb at treasures hidden for 3,000 years, he also unleashed a global passion for ancient Egypt.

Tut's glittering treasures made great headlines—and so did sensationalistic accounts of the subsequent death of expedition sponsor Lord Carnarvon.

In reality, Carnarvon died of blood poisoning and only six of the 26 people present when the tomb was opened died within a decade. Carter, surely any curse's prime target, lived until 1939.

But while the pharaoh's curse may lack bite, it hasn't lost the ability to fascinate audiences—which may be how it originated in the first place.

Birth of the Curse

The late Egyptologist Dominic Montserrat conducted a comprehensive search and concluded that the concept began with a strange "striptease" in 19th-century London.

"My work shows quite clearly that the mummy's curse concept predates Carnarvon's Tutankhamen discovery and his death by a hundred years," Montserrat told the Independent (U.K.) in an interview some years before his own death.

Montserrat believed that a lively stage show in which real Egyptian mummies were unwrapped inspired first one writer, and subsequently several others, to pen tales of mummy revenge.

The thread was even picked up by Little Women author Louisa May Alcott in her nearly unknown volume Lost in a Pyramid; or, The Mummy's Curse.

"My research has not only confirmed that there is, of course, no ancient Egyptian origin of the mummy's curse concept, but, more importantly, it also reveals that it didn't originate in the 1923 press publicity about the discovery of Tutankhamen's tomb either," Montserrat stressed to the Independent.

But Salima Ikram, an Egyptologist at the American University in Cairo and a National Geographic Society grantee, believes the curse concept did exist in ancient Egypt as part of a primitive security system.

She notes that some mastaba (early non-pyramid tomb) walls in Giza and Saqqara were actually inscribed with "curses" meant to terrify those who would desecrate or rob the royal resting place.

Enlarge

King Tutankhamun's coffin

Photograph by Kenneth Garrett

"They tend to threaten desecrators with divine retribution by the council of the gods," Ikram said. "Or a death by crocodiles, or lions, or scorpions, or snakes."

Tomb Toxin Threat?

In recent years some have suggested that the pharaoh's curse was biological in nature.

Could sealed tombs house pathogens that can be dangerous or even deadly to those who open them after thousands of years—especially people like Lord Carnarvon with weakened immune systems?

The mausoleums house not only the dead bodies of humans and animals but foods to provision them for the afterlife.

Lab studies have shown some ancient mummies carried mold, including Aspergillus niger and Aspergillus flavus, which can cause congestion or bleeding in the lungs. Lung-assaulting bacteria such as PseudomonasStaphylococcus may also grow on tomb walls. and

These substances may make tombs sound dangerous, but scientists seem to agree that they are not.

F. DeWolfe Miller, professor of epidemiology at the University of Hawaii at Manoa, concurs with Howard Carter's original opinion: Given the local conditions, Lord Carnarvon was probably safer inside Tut's tomb than outside.

"Upper Egypt in the 1920s was hardly what you'd call sanitary," Miller said. "The idea that an underground tomb, after 3,000 years, would have some kind of bizarre microorganism in it that's going to kill somebody six weeks later and make it look exactly like [blood poisoning] is very hard to believe."

In fact, Miller said, he knows of no archaeologist—or a single tourist, for that matter—who has experienced any afflictions caused by tomb toxins.

But like the movie mummies who invoke the malediction, the legend of the mummy's curse seems destined never to die.

source : national geographic

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Fibonacci Numbers in Nature

The sequence, in which each number is the sum of the two preceding numbers is known as the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ... (each number is the sum of the previous two).

The ratio of successive pairs tends to the so-called golden section
(GS) - 1.618033989 . . . . . whose reciprocal is 0.618033989 . . . . . so that we have 1/GS = 1 + GS.

The Fibonacci sequence, generated by the rule f1 = f2 = 1 , fn+1 = fn + fn-1,
is well known in many different areas of mathematics and science.
However, it is quite amazing that the Fibonacci number patterns occur so frequently in nature ( flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal "laws of nature".

History of Fibonacci Number

Fibonacci was known in his time and is still recognized today as the "greatest European mathematician of the middle ages." He was born in the 1170's and died in the 1240's and there is now a statue commemorating him located at the Leaning Tower end of the cemetery next to the Cathedral in Pisa. Fibonacci's name is also perpetuated in two streetsthe quayside Lungarno Fibonacci in Pisa and the Via Fibonacci in Florence.
His full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa. He called himself Fibonacci which was short for Filius Bonacci, standing for "son of Bonacci", which was his father's name. Leonardo's father( Guglielmo Bonacci) was a kind of customs officer in the North African town of Bugia, now called Bougie. So Fibonacci grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He then met with many merchants and learned of their systems of doing arithmetic. He soon realized the many advantages of the "Hindu-Arabic" system over all the others. He was one of the first people to introduce the Hindu-Arabic number system into Europe-the system we now use today- based of ten digits with its decimal point and a symbol for zero: 1 2 3 4 5 6 7 8 9. and 0
His book on how to do arithmetic in the decimal system, called Liber abbaci (meaning Book of the Abacus or Book of calculating) completed in 1202 persuaded many of the European mathematicians of his day to use his "new" system. The book goes into detail (in Latin) with the rules we all now learn in elementary school for adding, subtracting, multiplying and dividing numbers altogether with many problems to illustrate the methods in detail.
( http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits )

Pascal's Triangle and Fibonacci Numbers

The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui (about 500 years earlier, in fact) and the Persian astronomer-poet Omar Khayyám.

Pascal's Triangle is described by the following formula:

where is a binomial coefficient.

The "shallow diagonals" of Pascal's triangle
sum to Fibonacci numbers.

Fibonacci and Nature

Flower Patterns and Fibonacci Numbers

Why is it that the number of petals in a flower is often one of the following numbers: 3, 5, 8, 13, 21, 34 or 55? For example, the lily has three petals, buttercups have five of them, the chicory has 21 of them, the daisy has often 34 or 55 petals, etc. Furthermore, when one observes the heads of sunflowers, one notices two series of curves, one winding in one sense and one in another; the number of spirals not being the same in each sense. Why is the number of spirals in general either 21 and 34, either 34 and 55, either 55 and 89, or 89 and 144? The same for pinecones : why do they have either 8 spirals from one side and 13 from the other, or either 5 spirals from one side and 8 from the other? Finally, why is the number of diagonals of a pineapple also 8 in one direction and 13 in the other?

Passion Fruit
© All rights reserved Image Source >>
Are these numbers the product of chance? No! They all belong to the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (where each number is obtained from the sum of the two preceding). A more abstract way of putting it is that the Fibonacci numbers fn are given by the formula f1 = 1, f2 = 2, f3 = 3, f4 = 5 and generally f n+2 = fn+1 + fn . For a long time, it had been noticed that these numbers were important in nature, but only relatively recently that one understands why. It is a question of efficiency during the growth process of plants.
The explanation is linked to another famous number, the golden mean, itself intimately linked to the spiral form of certain types of shell. Let's mention also that in the case of the sunflower, the pineapple and of the pinecone, the correspondence with the Fibonacci numbers is very exact, while in the case of the number of flower petals, it is only verified on average (and in certain cases, the number is doubled since the petals are arranged on two levels).

© All rights reserved.
Let's underline also that although Fibonacci historically introduced these numbers in 1202 in attempting to model the growth of populations of rabbits, this does not at all correspond to reality! On the contrary, as we have just seen, his numbers play really a fundamental role in the context of the growth of plants


THE EFFECTIVENESS OF THE GOLDEN MEAN

The explanation which follows is very succinct. For a much more detailed explanation, with very interesting animations, see the web site in the reference.

In many cases, the head of a flower is made up of small seeds which are produced at the centre, and then migrate towards the outside to fill eventually all the space (as for the sunflower but on a much smaller level). Each new seed appears at a certain angle in relation to the preceeding one. For example, if the angle is 90 degrees, that is 1/4 of a turn, the result after several generations is that represented by figure 1.

Of course, this is not the most efficient way of filling space. In fact, if the angle between the appearance of each seed is a portion of a turn which corresponds to a simple fraction, 1/3, 1/4, 3/4, 2/5, 3/7, etc (that is a simple rational number), one always obtains a series of straight lines. If one wants to avoid this rectilinear pattern, it is necessary to choose a portion of the circle which is an irrational number (or a nonsimple fraction). If this latter is well approximated by a simple fraction, one obtains a series of curved lines (spiral arms) which even then do not fill out the space perfectly (figure 2).
In order to optimize the filling, it is necessary to choose the most irrational number there is, that is to say, the one the least well approximated by a fraction. This number is exactly the golden mean. The corresponding angle, the golden angle, is 137.5 degrees. (It is obtained by multiplying the non-whole part of the golden mean by 360 degrees and, since one obtains an angle greater than 180 degrees, by taking its complement). With this angle, one obtains the optimal filling, that is, the same spacing between all the seeds (figure 3).

This angle has to be chosen very precisely: variations of 1/10 of a degree destroy completely the optimization. (In fig 2, the angle is 137.6 degrees!) When the angle is exactly the golden mean, and only this one, two families of spirals (one in each direction) are then visible: their numbers correspond to the numerator and denominator of one of the fractions which approximates the golden mean : 2/3, 3/5, 5/8, 8/13, 13/21, etc.

These numbers are precisely those of the Fibonacci sequence (the bigger the numbers, the better the approximation) and the choice of the fraction depends on the time laps between the appearance of each of the seeds at the center of the flower.

This is why the number of spirals in the centers of sunflowers, and in the centers of flowers in general, correspond to a Fibonacci number. Moreover, generally the petals of flowers are formed at the extremity of one of the families of spiral. This then is also why the number of petals corresponds on average to a Fibonacci number.

REFERENCES:

1. An excellent Internet site of Ron Knot's at the University of Surrey on this and related topics.
2. S. Douady et Y. Couder, La physique des spirales végétales, La Recherche, janvier 1993, p. 26 (In French).

Human Hand

Every human has two hands, each one of these has five fingers, each finger has three parts which are separated by two knuckles. All of these numbers fit into the sequence. However keep in mind, this could simply be coincidence.

Human Face

Knowledge of the golden section, ratio and rectangle goes back to the Greeks, who based their most famous work of art on them: the Parthenon is full of golden rectangles. The Greek followers of the mathematician and mystic Pythagoras even thought of the golden ratio as divine.

Later, Leonardo da Vinci painted Mona Lisa's face to fit perfectly into a golden rectangle, and structured the rest of the painting around similar rectangles.

Mozart divided a striking number of his sonatas into two parts whose lengths reflect the golden ratio, though there is much debate about whether he was conscious of this. In more modern times, Hungarian composer Bela Bartok and French architect Le Corbusier purposefully incorporated the golden ratio into their work.

Even today, the golden ratio is in human-made objects all around us. Look at almost any Christian cross; the ratio of the vertical part to the horizontal is the golden ratio. To find a golden rectangle, you need to look no further than the credit cards in your wallet.
Despite these numerous appearances in works of art throughout the ages, there is an ongoing debate among psychologists about whether people really do perceive the golden shapes, particularly the golden rectangle, as more beautiful than other shapes. In a 1995 article in the journal Perception, professor Christopher Green,
of York University in Toronto, discusses several experiments over the years that have shown no measurable preference for the golden rectangle, but notes that several others have provided evidence suggesting such a preference exists.
Regardless of the science, the golden ratio retains a mystique, partly because excellent approximations of it turn up in many unexpected places in nature. The spiral inside a nautilus shell is remarkably close to the golden section, and the ratio of the lengths of the thorax and abdomen in most bees is nearly the golden ratio. Even a cross section of the most common form of human DNA fits nicely into a golden decagon. The golden ratio and its relatives also appear in many unexpected contexts in mathematics, and they continue to spark interest in the mathematical community.
Dr. Stephen Marquardt, a former plastic surgeon, has used the golden section, that enigmatic number that has long stood for beauty, and some of its relatives to make a mask that he claims is the most beautiful shape a human face can have.

The Mask of a perfect human face

Egyptian Queen Nefertiti (1400 B.C.)

Fibonacci's Rabbits

The original problem that Fibonacci investigated, in the year 1202, was about how fast rabbits could breed in ideal circumstances. "A pair of rabbits, one month old, is too young to reproduce. Suppose that in their second month, and every month thereafter, they produce a new pair. If each new pair of rabbits does the same, and none of the rabbits dies, how many pairs of rabbits will there be at the beginning of each month?"

1. At the end of the first month, they mate, but there is still one only 1 pair.
2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
4. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. (http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibBio.html)

The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, etc.

The Fibonacci Rectangles and Shell Spirals

We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1).

Phi pendant gold - a Powerful Tool for Finding Harmony and Beauty

We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles.
The next diagram shows that we can draw a spiral by putting together quarter circles, one in each new square. This is a spiral (the Fibonacci Spiral). A similar curve to this occurs in nature as the shape of a snail shell or some sea shells. Whereas the Fibonacci Rectangles spiral increases in size by a factor of Phi (1.618..) in a quarter of a turn (i.e. a point a further quarter of a turn round the curve is 1.618... times as far from the centre, and this applies to all points on the curve), the Nautilus spiral curve takes a whole turn before points move a factor of 1.618... from the centre.

A slice through a Nautilus shell

These spiral shapes are called Equiangular or Logarithmic spirals. The links from these terms contain much more information on these curves and pictures of computer-generated shells.

Here is a curve which crosses the X-axis at the Fibonacci numbers

The spiral part crosses at 1 2 5 13 etc on the positive axis, and 0 1 3 8 etc on the negative axis. The oscillatory part crosses at 0 1 1 2 3 5 8 13 etc on the positive axis. The curve is strangely reminiscent of the shells of Nautilus and snails. This is not surprising, as the curve tends to a logarithmic spiral as it expands.

Read More

Fibonacci Numbers in Nature

The sequence, in which each number is the sum of the two preceding numbers is known as the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ... (each number is the sum of the previous two).

The ratio of successive pairs tends to the so-called golden section
(GS) - 1.618033989 . . . . . whose reciprocal is 0.618033989 . . . . . so that we have 1/GS = 1 + GS.

The Fibonacci sequence, generated by the rule f1 = f2 = 1 , fn+1 = fn + fn-1,
is well known in many different areas of mathematics and science.
However, it is quite amazing that the Fibonacci number patterns occur so frequently in nature ( flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal "laws of nature".

History of Fibonacci Number

Fibonacci was known in his time and is still recognized today as the "greatest European mathematician of the middle ages." He was born in the 1170's and died in the 1240's and there is now a statue commemorating him located at the Leaning Tower end of the cemetery next to the Cathedral in Pisa. Fibonacci's name is also perpetuated in two streetsthe quayside Lungarno Fibonacci in Pisa and the Via Fibonacci in Florence.
His full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa. He called himself Fibonacci which was short for Filius Bonacci, standing for "son of Bonacci", which was his father's name. Leonardo's father( Guglielmo Bonacci) was a kind of customs officer in the North African town of Bugia, now called Bougie. So Fibonacci grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He then met with many merchants and learned of their systems of doing arithmetic. He soon realized the many advantages of the "Hindu-Arabic" system over all the others. He was one of the first people to introduce the Hindu-Arabic number system into Europe-the system we now use today- based of ten digits with its decimal point and a symbol for zero: 1 2 3 4 5 6 7 8 9. and 0
His book on how to do arithmetic in the decimal system, called Liber abbaci (meaning Book of the Abacus or Book of calculating) completed in 1202 persuaded many of the European mathematicians of his day to use his "new" system. The book goes into detail (in Latin) with the rules we all now learn in elementary school for adding, subtracting, multiplying and dividing numbers altogether with many problems to illustrate the methods in detail.
( http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits )

Pascal's Triangle and Fibonacci Numbers

The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui (about 500 years earlier, in fact) and the Persian astronomer-poet Omar Khayyám.

Pascal's Triangle is described by the following formula:

where is a binomial coefficient.

The "shallow diagonals" of Pascal's triangle
sum to Fibonacci numbers.

Fibonacci and Nature

Flower Patterns and Fibonacci Numbers

Why is it that the number of petals in a flower is often one of the following numbers: 3, 5, 8, 13, 21, 34 or 55? For example, the lily has three petals, buttercups have five of them, the chicory has 21 of them, the daisy has often 34 or 55 petals, etc. Furthermore, when one observes the heads of sunflowers, one notices two series of curves, one winding in one sense and one in another; the number of spirals not being the same in each sense. Why is the number of spirals in general either 21 and 34, either 34 and 55, either 55 and 89, or 89 and 144? The same for pinecones : why do they have either 8 spirals from one side and 13 from the other, or either 5 spirals from one side and 8 from the other? Finally, why is the number of diagonals of a pineapple also 8 in one direction and 13 in the other?

Passion Fruit
© All rights reserved Image Source >>
Are these numbers the product of chance? No! They all belong to the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (where each number is obtained from the sum of the two preceding). A more abstract way of putting it is that the Fibonacci numbers fn are given by the formula f1 = 1, f2 = 2, f3 = 3, f4 = 5 and generally f n+2 = fn+1 + fn . For a long time, it had been noticed that these numbers were important in nature, but only relatively recently that one understands why. It is a question of efficiency during the growth process of plants.
The explanation is linked to another famous number, the golden mean, itself intimately linked to the spiral form of certain types of shell. Let's mention also that in the case of the sunflower, the pineapple and of the pinecone, the correspondence with the Fibonacci numbers is very exact, while in the case of the number of flower petals, it is only verified on average (and in certain cases, the number is doubled since the petals are arranged on two levels).

© All rights reserved.
Let's underline also that although Fibonacci historically introduced these numbers in 1202 in attempting to model the growth of populations of rabbits, this does not at all correspond to reality! On the contrary, as we have just seen, his numbers play really a fundamental role in the context of the growth of plants


THE EFFECTIVENESS OF THE GOLDEN MEAN

The explanation which follows is very succinct. For a much more detailed explanation, with very interesting animations, see the web site in the reference.

In many cases, the head of a flower is made up of small seeds which are produced at the centre, and then migrate towards the outside to fill eventually all the space (as for the sunflower but on a much smaller level). Each new seed appears at a certain angle in relation to the preceeding one. For example, if the angle is 90 degrees, that is 1/4 of a turn, the result after several generations is that represented by figure 1.

Of course, this is not the most efficient way of filling space. In fact, if the angle between the appearance of each seed is a portion of a turn which corresponds to a simple fraction, 1/3, 1/4, 3/4, 2/5, 3/7, etc (that is a simple rational number), one always obtains a series of straight lines. If one wants to avoid this rectilinear pattern, it is necessary to choose a portion of the circle which is an irrational number (or a nonsimple fraction). If this latter is well approximated by a simple fraction, one obtains a series of curved lines (spiral arms) which even then do not fill out the space perfectly (figure 2).
In order to optimize the filling, it is necessary to choose the most irrational number there is, that is to say, the one the least well approximated by a fraction. This number is exactly the golden mean. The corresponding angle, the golden angle, is 137.5 degrees. (It is obtained by multiplying the non-whole part of the golden mean by 360 degrees and, since one obtains an angle greater than 180 degrees, by taking its complement). With this angle, one obtains the optimal filling, that is, the same spacing between all the seeds (figure 3).

This angle has to be chosen very precisely: variations of 1/10 of a degree destroy completely the optimization. (In fig 2, the angle is 137.6 degrees!) When the angle is exactly the golden mean, and only this one, two families of spirals (one in each direction) are then visible: their numbers correspond to the numerator and denominator of one of the fractions which approximates the golden mean : 2/3, 3/5, 5/8, 8/13, 13/21, etc.

These numbers are precisely those of the Fibonacci sequence (the bigger the numbers, the better the approximation) and the choice of the fraction depends on the time laps between the appearance of each of the seeds at the center of the flower.

This is why the number of spirals in the centers of sunflowers, and in the centers of flowers in general, correspond to a Fibonacci number. Moreover, generally the petals of flowers are formed at the extremity of one of the families of spiral. This then is also why the number of petals corresponds on average to a Fibonacci number.

REFERENCES:

1. An excellent Internet site of Ron Knot's at the University of Surrey on this and related topics.
2. S. Douady et Y. Couder, La physique des spirales végétales, La Recherche, janvier 1993, p. 26 (In French).

Human Hand

Every human has two hands, each one of these has five fingers, each finger has three parts which are separated by two knuckles. All of these numbers fit into the sequence. However keep in mind, this could simply be coincidence.

Human Face

Knowledge of the golden section, ratio and rectangle goes back to the Greeks, who based their most famous work of art on them: the Parthenon is full of golden rectangles. The Greek followers of the mathematician and mystic Pythagoras even thought of the golden ratio as divine.

Later, Leonardo da Vinci painted Mona Lisa's face to fit perfectly into a golden rectangle, and structured the rest of the painting around similar rectangles.

Mozart divided a striking number of his sonatas into two parts whose lengths reflect the golden ratio, though there is much debate about whether he was conscious of this. In more modern times, Hungarian composer Bela Bartok and French architect Le Corbusier purposefully incorporated the golden ratio into their work.

Even today, the golden ratio is in human-made objects all around us. Look at almost any Christian cross; the ratio of the vertical part to the horizontal is the golden ratio. To find a golden rectangle, you need to look no further than the credit cards in your wallet.
Despite these numerous appearances in works of art throughout the ages, there is an ongoing debate among psychologists about whether people really do perceive the golden shapes, particularly the golden rectangle, as more beautiful than other shapes. In a 1995 article in the journal Perception, professor Christopher Green,
of York University in Toronto, discusses several experiments over the years that have shown no measurable preference for the golden rectangle, but notes that several others have provided evidence suggesting such a preference exists.
Regardless of the science, the golden ratio retains a mystique, partly because excellent approximations of it turn up in many unexpected places in nature. The spiral inside a nautilus shell is remarkably close to the golden section, and the ratio of the lengths of the thorax and abdomen in most bees is nearly the golden ratio. Even a cross section of the most common form of human DNA fits nicely into a golden decagon. The golden ratio and its relatives also appear in many unexpected contexts in mathematics, and they continue to spark interest in the mathematical community.
Dr. Stephen Marquardt, a former plastic surgeon, has used the golden section, that enigmatic number that has long stood for beauty, and some of its relatives to make a mask that he claims is the most beautiful shape a human face can have.

The Mask of a perfect human face

Egyptian Queen Nefertiti (1400 B.C.)

Fibonacci's Rabbits

The original problem that Fibonacci investigated, in the year 1202, was about how fast rabbits could breed in ideal circumstances. "A pair of rabbits, one month old, is too young to reproduce. Suppose that in their second month, and every month thereafter, they produce a new pair. If each new pair of rabbits does the same, and none of the rabbits dies, how many pairs of rabbits will there be at the beginning of each month?"

1. At the end of the first month, they mate, but there is still one only 1 pair.
2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
4. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. (http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibBio.html)

The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, etc.

The Fibonacci Rectangles and Shell Spirals

We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1).

Phi pendant gold - a Powerful Tool for Finding Harmony and Beauty

We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles.
The next diagram shows that we can draw a spiral by putting together quarter circles, one in each new square. This is a spiral (the Fibonacci Spiral). A similar curve to this occurs in nature as the shape of a snail shell or some sea shells. Whereas the Fibonacci Rectangles spiral increases in size by a factor of Phi (1.618..) in a quarter of a turn (i.e. a point a further quarter of a turn round the curve is 1.618... times as far from the centre, and this applies to all points on the curve), the Nautilus spiral curve takes a whole turn before points move a factor of 1.618... from the centre.

A slice through a Nautilus shell

These spiral shapes are called Equiangular or Logarithmic spirals. The links from these terms contain much more information on these curves and pictures of computer-generated shells.

Here is a curve which crosses the X-axis at the Fibonacci numbers

The spiral part crosses at 1 2 5 13 etc on the positive axis, and 0 1 3 8 etc on the negative axis. The oscillatory part crosses at 0 1 1 2 3 5 8 13 etc on the positive axis. The curve is strangely reminiscent of the shells of Nautilus and snails. This is not surprising, as the curve tends to a logarithmic spiral as it expands.

Read More

Black Holes and Time Machines

If we are to get to travel to distant regions of space and time then first we need to get to grips with that most exotic of phenomena…the Black Hole…

Ever since the beginning, gravity has been making our universe less and less uniform and building up ever-larger contrasts of density and temperature. In the end, gravity overwhelms all the other forces in stars, and in anything larger, even though the effects of rotation and nuclear energy delay its final victory.

There are some entities in which gravity has already triumphed over all other forces. These are black holes - objects that have collapsed so far that no light or any other signal can escape them, but that nonetheless leave imprints, distortions of space and time, frozen in the space they've left.

An Astronaut who ventured too close to a black hole would pass into a region from which there is no return and from where no light signals can be transmitted to the external world; it is as though space itself were being sucked inward faster than light moves through it. An external observer would never witness the falling astronaut's final fate: any clock would appear to run slower and slower as it fell inward, into the hole, so the astronaut would appear impaled at a horizon, frozen in time.

The Russian theorists Yakov Zeldovich and Igor Novikov, who studied how time was distorted near collapsed objects, coined the term 'frozen star' for such objects. Zeldovich, one of the last polymaths of physics, holds a prominent place in modern cosmology. He was a dynamic and charismatic personality; from the 1960s onward, his research school in Moscow spearheaded many key discoveries (even though cosmology and relativity had previously been ideologically tainted in the USSR). The term 'black hole' itself was not coined until 1968, when John Wheeler described how an infalling object 'becomes dimmer millisecond by millisecond…light and particles incident from outside …go down the black hole only to add to its mass and increase its gravitational attraction."

Artists impression of a Black Hole - Possibly as heavy as 2.6 million Suns!

Black holes, the most remarkable consequences of Einstein's theory, are not just theoretical constructs. There are huge numbers of them in our Galaxy and in every other galaxy, each being the remnant of a star and weighing several times as much as the Sun. There are much larger ones, too, in the centres of galaxies. Near our own galactic centre, stars are orbiting ten times faster than their normal speeds within a galaxy. They are feeling, close up, the gravity of a dark object, presumably a black hole, as heavy as 2.6 million suns. Yet our Galaxy is poorly endowed compared to some others, in whose centres lurk holes more massive than a billion suns, betraying their presence by the high speed motions of surrounding stars and gas, induced by their gravitational pull.

Black holes are among the most exotic entities in the cosmos. But they are actually among the best understood. They are constructed from the fabric of space itself and are as simple in structure as elementary particles. A newly formed black hole quickly settles down to a standardised stationary state characterised stationary state characterised by just two numbers: those that measure its mass and its spin. (In principle, electric charge is a third such number, but stars can never acquire enough electric charge for this factor to be relevant to real collapse). The distorted space and time around black holes is described exactly by a solution of Einstein's general relativity equations that was first discovered in 1963 by Roy Kerr, a mathematician who later forsook research to become an internationally recognised bridge player. In general, macroscopic objects seem more and more complicated as we view them closer up, and we can't expect to explain their every detail; but black holes are an exception to this rule.

Viewed from outside, no traces remain that distinguish how a particular hole formed, nor what kind of object it swallowed. The great Indian astrophysicist Subrahmanyan Chandrasekhar was deeply impressed by this realisation, aesthetically as well as scientifically: " In my entire scientific life," he wrote, "the most shattering experience has been the realisation that an exact solution of Einstein's equations of general relativity, discovered by the New Zealand mathematician Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the Universe." Roger Penrose, the theorist who perhaps did most of to stimulate the renaissance in relativity theory that occurred in the 1960s, has remarked. "It is ironic that the astrophysical object which is strangest and least familiar, the black hole, should be the one for which our theoretical picture is most complete". The discovery of black holes thus opened the way to testing the most remarkable consequences of Einstein's theory.

Black Holes interest astronomers because the flow patterns and magnetic fields around them generate some of the most spectacular pyrotechnics in the universe. But they challenge basic physics as well. Around any black holes is a horizon, a surface shrouding from view an interior from which not even light can escape. A hole's size is proportional to its mass: if the sun became a black hole, its radius would be 3 kilometres, but some of the supermassive holes in galactic centres are as big as our whole solar system. If you fell inside one of these monster holes, you would be treated to several hours of leisurely observation before you approach the centre, where increasingly violent tidal forces would shred you apart. Right at the centre, you, or your remains, would encounter the singularity where the physics transcends what we yet understand. The new physics that we'll need is the same that governs the initial instants of the Big Bang.

Fast-Forward and Backward in Time?

Good science fiction should respect the fundamental constraints of physical law. In that sprit, it is worth mentioning that an observer could, in principle, observe the far future in what, subjectively, seemed quiet a short time. According to Einstein, the speed of a clock depends on where you are and how you're moving. If your subjective clock ran very slowly compared to the cosmic clock, you could travel "fast forward" into the future. This would happen if you were moving at a velocity close to the speed of light. Furthermore, strong gravity would distort time; clocks on a neutron star would run 20 or 30 percent slower. Near a black hole, the distortions would be even greater. If you were to fall into one, your future would be finite; you would be ripped apart - spaghettified - by ever more violent gravitational forces. However, a more prudent astronaut who managed to get into the closest possible orbit around a rapidly spinning hole without falling into it would also have interesting experiences, space-time is so distorted there that his clock would run arbitrarily slow and he could, therefore, in a subjectively short period, view an immensely long future timespan in the external universe.

This elasticity in the rate of passage of time may seem counter to our intuition. But such intuition is acquired from our everyday environment (and perhaps, even more, that of our remote ancestors), which has offered us no experience of such effects. Few of us have travelled faster than a millionth of the speed of light (the speed of a jet airliner); we live on a planet where the pull of gravity is 1000 billion times weaker than on a neutron star. But time dilation entails no inconsistency or paradox.

More problematic, of course, would be time travel back into the past. More than fifty years ago, the great logician Kurt Godel discovered that the theory of general relativity did not in itself preclude a time machine. He discovered a valid solution of Einstein's equations that described a bizarre universe where some of the worldlines were close loops - in other words, you could come back into your own past. But Godel's solution was not realistic: it described a universe that was rotating and not expanding.

Other theoretical examples of systems that seem to obey the laws of physics but which allow closed loops in time have been proposed. For example, Princeton theorist Richard Gott showed that a time machine could be constructed from two so called cosmic strings - long microscopically thin tubes of hyperdense material, heavy enough to distort space. Gott and his colleague Li-Xin Li also devised a cosmological model even stranger than Godel's in which an entire universe, with a finite life cycle, traces out a loop in time so that its end is also its beginning.

One much-discussed design for a time machine involves a "wormhole": two black holes linked together by a tunnel or "spacewarp". The tunnel could exist only if it were made of a substance that has very large negative pressure (or tension). Theorists speculate that exotic stuff of this kind did exist in the early universe, but even if such material still existed, the mass needed in order to make a wormhole wide enough to be comfortably traversed by a human would be 10,000 times that of the Sun!

Wormhole Travel?

Godel's discovery and its aftermath opened up a debate. Is there a future law of physics, more restrictive than Einstein's equations that rule out such effects? One might call it a "chronology protection law". Or could a time machine in principle exist? Such an artefact plainly still lies in the hypothetical reaches of science fiction, but we can still ask whether the barriers to constructing a time machine are merely technological, or whether there is a fundamental physical law that prohibit them. (To clarify the distinction, most physicists would say that a large spaceship travelling at 99.99 percent of the speed of light is in the first category, but one that travels faster than light is in the second.)

The events on the time loop must close up self-consistently, as in a movie whose last scene recapitulates its first. Paradoxes arise if you come back into the past and undo something that was a precondition of your existence: for instance, murdering your grandmother in her cradle would raise issues of logical consistency, not just of ethics. Time travel makes sense only if some law of nature precludes inconsistency of this kind. The implication that there must be "time police" to constrain our free will might seem paradoxical. But I am convinced by the robust retort of Igor Novikov, a leading physicist who has explored these ideas, that physical laws already constrain our choices: we cannot, for instance, exercise our free will by walking on the ceiling. The prohibition on violating the consistency of a time loop is, in a sense, analogous.

Even if a time machine could be built, it would not enable us to travel back prior to the date of its construction. So the fact that we have not been invaded by tourists from the future may tell us only that no time machine has yet been made, not that it is impossible.

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11 How to Become a Smarter Brain

Scientists from the University of California, Berkeley, USA, has examined the brains of rats. They found that the brains of mice grew by 4 percent when they were forced to perform mental tasks every day, for example, find a way out of a winding alley, climbing stairs, and socialize with other rats.

Well, the mouse brain can be trained to grow, let alone the human brain. The more trained, our brains must be more sharply. Memory loss in a certain amount at any age is natural, just as changes in other organs. Importantly, do not be lazy to diligently train our brain to stay strong memory of all time.

Here's 11 Ways to make your brain Smarter:

1. Practice your ability to observe. Note the surrounding environment. Record in your mind what you see, ranging from the simplest and forwarded with the observation that more complicated.

2. Sharpen your senses. Can be trained to distinguish the taste of food likes and what does not. Recognizing the smell and aroma in the vicinity or the sounds that exist in the way or may feel hot or cold air around you.

3. Memorize the names of friends and the pair number. How many can remember? Rehearse so I could remember more.

4. Learn something new. Many read and get acquainted with other things that may not be your field, can a foreign language, knowledge of computers, and others.

5. Use your hands to follow the instructions of the brain. Such as playing guitar, typing without looking at the keys, doing crafts out of wood, or practicing penmanship.

6. Tekuni hobby. Use the opportunity to develop your hobby.

7. Learn and memorize important dates, involving family members, friends, or a particular celebration.

8. Memorize something you like. Could be it poetry, song, words from a book or someone's words. As much as possible also try to phrase that is used is a foreign language.

9. Exercise memorize a long sequence of numbers lined up, for example 32145687390282930498. This is a form of exercise to improve short-term memory. Do it by grouping or splitting it into several parts numbers, eg 7390282 and 3214568 then the last 930 498.

10. Remember personal journey. What you are doing an hour ago, last week on Wednesday at 10.00, for example. With whom, where, and so on.

11. Remember and re-spending meticulous daily. What did you buy yesterday? How much money is in your wallet right now? When did you last take the cash, and so on.

These exercises will allow brain cells remain active and intercellular connective tissue of the brain the meeting. Challenging mental activities increase the number of active circuits or synapses in the brain. The more circuits, more and more associations, the greater the ability to remember

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