The Gates of Horn and Ivory

For two are the gates of shadowy dreams, and one is fashioned of horn and one of ivory. But in my case it was not from thence, methinks, that my strange dream came.

The Gates of Horn & Ivory

Stranger, dreams verily are baffling and unclear of meaning, and in no wise do they find fulfilment in all things for men. For two are the gates of shadowy dreams, and one is fashioned of horn and one of ivory. Those dreams that pass through the gate of sawn ivory deceive men, bringing words that find no fulfilment. But those that come forth through the gate of polished horn bring true issues to pass, when any mortal sees them. But in my case it was not from thence, methinks, that my strange dream came.


the Odyssey, book 19, lines 560-569


Have Neuroscientists Built a Dream-Control Device?

Have you ever tried to control your dreams?

Back in my college days I became obsessed with lucid dreaming — having dreams in which I was aware I was dreaming and could shape my internal world with thought alone. After a few weeks of practice, I’d trained myself to ask, “Am I dreaming? How do I know?” whenever something in my environment seemed “off.” One night I found myself chatting with Kurt Cobain in my living room. “Wait a minute,” I said, “Kurt Cobain’s been dead for years!” And suddenly I could fly through the roof and off into the sky. For the next few months I spent a few nights every week hurtling through space, defeating mobs of ninjas and sampling the affections of famous actresses. Sue me — I was 18, and The Matrix was the coolest thing on Earth.

According to a recent paper, though, it may be possible to control someone else’s dreams from the outside.

The study, published in September in the journal Nature, investigated whether a specific memory could be activated in a dream, if animals were taught to associate that memory with a particular sound. The scientists put some mice in a maze and played them two kinds of sounds: one that meant “turn right,” and one that meant “turn left.” Once the mice had learned to run the maze, the researchers gave them some chill time — or so the mice thought!

In fact, the researchers had made detailed recordings of each mouse’s brain activity as it ran through the maze. Then, as the mice drifted off to sleep, the researchers waited for those same brain-activity patterns to reappear in their furry dreams. As each Little Rodent Nemo ran its maze in slumberland, the scientists played each of the sounds again. Not only did the dreaming mice respond to the tones, but their brains reacted exactly as they had when the mice were awake (minus the actual scurrying, of course). In short, these scientists have discovered a way to track and control dreams.

“Seriously?” I can hear some of you saying. “How can we possibly know what a mouse is dreaming about?” And that’s a fair question. It’s not like we can enter their minds and see what they’re seeing. But, as eerie as it seems, we’re getting pretty damn close. We know that specific nerve cells in the hippocampus (a brain structure crucial for memory) fire only when an animal returns to a specific place it remembers. Other hippocampal cells fire only when an animal faces in a particular direction, or when it focuses on a specific area of its visual field.

While this isn’t exactly Matrix territory, it’s enough for scientists to consistently predict, based solely on brain activity, that a mouse is, say, standing in the upper-left corner of the maze, facing left, looking at an object it recognizes in the right-hand corner. So when researchers detect a telltale hippocampal activation pattern when a mouse is dreaming, it’s a safe bet that the same predictive power holds true.

It’ll probably be a while before we can program our dreams as we would a computer, but even so, it’s not hard to see how simple technologies like this could have profound implications for people who experience recurrent nightmares as a result of traumatic memories. And what about you? If you could pre-program a device like this to trigger specific scenes you’d memorized, would you try it? Or do you prefer to let your dreams take you wherever they please?



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Scientists say Google Earth island in Pacific doesn’t exist…

Sandy Island on Google Earth might be a perfect Robinson Crusoe spot. But Australian researchers who went there found only ocean.

Tim Hornyak

November 22, 2012 10:26 AM PST

Sandy Island lies between Australia and New Caledonia, according to Google Earth. Reality is different.

(Credit: Google Earth)

If you thought Apple’s Maps app might steer you wrong, just watch out if you’re navigating the South Pacific with Google Maps.

It and Google Earth, as well as marine maps and charts, show a feature west of New Caledonia that Australian scientists say is a phantom island.

Sandy Island looks like a gaping hole in the Coral Sea. About 16 miles long, north to south, it could make the perfect beach nirvana.

But the University of Sydney scientists found only ocean 4,620 feet deep when they went to the site while on a research expedition. The depth would preclude the island sinking.

“We wanted to check it out because the navigation charts on-board the ship showed a water depth of 1,400 meters in that area — very deep,” geoscience postdoctoral fellow Maria Seton was quoted as saying by AFP news.

“It’s on Google Earth and other maps so we went to check and there was no island. We’re really puzzled. It’s quite bizarre.

“How did it find its way onto the maps? We just don’t know, but we plan to follow up and find out.”

“We work with a wide variety of authoritative public and commercial data sources to provide our users with the richest, most up-to-date maps possible,” a Google spokesperson said in response to a CNET inquiry about the island. “One of the exciting things about maps and geography is that the world is a constantly changing place, and keeping on top of these changes is a never-ending endeavor.”

Exciting yes, but islands don’t come and go very often.

Sandy Island has featured on maps for at least a decade, but not French government charts. If it did exist, it would be within French territorial waters as New Caledonia is a French territory.

Travel guidebook publishers deliberately include small errors on city maps, such as nonexistent lanes, to protect copyright. The island, however, could be the result of human error, repeated many times. But that wouldn’t explain why it’s on other maps, too.

For instance, it seems to appear on an online map at NOAA’s National Geophysical Data Center.

If Sandy Island does exist, it would make the perfect hideaway. Invisible, undetectable, it could be a Bond villain’s lair, offshore data center, or pirate’s cove. I’ll bet Google staff go there for secret parties.

(Via BBC News)


November 25th 2012

Sleep: 9pm – 5am

Sleep/Dream Aid(s): [2] 3mg Melatonin, [1] 100mg B-6

I am at the school that I have visited many times before. It is very busy at the moment. There are groups of people everywhere and as far as I can see down the hall. I am with a group of friends. Hard to tell who but I believe I know them from RL. I enter a basketball game as a player. I remember doing very well. I look down at my legs and I am wearing over-sized, red gym shorts. My legs are very hairy. I immediately become self-conscious of this and head for the locker room downstairs. It is different than the last time I visited. There is a lot of white and mirror everywhere. I meet a very tall, manly teenage girl down here who reminds me of someone I know from high school. She has black hair and olive skin. We talk a bit and she ask me,

“Are you going up to shower?”

To my right I see a light colored spiral staircase leading up to some place I surely don’t remember bing here in dreams past. The positioning reminds me of the bedroom entry in the Gryffindor common room.

I respond,  “No.”

This question makes me remember the rest of my meal. Earlier, before the game, I stopped to get some food from the cafeteria. A basket of very large blue fruit was displayed at the check out counter and I couldn’t resist. I figured that these fruit were some genetically engineered monster blueberry variety. I’m getting flashes of the city house, the one that could be in SE Portland. My home or a close friends) I go back for it but the lunch lady isn’t there. I contemplate finding some way to break in but decide against it. I fear someone will see me. Within this school function is a huge set of bleachers ( I am very familiar with). A group of us sit here waiting for a moment to get our soup.  I explain to one of the girls about no-shave-November. they have obviously never heard of such a thing and I feel somewhat embarrassed at the state of my prickly, unshaven legs. At one point I am in another crowd within this event. Nadra silently makes her way toward me and I edge away. I’m sitting near/under this round shaped mushroom type structure, on the lawn. It sits at the edge of a path and I make my get away. I avoid making eye contact or contact of any kind for that matter. I do not wish for her to speak to me. I’m not sure if it’s is the same dream, but at some point we are watching Eddie Murphy do stand-up comedy. Everyone is laughing. I am sitting toward the back of the crowd and I see my RL boyfriend sitting a few rows in front of me. He is enjoying himself fully and sitting next to a lovely young woman. I see him put his head into her lap.

I am a  young native girl and I have a baby. I found this baby on the beach. People are looking for it. White men. The one’s who shipwrecked here earlier. This baby is important and I must protect it. The white men are dangerous and deadly.


The Fibonacci numbers are Nature’s numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.

Stan Grist




Golden Ratio & Golden Section : : Golden Rectangle : : Golden Spiral


Golden Ratio & Golden Section

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.

Expressed algebraically:

The golden ratio is often denoted by the Greek letter phi (Φ or φ).
The figure of a golden section illustrates the geometric relationship that defines this constant. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

Golden Rectangle

A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: j (one-to-phi),
that is, 1 :  or approximately 1:1.618.

A golden rectangle can be constructed with only straightedge
and compass by this technique:

  1. Construct a simple square

  2. Draw a line from the midpoint of one side of the square to an opposite corner

  3. Use that line as the radius to draw an arc that defines the height of the rectangle

  4. Complete the golden rectangle

Golden Spiral

In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to j, the golden ratio. Specifically, a golden spiral gets wider (or further from its origin) by a factor of j for every quarter turn it makes.

Successive points dividing a golden rectangle into squares lie on
a logarithmic spiral which is sometimes known as the golden spiral.
Image Source:

Golden Ratio in Architecture and Art

Many  architects and artists have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. [Source:]

Here are few examples:

Parthenon, Acropolis, Athens.
 This ancient temple fits almost precisely into a golden rectangle.

The Vetruvian Man”(The Man in Action)” by Leonardo Da Vinci
We can draw many lines of the rectangles into this figure.
Then, there are three distinct sets of Golden Rectangles:
Each one set for the head area, the torso, and the legs.
Image Source >>

Leonardo’s Vetruvian Man is sometimes confused with principles of  “golden rectangle”, however that is not the case. The construction of Vetruvian Man is based on drawing a circle with its diameter equal to diagonal of the square, moving it up so it would touch the base of the square and drawing the final circle between the base of the square and the mid-point between square’s center and center of the moved circle:

Detailed explanation about  geometrical construction of the Vitruvian Man by Leonardo da Vinci >>

Golden Ratio in Nature

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.[38] Zeising wrote in 1854:

The Golden Ratio is a universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.


Click on the picture for animation showing more examples of golden ratio.


A slice through a Nautilus shell reveals
golden spiral construction principle.


About Fibonacci

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.  ( )

Fibonacci Numbers

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 is 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.  

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.

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”. Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves  The Fibonacci numbers are also found in the family tree of honeybees.

Fibonacci and Nature

Plants do not know about this sequence – they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144. Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks.

Why do these arrangements occur? In the case of leaf arrangement, or phyllotaxis, some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. In the case of close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of space.  This is well described in several books listed here >>

So nature isn’t trying to use the Fibonacci numbers: they are appearing as a by-product of a deeper physical process. That is why the spirals are imperfect. 
The plant is responding to physical constraints, not to a mathematical rule.

The basic idea is that the position of each new growth is about 222.5 degrees away from the previous one, because it provides, on average, the maximum space for all the shoots. This angle is called the golden angle, and it divides the complete 360 degree circle in the golden section, 0.618033989 . . . .

Examples of the Fibonacci sequence in nature.

Petals on flowers*

Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number: 

  • 3 petals: lily, iris
  • 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
  • 8 petals: delphiniums
  • 13 petals: ragwort, corn marigold, cineraria,
  • 21 petals: aster, black-eyed susan, chicory
  • 34 petals: plantain, pyrethrum
  • 55, 89 petals: michaelmas daisies, the asteraceae family

Some species are very precise about the number of petals they have – e.g. buttercups, but others have petals that are very near those above, with the average being a Fibonacci number.

 One-petalled …

white calla lily

Two-petalled flowers are not common.





Three petals are more common.





Five petals – there are hundreds of species, both wild and cultivated, with five petals.




Eight-petalled flowers are not so common as five-petalled, but there are quite a number of well-known species with eight.




Thirteen, …




black-eyed susan

Twenty-one and thirty-four petals are also quite common. The outer ring of ray florets in the daisy family illustrate the Fibonacci sequence extremely well.  Daisies with 13, 21, 34, 55 or 89 petals are quite common.


shasta daisy with 21 petals
Ordinary field daisies have 34 petals … 
a fact to be taken in consideration when playing “she loves me, she loves me not”. In saying that daisies have 34 petals, one is generalizing about the species – but any individual member of the species may deviate from this general pattern. There is more likelihood of a possible under development than over-development, so that 33 is more common than 35.

* Read the entire article here:

Related Links:

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 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.


  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).

Source of the above segment:
© Mathematics and Knots, U.C.N.W.,Bangor, 1996 – 2002


Fibonacci numbers in vegetables and fruit

Romanesque Brocolli/Cauliflower (or Romanesco) looks and tastes like a cross between brocolli and cauliflower. Each floret is peaked and is an identical but smaller version of the whole thing and this makes the spirals easy to see.

© All rights reserved Image Source >>

* * *

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.

To view more examples of Fibonacci numbers in Nature explore our selection of related links>>.


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. 

Mona Lisa

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.)

An artist’s impression of the face of Jesus 
based on the Shroud of Turin and corrected 
to match Dr. Stephen Marquardt’s mask.
Click here for more detailed analysis.

“Averaged” (morphed) face of few celebrities.
Related website: 

You can overlay the Repose Frontal Mask (also called the RF Mask or Repose Expression – Frontal View Mask) over a photograph of your own face to help you apply makeup, to aid in evaluating your face for face lift surgery, or simply to see how much your face conforms to the measurements of the Golden Ratio. 

Visit Dr. Marquardt’s Web site for more information on the beauty mask.

Source of the above article (with exception of few added photos):

Related links:

Related websites


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. (

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.

Nautilus shell (cut)
© All rights reserved. Image source >>

Nautilus jewelry pendant gold – A Symbol of Nature’s Beauty

Proportion – Golden Ratio and Rule of Thirds

© R. Berdan 20/01/2004. Published with permission of the author 


Proportion refers the size relationship of visual elements to each other and to the whole picture. One of the reasons proportion is often considered important in composition is that viewers respond to it emotionally. Proportion in art has been examined for hundreds of years, long before photography was invented. One proportion that is often cited as occurring frequently in design is the Golden mean or Golden ratio.

Golden Ratio: 1, 1, 2, 3, 5, 8, 13, 21, 34 etc. Each succeeding number after 1 is equal to the sum of the two preceding numbers. The Ratio formed 1:1.618 is called the golden mean – the ratio of bc to ab is the same as ab to ac. If you divide each smaller window again with the same ratio and joing their corners you end up with a logarithmic spiral. This spiral is a motif found frequently throughout nature in shells, horns and flowers (and my Science & Art logo).

The Golden Mean or Phi occurs frequently in nature and it may be that humans are genetically programmed to recognize the ratio as being pleasing. Studies of top fashion models revealed that their faces have an abundance of the 1.618 ratio.

Many photographers and artists are aware of the rule of thirds, where a picture is divided into three sections vertically and horizontally and lines and points of intersection represent places to position important visual elements. The golden ratio and its application are similar although the golden ratio is not as well known and its’ points of intersection are closer together. Moving a horizon in a landscape to the position of one third is often more effective than placing it in the middle, but it could also be placed near the bottom one quarter or sixth. There is nothing obligatory about applying the rule of thirds. In placing visual elements for effective composition, one must assess many factors including color, dominance, size and balance together with proportion. Often a certain amount of imbalance or tension can make an image more effective. This is where we come to the artists’ intuition and feelings about their subject. Each of us is unique and we should strive to preserve those feelings and impressions about our chosen subject that are different.

Rule of thirds grid applied to a landscape
Golden mean grid applied a simple composition

On analyzing some of my favorite photographs by laying down grids (thirds or golden ratio in Adobe Photoshop) I find that some of my images do indeed seem to correspond to the rule of thirds and to a lesser extent the golden ratio, however many do not. I suspect an analysis of other photographers’ images would have similar results. There are a few web sites and references to scientific studies that have studied proportion in art and photography but I have not come across any systematic studies that quantified their results- maybe I just need to look harder (see link for more information about the use of the golden ratio:
In summary, proportion is an element of design you should always be aware of but you must also realize that other design factors along with your own unique sensitivity about the subject dictates where you should place items in the viewfinder. Understanding proportion and various elements of design are guidelines only and you should always follow your instincts combined with your knowledge. Never be afraid to experiment and try something drastically different, and learn from both your successes and failures. Also try to be open minded about new ways of taking pictures, new techniques, ideas – surround yourself with others that share an open mind and enthusiasm and you will improve your compositional skills quickly.



35 mm film has the dimensions 36 mm by 24 mm (3:2 ratio) – golden mean ration of 1.6 to 1 Points of intersection are recommended as places to position important elements in your picture.

An Atheist’s Year in Jail.

A Year in Jail for Not Believing in God? How Kentucky is Persecuting Atheists

From AlterNet

In Kentucky, a homeland security law requires the state’s citizens to acknowledge the security provided by the Almighty God–or risk 12 months in prison.
November 21, 2012  |  

In Kentucky, a homeland security law requires the state’s citizens to acknowledge the security provided by the Almighty God–or risk 12 months in prison.

The law and its sponsor, state representative Tom Riner, have been the subject of controversy since the law first surfaced in 2006, yet the Kentucky state Supreme Court has refused to review its constitutionality, despite clearly violating the First Amendment’s separation of church and state.
“This is one of the most egregiously and breathtakingly unconstitutional actions by a state legislature that I’ve ever seen,” said Edwin Kagin, the legal director of American Atheists’, a national organization focused defending the civil rights of atheists. American Atheists’ launched a lawsuit against the law in 2008, which won at the Circuit Court level, but was then overturned by the state Court of Appeals.
The law states, “The safety and security of the Commonwealth cannot be achieved apart from reliance upon Almighty God as set forth in the public speeches and proclamations of American Presidents, including Abraham Lincoln’s historic March 30, 1863, presidential proclamation urging Americans to pray and fast during one of the most dangerous hours in American history, and the text of President John F. Kennedy’s November 22, 1963, national security speech which concluded: “For as was written long ago: ‘Except the Lord keep the city, the watchman waketh but in vain.'”
The law requires that plaques celebrating the power of the Almighty God be installed outside the state Homeland Security building–and carries a criminal penalty of up to 12 months in jail if one fails to comply. The plaque’s inscription begins with the assertion, “The safety and security of the Commonwealth cannot be achieved apart from reliance upon Almighty God.”
Tom Riner, a Baptist minister and the long-time Democratic state representative, sponsored the law.
“The church-state divide is not a line I see,” Riner told The New York Times shortly after the law was first challenged in court. “What I do see is an attempt to separate America from its history of perceiving itself as a nation under God.”
A practicing Baptist minister, Riner is solely devoted to his faith–even when that directly conflicts with his job as state representative. He has often been at the center of unconstitutional and expensive controversies throughout his 26 years in office. In the last ten years, for example, the state has spent more than $160,000 in string of losing court cases against the American Civil Liberties Union over the state’s decision to display the Ten Commandments in public buildings, legislation that Riner sponsored.
Although the Kentucky courts have yet to strike down the law, some judges have been explicit about its unconstitutionality.
“Kentucky’s law is a legislative finding, avowed as factual, that the Commonwealth is not safe absent reliance on Almighty God. Further, (the law) places a duty upon the executive director to publicize the assertion while stressing to the public that dependence upon Almighty God is vital, or necessary, in assuring the safety of the commonwealth,” wrote Judge Ann O’Malley Shake in Court of Appeals’ dissenting opinion. 
This rational was in the minority, however, as the Court of Appeals reversed the lower courts’ decision that the law was unconstitutional.
Last week, American Atheists submitted a petition to the U.S. Supreme Court to review the law.
Riner, meanwhile, continues to abuse the state representative’s office, turning it into a pulpit for his God-fearing message.
“The safety and security of the state cannot be achieved apart from recognizing our dependence upon God,” Riner recently t old Fox News.
“We believe dependence on God is essential. … What the founding fathers stated and what every president has stated, is their reliance and recognition of Almighty God, that’s what we’re doing,” he said.

Laura Gottesdiener is a freelance journalist and activist in New York City.

The Video Tape

From the blog Fascist Soup

“Update: The Wiki Leaks video has been released, see the linked blog post for details.”

WikiLeaks tweet:

Finally cracked the encryption to US military video in which journalists, among others, are shot. Thanks to all who donated $/CPUs.


If this turns out to be true (which I have no doubt it is) – this could undermine all the propaganda they’ve put out to-date. If the US government is capable of this, do you think they might be capable of pulling a 9/11 false flag event? We already know the Gulf of Tonkin was staged as well as the attack on the USS Liberty.

As you may or may not know, the current head of the CIA is also the chair of the New York Stock Exchange – with previous heads including George HW Bush.

From Techie-buzz:

WikiLeaks has promised to reveal a Pentagon murder cover-up at the US National Press Club on April 5. Although, very little information is available about the nature of the information available to WikiLeaks, it is definitely sensitive enough to prompt the US government into action.

A flurry of recent updates on the official WikiLeaks Twitter account suggests that they are currently under an aggressive US and Icelandic surveillance operation. Here are the recent tweets by WikiLeaks in chronological order:

      WikiLeaks is currently under an aggressive US and Icelandic surveillance operation. Following/photographing/filming/detaining.

      If anything happens to us, you know why: it is our Apr 5 film. And you know who is responsible.

      Two under State Dep diplomatic cover followed our editor from Iceland to…

      on Thursday.

      One related person was detained for 22 hours. Computer’s seized.That’s…

      We know our possession of the decrypted airstrike video is now being discussed at the highest levels of US command.

      If you know more about the operations against us, contact…

      We have been shown secret photos of our production meetings and been asked specific questions during detention related to the airstrike.

    We have airline records of the State Dep/CIA tails. Don’t think you can get away with it. You cannot. This is WikiLeaks.

In the past, WikiLeaks has played a pivotal role in exposing wrongdoing. It has managed to remain true to its principle of only publishing documents of political, diplomatic, historical or ethical interest. Over the years, it has managed to gain both respect and reputation for making a positive impact.

At the moment, the big question is, did WikiLeaks byte off more than it can chew?

Not if honest bloggers like myself have anything to say about it.

If anything happens to the editors of WikiLeaks, the world will know who did it thanks to citizen journalists not afraid to stand up to the machine.

In related news – WikiLeaks cracks the NATO Narrative for Afghanistan:

Wikileaks has cracked the encryption to a key document relating to the war in Afghanistan. The document, titled “NATO in Afghanistan: Master Narrative”, details the “story” NATO representatives are to give to, and to avoid giving to, journalists.

An unrelated leaked photo from the war: a US soldier poses with a dead Afghani man, in the hills of Afghanistan

The encrypted document, which is dated October 6, and believed to be current, can be found on the Pentagon Central Command website “”: [UPDATE Fri Feb 27 15:18:38 GMT 2009: the entire Pentagon site is now down–probably in response to this editorial]

The encryption password is progress, which perhaps reflects the Pentagon’s desire to stay on-message, even to itself.

Among the revelations, which we encourage the press to review in detail, is Jordan’s presense as secret member of the US lead occupation force, the ISAF.

Jordan is a middle eastern monarchy, backed by the US, and historically the CIA’s closest partner in its extraordinary renditions program. “the practice of torture is routine” in the country, according to a January 2007 report by UN special investigator for torture, Manfred Nowak.[1]

The document states NATO spokespersons are to keep Jordan’s involvement secret. Publicly, Jordan withdrew in 2001 and the country does not appear on this month’s public list of ISAF member states.[2]

Some other notes on matters to treat delicately are:

Any decision on the end date/end state will be taken by the respective national and/or Alliance political committee. Under no circumstances should the mission end-date be a topic for speculation in public by any NATO/ISAF spokespeople.

The term “compensation” is inappropriate and should not be used because it brings with it legal implications that do not apply.

Any talk of stationing or deploying Russian military assets in Afghanistan is out of the question and has never been the subject of any considerations.
Only if pressed: ISAF forces are frequently fired at from inside Pakistan, very close to the border. In some cases defensive fire is required, against specific threats. Wherever possible, such fire is pre-coordinated with the Pakistani military.

Altogether four classified or restricted NATO documents on the Pentagon Central Command (CENTCOM) site were discovered to share the ‘progress’ password. Wikileaks has decrypted the documents and released them in full:

NATO Media Operations Centre: NATO in Afghanistan: Master Narrative, 6 Oct 2008
ISAF Afghanistan Theatre Strategic Communications Strategy, 25 Oct 2008
NATO-ISAF Afghanistan Strategic Communications External Linkages, 20 Oct 2008
NATO-ISAF Strategic Communications Ends, Ways and Means, slide, 20 Oct 2008

Now that’s progress.

Unearthing the City of Wu

It is probably the largest ancient city relics from the Spring and Autumn Period (770-476 BC) ever known in China, according to Chen Jun, director of Suzhou Archaeological Institute, though he adds it is still too soon for a final conclusion.

“It is extremely rare to see such complete relics of a Spring and Autumn city in this highly developed area of southern Jiangsu province,” Chen says.

Farmers laboring here have taken them for granted for generations. However, when an archaeological team took a look in 2009, they found this to be the remains of an ancient city moat. Further investigation unveiled an underground city with more than 2,400 years’ history.

The team, from the Chinese Academy of Social Sciences and the Suzhou Archaeological Institute, has recently finished the first stage of study. Based on that, the expanse of the site ranges 6,728 meters from north to south and 6,820 meters from east to west, covering an area of 24.79 sq km.

Exciting news.

This is a very interesting and fascinating story out of China.

For many who follow such breaking stories from around the globe, China’s release of information always seems to be painstakingly slow at times. This find should give us a wonderful glimpse into many aspects of this period in China’s history. Here we have a whole city from the period to explore. I’m looking forward to viewing relics and learning what else they will undoubtedly uncover.

As always stay tuned.