An Interesting painting

A few days ago, a friend sent me an email with a picture of an interesting painting.

Beijing2008

The painting is a political allegory, for an explanation see below. But first more about the painting and the painter.

The painting is named Beijing 2008 and was created in 2005 by the Chinese-Canadian painter Lui Liu. He was born in 1957 in North China and is now living in Toronto. This painting was sold at an auction last year for more than 3 million US$.

Here is another painting by Lui Liu, “Towards de Future” (2008)

Towards the future

Personally I find his style of painting an interesting mixture of Magic Realism and Surrealism, similar to the paintings of Paul Delvaux (1897-1994), one of my favourite painters. See my Virtual Museum

Here is a painting by Delvaux: The Entrance to the City (1940)

The Entrance of the City

About the allegory, the title refers to the Olympic Games in Beijing, 2008. But in this painting another game is played, Mahjong. Here is the explanation as given in the email.

The woman with the tattoos on her back is China. On the left, focused intensely on the game, is Japan. The one with the shirt and head cocked to the side is America. Lying provocatively on the floor is Russia. And the little girl standing to the side is Taiwan.

China’s visible set of tiles “East Wind” has a dual meaning. First, it signifies China’s revival as a world power. Second, it signifies the military might and weaponry that China possesses has already been placed on the table. On one hand, China appears to be in a good position, but we cannot see the rest of her hand. Additionally, she is also handling some hidden tiles below the table.

America looks confident, but is glancing at Taiwan, trying to read something off of Taiwan’s expression, and at the same time seems to be hinting something at Taiwan.

Russia appears to be disinterested in the game, but this is far from the truth. One foot hooks coyly at America, while her hand passes a hidden tile to China, both countries can be said to be exchanging benefits in secret. Japan is all seriousness while staring at her own set of tiles, and is oblivious to the actions of the others in her self-focused state.

Taiwan wears a traditional red slip, symbolizing that she is the true heir of Chinese culture and civilization. In one hand she has a bowl of fruit, and in the other, a paring knife. Her expression as she stares at China is full of anger, sadness, and hatred, but to no avail; unless she enters the game, no matter who ends up as the victor, she is doomed to a fate of serving fruit.

Outside the riverbank is darkened by storm clouds, suggesting the high tension between the two nations is dangerously explosive. The painting hanging on the wall is also very meaningful; Mao’s face, but with Chiang Kai Shek’s bald head, and Sun Yat-Sen’s mustache.

At first glance, America appears to be most composed and seems to be the best position, as all the others are in various states of nakedness. However, while America may look radiant, her vulnerability has already been exposed. China and Russia may look naked, yet their key private parts remain hidden.

If the stakes of this game is that the loser strips off a piece of clothing, then if China loses, she will be in the same state as Russia (similar to when the USSR dissolved). If America loses, she also ends up in the same state as Russia. If Russia loses, she loses all. Japan has already lost everything.

Russia seems to be a mere “filler” player, but in fact is exchanging tiles with China. The real “filler” player is Japan, for Japan has nothing more to lose, and if she loses just once more she is immediately out of the game.

America may look like she is in the best position, but in fact is in a lot of danger, if she loses this round, she will give up her position as a world power. Russia is the most sinister, playing along with both sides, much like when China was de-occupied, she leaned towards the USSR and then towards America; as she did not have the ability to survive on her own, she had to weave between both sides in order to survive and develop.

There are too many of China’s tiles that we cannot see. Perhaps suggesting that China has several hidden aces? Additionally China is also exchanging tiles with Russia, while America can only guess from Taiwan’s expression of what actions have transpired between Russia and China. Japan on the other hand is completely oblivious, still focused solely on her own set of tiles.

Taiwan stares coldly at the game from aside. She sees everything that the players at the table are doing, she understands everything that is going on. But she doesn’t have the means or permission to join the game, she isn’t even given the right to speak. Even if she has a dearth of complaints, she cannot voice it to anyone, all she can do is to be a good page girl, and bring fresh fruit to the victor.

The final victor lies between China and America, this much is apparent. But look closely; while America is capable, they are playing Chinese Mahjong, not Western Poker. Playing by the rules of China, how much chance at victory does America really have?

Fibonacci (part 2)

In part 1 we have seen that the Fibonacci series can be generated by a simple rule: start with 1, 1 then adding the last two numbers of the series gives the next one

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765 ...

When you take the ratio of two consecutive Fibonacci numbers, this ratio approaches the Golden Ratio φ (phi) = 1.61803.. as you proceed in the series. And we have seen the close relation between the Fibonacci numbers and the Golden Spiral. Take note in the picture below of the rectangle containing the spiral. The ratio between width and height is φ. You will not be surprised that it is called the Golden Rectangle..:-)

Golden spiral

Has all or any of this a relation with the world around us? It is said that the Golden Rectangle is pleasing to the eye and that therefore many paintings have that format. It is said that the Parthenon temple in Athens was designed with φ in mind. There are people who believe that φ ( like π ) can be found in the dimensions of the  Pyramid of Cheops. And that the spiral arms of galaxies are Golden Spirals.

parthenonMona Lisagalaxy

The above statements are unfounded (the Parthenon) , false (painters had no preference for the Golden Rectangle) or only partly true (spiral arms of galaxies are logarithmic, but in general not golden). People who know me, will not be surprised that I am a skeptic..:-). When you are a believer, you will be pleased with the Golden Number website, maintained by “Phi Guy:” who introduces himself with “The inspiration for the site came as a result of coming to faith in God through Jesus Christ.” The guy must have taken the alternative name for φ (Divine Proportion) very literally.

This site is more to my liking: The Myth That Will Not Go Away

Often in a discussion about the Golden Spiral, the Nautilus shell is mentioned. Here it is,  a beautiful example of a logarithmic spiral in nature. But is it a Golden Spiral, does it grow with a factor 1.618 each quarter turn? The answer is negative. The golden spiral is the blue line. It is clear that it grows faster than the Nautilus! The average growth factor for a Nautilus is ~ 1.33.

Nautilus with golden spiral

Many references to Fibonacci and the Golden Ratio in  the world around us, are incorrect.

Here is one more, funny example. Recently I found on FB a picture of Fibonacci Pigeons. The location of the pigeons has been indicated with yellow arrows. Nice how the distance between the pigeons increases from left to right. But is it Fibonacci?

pigeons1

The answer is again negative. In the upper part of the picture I have indicated the Fibonacci positions. Compare the regular increase in spacing with the very irregular spacing of the yellow arrows!

Ok, time to become more positive…:-) Several painters (Dali, Seurat) were intrigued by Fibonacci numbers and the Golden Ratio. One 20th century artist, Mario Merz, was fascinated by it and you can find references in many of his works. Here are a few.

TablesMario MerzSpiral

But the most beautiful examples come from nature, from the Kingdom of Plants.

Here is a sunflower. What we call the flower is actually a combination of hundreds of small flowers (florets) surrounded by petals. As you see the florets are arranged in two series of spirals, starting from the center. One series is curving clockwise (red), the other one anticlockwise (blue). Can you count the number of blue spirals and red spirals?

Sunflower

The answer is : 55 red spirals and 34 blue ones. Two Fibonacci numbers!

Here is a pine cone, that I found in the forest a couple of years ago. Also here the scales form  two series of spirals. I counted how many and found 13 green ones and 8 yellow ones. Again two Fibonacci numbers! Isn’t it amazing? Try it yourself with a pineapple..:-)

pine cone

Is this Intelligent design? The Divine Proportion in action? The explanation is very interesting: by arranging the florets/scales this way, the available space is used the most efficiently. The idea is that new flowers/seeds are not formed in line with the old one, but rotated. If we express this rotation as part of a full turn (360 degrees), then a rotation of 0.5 would mean that each new cell is rotated 180 degrees. A rotation of 0.25 gives a rotation of 90 degrees, etc. Here are a few examples for various rotations. As you see, the result depends strongly on the chosen rotation value. For a value of 1.610, the space is used already quite efficiently.

rotations

So, let us try 1.6180, the golden ratio. We see an almost perfect filling of the space. Notice the two series of spirals and count the number in each series and you will find 13 and 21 !

rotphi

The pictures above have been created with a beautiful app, which can be found on the website Math Is Fun . Have a look, and try out other values of the rotation. You will also find there an explanation why φ is so special.

Much more can be said about the Fibonacci numbers and the Golden Ratio. A Google search gives numerous hits. Also a book has been published: The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number

Fibonacci (part 1)

Let me start with a rabbit tale..:-)

Rabbits

Once upon a time a pair of rabbits was born. One month later they were mature and mated. After a pregnancy of 1 month, a new pair of rabbits was born (and being rabbits, the parents mated again immediately). In this tale rabbits have eternal life and keep mating. The question is, how does the number of rabbit pairs grow with time?

Let us do some simple arithmetic (if you are afraid of numbers, skip this paragraph)

  1. At the start there is 1 pair, just born. We call it pair A
  2. After 1 month this pair A is mature and mates, so still 1 pair
  3. After two months, pair A delivers a new pair B, so we have now 2 pairs
  4. After three months pair A delivers again a new pair C, pair B has matured and mated, so 3 pairs
  5. After four months pair A delivers pair D, pair C has matured and mates, but now also pair B produces pair E, so totally 5 pairs
  6. After five months pair A delivers pair F, pair C delivers pair G, pair B delivers pair H, totally 8 pairs of rabbits.

Complicated..:-)?  Maybe the graph below will help. Red vertical lines show mating (M), yellow diagonal ones indicate gestation and delivery  (N), black vertical ones are stable (each month producing a new pair)

rabbits

So the sequence gives (in rabbit pairs) 1,1,2,3,5,8… can you guess how it continues?

Here is the answer, each number is the sum of the two preceding ones! 2+3=5, 3+5=8, so the next term should be 5+8=13, then 8+13=21, 13+21=34. A very simple rule..:-)

Still you may wonder, who came with such a funny story. Well, that was Fibonacci, an Italian mathematician, living from c. 1170 – c. 1250.

Fibonacci

The numbers are now called Fibonacci numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765 ...

This Fibonacci series has several remarkable properties. To explain one of them, I first have to introduce the Golden Ratio, also known as the Divine Proportion!

Hm, a little bit more mathematics (skip it if you are afraid of formulas). We have to go back to the old Greek philosophers. They were fascinated with numbers and ratios between numbers. You you may remember the Pythagorean triangles from your school days! Here is one of the problems they were interested in. Suppose you have a stick you like to divide in two parts a and b in such a way that the the ratio (a + b) / a is the same as the ratio a/b

220px-Golden_ratio_line

Hm, let’s try. We have a stick of 100 cm long and we divide it in a =50 cm and b = 50 cm. Then (50+50)/50 = 2 and 50/50 = 1. Not at all the same. Next we try a = 60 and b = 40. This results in (60+40)/60 = 1.6666.. and 60/40 = 1.500. Not yet equal. Next a =62 and b =38. Result: (62+38)/62 = 1.6129.. and 62/38 = 1.6315.. Getting closer! Refining this, we finally find a division of a = 61.8034.. and=38.1966.. with a common ratio of 1.6180..  The Greek mathematicians named this ratio the Golden Ratio with  φ (phi) as its symbol.

Back to our Fibonacci numbers. Let us take the ratio of two consecutive numbers. We start with 1/1 = 1, then 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615, 34/21 = 1.6190, 55/34 = 1.6176, 89/55 = 1.6182, 144/89 = 1.6180 etc

Surprise! The Fibonacci numbers and the Golden Ratio are related!

The ratio of two consecutive Fibonacci numbers approaches the Golden Ratio !

For the second remarkable property we will visualise the Fibonacci numbers in two dimensions, as squares. We add each square so that the result will be a rectangle. The last rectangle in the picture below measures 13 by 8, ratio 13/8 = 1.625, already close to the Golden Ratio..:-)

Fibonacci.squares

Next step, in each square we draw a quarter-circle. This is the result, we get a beautiful spiral, the Golden Spiral!

golden spiral

Not surprisingly, artists throughout the ages have been fascinated by Fibonacci numbers and the Golden Ratio. And in nature we come across many examples of Golden Spirals.

But that will be the topic of a separate post.

Note for readers with a mathematical background. The spiral above is an approximation of the Golden Spiral. And the Golden Ratio can be easily calculated.

golden ratio

Finally: the Golden Spiral is a special case of a logarithmic spiral, with a growth factor of φ for every 90 degrees of rotation.

Journal 14-12-2012

As usual I am suffering from jet lag, waking up in the middle of the night, feeling sleepy in broad daylight. Sniffy too, and coughing a lot. Cold! But no snow.
On Tuesday I was invited for dinner by Inez. I found a fleece and an old winter jacket in a cupboard, gloves, and a head cap. I went on my bike to Inez, where I arrived half-frozen.

Winter in Holland

But the company of Inez and the nice, typical Dutch dinner (and a few glasses of wine) brought me back to life soon…:-)

IMG_1995

 

IMG_1996

 

 

The next morning it was raining, and I seriously considered to stay in bed the whole day…:-) Here is the view from my bedroom window.

Bedroom window view

But in the afternoon the rain stopped, fortunately, because I had planned to go to the University, where an ex-colleague of me was giving a farewell/retirement party. It was nice to meet many people I had been working with, ten years ago.

Farewell party

I could not stay long because I had invited Yolanda, Paul’s sister for dinner.

Dinner Yolanda

And the next day my brother Pim came to my place, also for dinner. As I wrote in an earlier post, I will gain weight during these few weeks…:-)

Dinner Pim

Today I have visited the recently re-opened Stedelijk (Municipal) Museum of Amsterdam, my favourite Modern Art Museum. It was closed for renovation for about nine years. What a pleasure to see again many of the famous paintings of this museum, for example this Mondriaan. Click here for a detailed report

Mondriaan

I will end this post with a bit of Malaysian news. Just before I came back to Amsterdam, I realised that my road tax and car insurance were about to expire. So I asked Aric to renew it for me. He came back to me with a shocking discovery!

Do you know that you have not  renewed it December last year? So for one full year you have been driving without insurance and without road tax. Quite a miracle that you have not been caught by the police!

Here is the old sticker, expiry date December 2011!

Car Sticker

The Hudson River School

Now that I have migrated my Multiply albums to my own website and also created my blog on the Stuif site , it is time to get my website out of its hibernation. Because for a long time I have not really maintained it.

So long that I had almost forgotten what I had put into it. Like for example my Virtual Museum. I created it in 2002 as a collection of some of my favourite painters. A very personal choice, often they have not created many works of art, so it is a pleasant surprise when you visit a museum and find out that they have a  Balthus or a Delvaux or a  Cole .

A Cole? Maybe you have never even heard about him.

Thomas Cole (1801-1848) was the founder of the Hudson River School, an American school of Romantic landscape painters. They started painting in the Hudson Valley (New York) although later they also visited other regions.

This is one of Cole’s paintings, created in 1826. The Falls of the Kaaterskill in the state of New York.

As romantic as it can be. Click on the picture to enlarge it and try to find the single person depicted, to show how fragile a human being is.

With my waterfall addiction I was of course curious to find out if waterfalls were a common topic for these landscape painters.

And indeed, they were..:-)

 

This is my favourite painting, by Albert Bierstadt, 1830-1902. He was born in Germany but emigrated, one year old, with his family to the USA.

It is called “Among the Sierra Nevada Mountains” and dates from around  1868. The Sierra Nevada is a mountain range in California, culminating in Mount Whitney at 4421 m.

California? It looks more exotic, like a landscape in Avatar..:-)

And that was indeed a criticism of his contemporaries, that he was not realistic enough. People were also critical about the size of his paintings, that they were dwarfing the works of his fellow-painters. This one measures (including the frame) 245 x 367 cm !
A beautiful painting.

Here is another one, by Thomas Hill (1829-1908). Also a landscape in California, named Great Canyon of the Sierra, Yosemite. It dates from 1872. Also here signs of human life, an Indian tipi. Keep in mind that this part of the USA had only been recently explored and basically was still Indian territory.

One more, by Thomas Moran (1837-1926). In 1871 he joined, as a guest artist, an expedition to Yellowstone, until then an almost unknown part of the USA.

Back to the East Coast, the Niagara Falls, painted in 1857 by Frederic Edwin Church (1826 – 1900). Church was one on the most influential painters of the Hudson River School.

A final one by Church, possibly his most famous one, an impression of the Andes in South America, where he had traveled to Equador and Columbia a few years earlier. It was completed in 1859 and a huge success on its first exhibition  in New York, with people queuing and paying money to view the painting. Zoom in on the painting and try to find the village with the church, and other details.