Neptune - The Blue Planet

Neptune, which was named after the Roman god of the sea, is a planet of blue. Its atmosphere consists of methane, hydrogen, helium, and water. It has very strong wind. The wind velocity could reach to 2100 kilometers per hour, which is almost supersonic. Though the name is ironically correct, the planet is as blue as sapphire. However, scientisits still do not understand completely why it is so blue!

No man is island

But how to connect to the continent? Is there any island shared by two?

On Aug 8, I just found the origin of 'No man is island'.
This is a quotation from John Donne (1572-1631). It appears in Devotions Upon Emergent Occasions, Meditation XVII:

All mankind is of one author, and is one volume; when one man dies, one chapter is not torn out of the book, but translated into a better language; and every chapter must be so translated...As therefore the bell that rings to a sermon, calls not upon the preacher only, but upon the congregation to come: so this bell calls us all: but how much more me, who am brought so near the door by this sickness....No man is an island, entire of itself...any man's death diminishes me, because I am involved in mankind; and therefore never send to know for whom the bell tolls; it tolls for thee.

Penrose Tiling

In one of the previous posts, I mentioned the surprise connection of Penrose rhombi tiling from 2-D to 3-D. In fact, the seemingly weird idea may review deeper meaning of Penrose tiling. Actually, de Bruijn, a Netherlands mathematican, found the Penrose tiling can be constructed from the projection of the Voronoi hypercubes of a plane in 5-D space onto the plane itself. The plane has to be the golden-ratio plane. Furthermore, at different dimensions, one obtains various kinds of colorful tilings. If we choose the 3-D space that is orthogonal to the 2-D golden plane in the 5-D space, we could obtain the 3-D Penrose rhombi tiling by projecting out the Voronoi hypercubes onto the 3-D space. The 3-D Penrose rhombi tiling is intimately connected with the semi-periodic crystal packing patterns which shows local 5-fold rotation symmetry. That is an interesting topoic itself in chemistry and physics. Penrose mused that the tiling is a reflection of the electron correlation effect at the quantum mechanics level.



Well how to understand the idea? Here is an example of one-dimensional tiling. On a 2-D space, we first draw a straight line with a slope of golden-ratio. The blue grid is with a gap of 1 and can be seen as a Z^2 space. The blocks in light green are the Voronoi square cells that enclose the straight line. If we project the Vornoi square sides onto the golden-ratio straight line, we will get a series on long and short segements that cover the whole line. And the ratio of the long and short lengths is again the gold-ratio. The tiling pattern is given by the Fibbonacci sequence. It is so called Fibbonacci tiling of reals.

The Voronoi cubes of a plane in 3-D space.


The following Penrose tilings are generated with a sub-division (deflation) algorthims found by de Bruijn and Han. In this algorthim, a half rhombi is divided into two or three smaller half rhombus scaled-down by golden-ratio.The division can go on infinitely and get more and more complex structures. The following pictures show the same tiling region but to different depths.

Depth = 3


Depth = 5


Depth = 8


I have not figured out the details on how to get the tiling from projection in 5-D space to 2-D plane or 3-D solid.

Tropical Curve

First we define the following "strange" operations on the set Union[Reals, -Infinity].
StPlus[a,b]:= Max[a,b];
StTimes[a,b]:= Plus[a,b].
For such new "+" and "*" operators, -Infinity is the addition zero and 0 is the multiplication unit, since StPlus[a, -Infinity] = a and StTimes[a, 0] = a. However, there is no substraction that could be defined on this field (since the new addition is idempotent, StPlus[x,x]=x. Therefore, such a field is called semifield.
More interesting is the polynomials defined on this semifield.
P[x,y]:=StSum[a[[j]][[k]]*(x^j)*(y^k), {j,1,m},{k,1,n}]
= Max[Sequence@@Table[j*x + ky + a[[j]][[k]],{j,1,m},{k,1,n}]]
The tropical curve C defined by P consists of those points (x,y) which belongs to R^2, where P is not differentiable. "In other words, C is the locus where the maximum is assumed by more than one of the monomials of p."
For example, the following graph represents two tropical curves.

The blue curve is defined by the polynomial, StPlus[x, y, 1] while the red curve is given by StPlus[StTimes[2,x], StTimes[3,y], 5]. This seems a very interesting topic. So far I could only understand these. For higher order polynomials, more interesting features appear and more useful in some applications.

2 Paintings by Picasso Stolen

Pablo Picasso's "Portrait of Jacqueline," left, and "Maya with Doll."
Image from NY Times.

Two paintings by Picasso, worthy $66 million were stolen by some professionals. The left oil, “Portrait of Jacqueline”, was done in 1961 and it is a vivid Cubist work of of Picasso's second wife, Jacqueline Rogue.
Once again, the theft raised the question: who should own and make profit from the great work of these great artists who have passed way? The whole society or their descendants?

Matuse

Matuse, Japan. Image from NY Times.

Since there is no snow at all in Chengdu, this picture of Matuse, famous for castles, appears right for this moment, the Chinese new year. In my memory for the past Spring festivals at younger ages, the village is almost covered with white snow. Sometimes, it is bitter cold, but it makes the holiday full of festival atomosphere. The sound from stamping feet on the paritally frozen snow is so delicate. Esp. during the nights when I return to the bedroom, the sound breaks the blue/white silence and it seems like there is a spirit accompanying me. Snow also makes a good time to catch sparrow, fish or weasel...

Matuse is known for castles. The story in the NY Times is about Lafcadio Hearn, a Greek-Irish writer who stayed in Japan in the time at the turn of 1900's. Hearn, who married a local Japanese woman, played a similar role of a foreign observer as Edgar Snow who interviewed Mao and many other leaders of the Chinese Communist Party. In contrast to the revolutionary Yan'an, Matuse is a "fairland", a "crimeless Utopian society". Hearn wrote down local folk tales in English told to him by his wife. Then his English stories were translated back to Japanese. They were very popular for a time.. Interestingly, many stories recorded by Hearn are ghost stories. That does not make Matuse a fairland, instead a frighten place. It seems that is quite a part of Japanese culture.

Last day of 2006

After finishing the problem set for N and stuffing the stomach with some food, took a walk in the streets. The weather was nice, at least there was sun in the sky, which is rarely seen in Chengdu. The temperature was warm, and many plants along the streets are blooming. There were not so many people in the streets though. Most of people are celebrating the Chinese new year at home. Buying a ticket of 5 RMB, I was able to get into the culture garden, which is ususally free of admission. It was said a flower show is going on. There are actually not many.. There are many Camilia trees along the streets and in the garden and it is high time for them to bloom. I started to miss Dresden and the story about Camilia..

It is almost 11 pm now. There are fireworks here and there and the firecracks from near and far.. The new year is full of good wishes, isn't it?

Rhombi Tiling

The graph is about rhombi tiling an octagon shape, created by R. Kenyon and A. Okounkov. Finally, I figured out how to read the graph.
The graph is planar 2-D shape. But there are three possible orientations for a rhombi, straight up, left and right. These three orientations are painted by three different colors. As a result, if you pick the color properly, the graph looks like a 3-D one (block buildings). The genius thing here is that the third coordinate does not come in a natural sense (for example, directly proportional to color or some other parameter). Rather it seems a visual effect only without deep meaning. They further proved that for the random tiling, the 3D graph shows some patterns: The borders of the 3-D building approaches to some functions. The idea seems crazy at its first sight. However, it works very beautifully. The understanding of the deeper connection seems not easy though.

Euler Characteristic

Königsberg City Map and 7 Bridges
(Königsberg is a city of Prussia where Goldbach (Yes, the Goldbach Conjecture!) is born.)





The Swiss mathematician Leonhard Euler is one of the great mathematicians of all times. He used to live in a city called, Königsberg, which is a city divided into a few islands by several rivers. See the picture. It was at there that he put forward the Königsberg bridges problem (i.e. the sales man problem): One cannot cross all seven bridges just once for each.
TWF also made up a story for Euler to find the Euler characteristic formula, V - E - F. It goes like this... There used to be an isolated island in the city and there was no brige connecting it with the rest of the city. But one day, the city finally decided to build a brige to the island and Euler saw the construction. Then he thought, the number of the isolated islands is decreased by one due to the new bridge. Everybody can see this... Well, Euler did one step further: The bridge was acting as a negtive bridge. So, # islands - # bridges. One can another step further.. If there is a dock builded between the two bridges, the dock is then acting as a negative bridge, i.e. a positive island, # islands - # bridges + # docks. So comes the Euler characteristic V - E - F, where V stands for the number of vertices of a polyhedra, E stands for the edge (connecting vertices) and F for the face (connecting edges). Well, the story does not stop here if you imagine a higher dimensional world than 3-D. You may add some structure to connect two faces in a 4-D world and the formula repeats the alternative sigh pattern. The Euler characteristic is applied into counting the size of categories...
Just like this idea. Simple and clear.

Ich liebe dich nicht du liebst mich nicht

Da da da I don’t love you you don’t love me
aha aha aha
(music und lyrics: Remmler/Kralle)

What you do and what you don't
What you will and what you won't
what you can and what you can’t
This is what you got to know
loved you though it didn’t show

Ich lieb dich nicht du liebst mich nicht

Da da da I don’t love you you don’t love me

I know why you went away
understand you couldn’t stay
wonder where you are today
after all is said and done
it was right for you to run

Ich lieb dich nicht du liebst mich nicht

Da da da I don’t love you you don’t love me

Guess what happens next?

It has been a while since the last post. Becuase the earthquake in Taiwan Strait broke the optic cable, the internet connection to the rest of the world from China has been almost completely lost. For that matter, I lost the time to earn more on my capital investment. At the same time, I have gained more recognition on other sides of life. You could never be exactly sure what will happen in the next second. The life stream may meet some rocks or merge with some creeks... The more important thing is to keep going and to execute what you wish you do before you lose the chance.
Keep on fighting, and let the drama of your life more colorful and exciting.
This post happens to be the first one in 2007. I wish myself and any vistors of this blog have a WONDERFUL year of 2007.