Sunday, October 22, 2006

Volume of four-dimensional objects.

There are two matters I would like to deal with this morning.

One is “Clifford Pickover`s question” about hyperspace and the other is 1820`s events in Turkey. The first job involves the elucidation of the relationship between dimensionality and volume and the second the impact of events that took place in Europe in the first half of 19th century and their determining effect in the fate of the nascent Greek state.

Since I am undecided, I will channel my thoughts on either of these questions, in confrontation, till either of these questions find a stronger hook to already existing thoughts, mental structures stored in my brain, schema already in place, drawing my thought resources on it and therefore dominate over the other.

I will go through the paragraphs in Clifford Pickover`s book relevant to the question. Let us go in a search of Hinton cubes. I found the Banubula blogspot where a definition of the four-dimensional object, tesseract, is thought provoking:

A Tesseract is a four-dimensional object which completes the following analogy: Square (2-d) is to Cube (3-d) as Cube (3-d) is to Tesseract (4-d).

So to take up the challenge, trying to visualize the relationship, the square is one of the 6, obviously flat, sides of the cube, and the cube is one the 6 sides of the tesseract. Each of the sides of the tesseract is a cube analogous to each side of a cube is a square. As the sides of the cube are two-dimensional, the sides of a tesseract are three-dimensional and this analogy can be carried over into the n+1 dimensionality. In the 5th dimension, to a “penteract” with four-dimensional sides and so on.

Deciding about the nature of the sides, it is followed by resolving the space bound by the sides; in cubes, tesseracts and so on. In a cube the six square sides bound space which can be measured in units of volume calculated by multiplying the surface of one side times the height of one of its adjacent sides. E.g. 1cm2 x 1 cm = 1 cm3. In a tesseract the space bound by the six cubes should, according to the rule, be calculated by multiplying its equivalent components, the cubes, in a similar analogy. And these are the volume of one of its sides times the surface of one of its adjacent sides. E.g. 1 cm3 x 1 cm2 = 1 cm5 . What is the significance of this calculation? What is the volume that is measured in cm5 units? Let carry on the calculation further for a “penteract”. This should be 1 cm5 x 1 cm3 = 1 cm8 and further 1 cm8 x 1 cm5 = 1 cm13. Can volume with units other than cm3 exist?

The classic tesseract design accommodate for the sides only but not for the space bound by the sides. The space that is enclosed by the six cube sides. The quantity of volume, quite carefully is defined, as in this website, as “the measure of space taken up by a three-dimensional object”. So the use of the word volume for four-dimensional objects and with cm5 units raises questions.

What is the connection of volume cm5 with volume cm3? Is it possible to convert cm5 to cm3?

Friday, October 20, 2006

Entering blogs

Let us see how it looks in the browser. Let, as well, enter body in this post. I am going to write about, what I have been thinking these past few days. I am currently reading a book, of Europe's history from 1815 to 1850. The reason I started to read this book, is to get a background idea of the situation prevailing in Europe, that led to the events that took place in Asia Minor, beginning of 20th century. As Turkey, emerged defeated, in her involvement in the first world war, managed to re-organise herself and pursued towards the establishement of a Turkish national identity, albeit controversial overly criticized and widely condemned methods. She could not do so, without taking advantage of the squabbles between the, then, powers in the european scene, triggered by their efforts to gain control over the oil-rich regions of Middle East. The powers (France, Italy, Britain, Russia, USA) employed, each on their own accord, policies which will strengthen in the future their influence in the region. Their commercial interests prevailed and Turkey, the not so long ago enemy, was seen as a potential friend and therefore supported. Turkey's struggle to survive as a nation, meant that she had to promote its turkishness and musulman religion. Turkish politicians, have long realised that turkish identity could not survive within a multicultural society, because the other members in the multicultural society were more powerful than the turks themselves. A pursuit of a multicultural society would have as a result the loss of the turkish national identity. This stems as well from the historical background of the modern turks themselves. All through the eons of turkish rule, there are numerous references of christians forcibly turkified. These turkifications, the extent of which might not be possible to estimate, would have been a problem within the bounds of a multicultural society, as it would have been impossible to maintain a national turkish identity. Even after the forced expulsion of all the Asia Minor inhabitants, who would not identify themselves as musulman Turks, it would have been impossible if it wasn't of the measures of Kemal Atatourk which institutionalise turkishness and forbade its subjects to speak or act against it. This is matter which is carried in our present era. be continued later.