Toshiba Laptop Charger Smoking

The natural measure of things

The relationship we seek to establish between two quantities can be misleading. Sometimes the logical values \u200b\u200bof these lead us away from reality and the phenomenon we are trying to study. Common sense can give us an approximation of the result that would lead us to find the right solution, which truly conforms to reality.



Suppose we want to relate two quantities which correspond to a tangible reality, for example, two lengths of a particular object, and give us the following steps: 2 and 1 / 2, 3 and 1 / 3, 4 and 1 / 4 ... n 1 / n. Where n is a natural number. The division between them does not offer any conflict, will be 4, 9, 16, ... n ² , is giving us the number of times a quantity is greater than another. However there relationships that can make mistakes if we are guided by purely mathematical result. For example, if you look at the figure representing the classic fractal called the Koch snowflake and construction, we see that in each iteration replace a segment of 3 units of a unit four segments: exactly the relationship between log 4 / log 3 gives the fractal dimension of the figure, which is 1.261859 ... If we want to relate the two lengths are represented by any natural number N and its inverse 1 / N, finding the relationship similar to the above, the Koch snowflake, we find a negative value, -1, a fractal dimension negative when it makes no sense physically, since the dimension fractal is always equal to the topological (or apparent size) plus a dimensional coefficient, the greater the more irregular is the fractal.


mathematical and logical Kronecker argued that arithmetic and analysis must be based on whole numbers apart from the irrational and imaginary. He authored a phrase well known to mathematicians: "God made the natives, the rest is man " (Eric Temple Bell 1986, p.477. Men of Mathematics).

That is the question, in our case we convert 1 / N and N in two new natural numbers that relate to express the value represents the dimension of the object, give us a result consistent with the reality we are watching. The figures that follow this paragraph we clarify the path to take to find a possible solution to this particular case.


see the construction of a figure when N = 3, N = 4 and N = 5. In the first figure if we give the value 3 to the side, its perimeter is 27 (3 ³ ), but if we give the value 1 / 3, its new perimeter is 3. This is true for N = 4 or N = 1 / 4, etc, and in general for any value N 1 / N (with N finite, even as big as we want). Always occur if the side is the perimeter will be N N 3 and if the side is 1 / N the perimeter is No, not that this changes the shape of the figure.


conversion shall be that which transforms pair of measures (1 / N, N) (N, N ³ ) and the irregular value, -1, we found for the fractal dimension of the curve would become 3. This value would give the curve the ability to fill the space. fractal dimension is a whole , similar to the case of a pure random motion, that for every N ² made only steps away from N, any arbitrary reference point to consider, and therefore has a dimension equal to 2 fractal that fills the plane.

Indeed, in our case (1 / N, N), there are endless conversions respond to the expression:

Dim fractal (*) = 1 + 2/logL (N) , where L (N) the value of side we consider, as a function of N. For L (N) = 1 / N have the value -1 for L (N) = N, it is for the value 3, as stated above. For natural exponent values \u200b\u200bmore negative (1 / N ² ) and higher dimension asymptotically approaches l. For higher values \u200b\u200bof N, as N 2, No. 3 , or much more exponential in the asymptotic value is also 1.

the end we can not blindly trust the value we give the mathematics, because the world they represent is much larger than the real world and we always need our common sense in the analysis of the results. Moreover, paradoxically, sometimes the opposite is true: common sense blinds us and prevents us from seeing a deeper reality behind the mathematical results .

(*) Taking logarithms to base N





T Duality (1 / N, N)



As simple curiosity on the exchange of values \u200b\u200b1 / N and N, and as Culturilla on string theory, all this may remind called duality-T:



The term representing the square the excitation energies of a string in a curved space with a dimension or compacted, K. Kikkawa and M. Yamanaka in 1984, noted that the formula remains the same aspect if we exchange R <--> 1 / R. Where R is the radius microscopic dimension curves.

From a physical point of view this indicates that the excitation energies of a rope, when there is an extra dimension of radius R, is the same as that of a rope when the radius is 1 / R. Not only energy, but all the physical properties of both systems are exactly the same. Striking, because when R increases by 1 / R decreases, contradicting the experience of daily life, which tells us small things differ from large ones. For a string is not the case.


About " Unification and duality in string theory , see the August 1998 Scientific American, Luis E. Ibáñez Santiago.