**4.2 The Sierpinski triangle, Menger sponge and their random analogues**

The generalization of the one-dimensional triadic Cantor set to two-dimensions is called the Sierpinski triangle. It is constructed as shown in Fig. 3 and the Hausdorff-Besicovitch dimension is given by the inverse of the triadic Cantor set [2]

90 Quantum Gravity

Similar work, but not identical, was carried out by the French cosmologist Laurent Nottale, fifteen years ago. Nottale connected scaling and Einstein's relativity to what is now called scale relativity theory [5]. Around 1990, M. S. El Naschie began to work on his Cantorian version of fractal space-time [6]. In M. S. El Naschie's work on high energy physics and

In the year 1995 Nobel laureate Prof. Ilya Prigogine, Otto Rössler and M. S. El Naschie edited an important book [7] in which the basic principles of fractal space-time were spelled out. Sometime later El Naschie using the work of Prigogine on irreversibility showed that the arrow of time may be explained in a fractal space-time. Recently El Naschie gave for the first time a geometrical explanation of quantum entanglement and calculated a probability of the

In this section, we give a very brief account of Cantor sets and fractals which are

The archetypal fractal is what is known as Cantor triadic set. We start by describing the fundamental construction. Consider a unit interval. Let us delete the middle third but leave the end points. We repeat the procedure with the two segments left and so on as shown in Fig. 3 infinitely many times. At the end we obtain an uncountable set of points of measure zero. This means adding all these points together and we obtain a zero length. However, from the point of view of transfinite set theory something very profound is left, namely a transfinite points set

ln 2 0.63

ln 3 *Cd*

Mauldin and Williams replaced the orderly triadic construction by a random construction. In their original paper [10] they said they used a uniform probabilistic distribution. The Mauldin-Williams theorem which states that with the probability equal to one, a one dimensional randomly constructed Cantor set will have the Hausdorff-Besicovitch

> (0) 5 1 <sup>2</sup> *Cd*

The Menger-Urysohn dimension of all Cantor sets is zero, while the empty set has the

The generalization of the one-dimensional triadic Cantor set to two-dimensions is called the Sierpinski triangle. It is constructed as shown in Fig. 3 and the Hausdorff-Besicovitch

**4.2 The Sierpinski triangle, Menger sponge and their random analogues** 

dimension is given by the inverse of the triadic Cantor set [2]

for the entanglement of two quantum particles [8, 9].

electromagnetic weak interactions the golden mean plays a very important role.

with a finite dimension, the so-called Hausdorff-Besicovitch dimension [2]

golden mean to the power of five <sup>5</sup>

fundamental to the Cantorian space-time theory.

**4.1 Triadic Cantor set and the random Cantor set** 

**4. Fractals** 

dimension

dimension minus one [11].

$$d\_S = \frac{\ln 3}{\ln 2} \cong 1.5849$$

It is important to note that the Sierpinski triangle is a curve and its dimension lies between the classical line and the classical area.


Fig. 3. In this figure we draw analogy between smooth spaces as a line, a square, a cube, a higher-dimensional cube and the Cantor set, the Sierpinski triangle, the Menger sponge and the Cantorian space-time which is difficult to draw. The calculation of the Hausdorff-Besicovitch dimension of classical fractals and their random version is presented [1, 13].

It was shown in the Cantorian space-time theory [12] that the generalization of the formula connecting the triadic Cantor set with the Sierpinski triangle is possible for *n* dimension and is given by the so-called bijection formula

$$d\_{\mathbb{C}}^{(n)} = \left(\frac{1}{d\_{\mathbb{C}}^{(0)}}\right)^{n-1}$$

mean

5 1 0.61803

2

these Cantor sets by (0)

be [1, 8, 11]

Since

and

one finds that

Quantum Gravity in Cantorian Space-Time 93

For a randomly constructed Cantor set on the other hand, the Hausdorff-Besicovitch dimension is found to take the surprising value of the inverse of the golden

In 1986 R. Mauldin and S. Williams proved a remarkable theorem which confirmed the main conclusion of the Hausdorff-Besicovitch dimension of the Cantorian space-time. To explain the Mauldin-Williams theorem let us construct a Cantor set of the interval *[0, 1]* via a random algorithm as follows. First we chose at random an *x* according to the uniform distribution on *[0, 1],* then between *x* and *1* we chose *y* at random according to the uniform distribution on *[x, 1].* That way we obtain two intervals *[0, X]* and *[Y, 1].* Next we repeat the same procedure on *[0, X]* and *[Y, 1]* independently and so on. Continuation of this procedure leads then to a random Cantor dust and the Hausdorff-Besicovitch dimension of

Cantorian space-time is made of an infinite number of intersections and unions of the randomly constructed Cantor sets. Let us denote the Hausdorff-Besicovitch dimension of

dimension *n=1* to and determine the average dimension *n* , i.e. the expectation value of *n*. This value is easy to find following the centre of gravity theorem of probability theory to

0

 

*n*

 

*C n C <sup>d</sup> n d*

2 (0)

*C n C d d n d*

*n*

*n*

0

0

0

(0) (0)

( ) (1 ) *n C*

(1 ) ( ) (1 ) *n C C*

> (0) (0) <sup>1</sup> . <sup>1</sup> *C C d*

*d* 

Next let us calculate the average Hausdorff-Besicovitch dimension . *Cd* We sum together all the Hausdorff-Besicovitch dimensions (0) (0) ( ) *Cd* , (0) (1) ( ) *Cd* , (0) (2) ( ) *Cd* ....., following the

*n*

2 (0)

*n d*

*n d*

( )

*n C*

*n C*

(0) 2

*d*

(0) (0)

*d*

(0) 3

(0)

( )

2

[10].

*Cd* as a statistical weight for the topological

*<sup>D</sup>* by virtue of the Mauldin-Williams theorem [10].

this set will be with a probability one equal to 5 1 0.61803

*Cd* . Next we use (0) ( )*<sup>n</sup>*

For (0) *Cd* , the random contra part of the Sierpinski triangle will have the Hausdorff-Besicovitch dimension equal to [13]

$$d\_{\mathbb{C}}^{(2)} = \left(\frac{1}{\phi}\right)^{2-1} = \frac{1}{\phi} \cong 1.61803$$

A most remarkable 3D fractal is the Menger sponge which is shown in Fig. 3. The Hausdorff-Besicovitch dimension of this fractal is given by [2]

$$d\_M = \frac{\ln 20}{\ln 3} \equiv 2.7268$$

The volume of the Menger sponge is zero. The random version of the Menger sponge has a Hausdorff- Besicovitch dimension equal to [1, 13]

$$d\_{\mathbb{C}}^{(3)} = \left(\frac{1}{\phi}\right)^{3-1} = \left(\frac{1}{\phi}\right)^2 = 2 + \phi^2$$

Using the bijection formula we can calculate any higher dimensional fractals [8, 11].

One of the most far reaching and fundamental discoveries using the zero measure Cantor sets is undoubtedly that of El Naschie probability of quantum entanglement. His result for two entangled particles is a generic and universal value of the golden mean to the power of five. This is exactly equal to the famous result of Lucien Hardy [8, 9]. Quantum entanglement is thus explained as a consequence of zero measure gravity. Similarly one could explain any velocity larger than the speed of light [8, 9].
