I.2.1 Fractals and Divergent Series

One of the main characteristics of fractals is that they have finite area but infinite perimeter. More generally, the measure of any object with  dimensions is  in dimensions larger than  and  in dimensions smaller than . For example, as shown in Figure I.2, the Koch Snowflake curve has dimension between one and two, so its measure in one-dimension is infinite, which is its perimeter length, but its area, as a measure of two-dimensions is zero, but that area is taken along the line and not as the two-dimensional shape filled within its perimeter.

Therefore, in studying fractals, geometric series are used to measure the perimeter, area, or volume of the self-similar figures. For example, the area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles, whose sides are exactly  the size of their parent triangle, and therefore it has exactly  the area, and so on for the child triangles, so the total area of the snowflake is: , which can be shown to converge to , so the total area of the snowflake is  the area of its original triangle, because the general formula for a geometric series is: , interpreted as Taylor s power series which converge when .

The convergence of geometric series means that the sum involving an infinite number of summands can be finite, and this is usually applied to resolve many of Zeno s paradoxes, but this solution is not complete because it still requires infinite number of steps, as we described in Chapter II of Volume II.

Divergent series are infinite sequence of the partial sums that do not have a finite limit. One of the most important divergent series, which is encountered in various physics theories, is the infinite series whose terms are the natural numbers  and its  partial sum is the triangular number: , which increases without bound as  goes to infinity. Nevertheless, although this series sums to infinity, it is typically manipulated to converge to  by many peculiar summation methods, such as zeta function regularization and Ramanujan summation, and this has various applications in Complex Analysis, Quantum Field Theory, and Strings Theory.

As we have already introduced at the end of Volume II, in order to attain wholeness, or personal individuation, in Alchemy, it is said that one must square the circle , which means to unite the dimensions, or to react indifferently and become equally conscious in all directions. Unlike the circle which smoothly curves to make a whole entity whose parts are indistinguishable, dimensions include sharp angles, so squaring the circle means to make the infinite equals the finite, exactly; without any extra spaces or any kind of approximation.

Squaring the circle is also a problem proposed by ancient mathematicians; which is a challenge for constructing a square with the same area as a given circle, by using only a finite number of steps using only a compass and straightedge. The task was proven to be impossible in 1882, as a consequence of the Lindemann-Weierstrass theorem which proved that  is a transcendental number, which means that it is non-algebraic and therefore non-constructible, rather than simply an algebraic irrational number. For this reason, the expression squaring the circle is often used as a metaphor for trying to do the impossible.

Using complex numbers and Euler s identity, the association between imaginary powers of the natural logarithm  and points on the unit circle centered at the origin in the complex plane is given by Euler s formula: . Setting  in Euler s formula results in Euler s identity: , or: . Similarly, using the split-complex numbers, we can write Euler s formula with the hyperbolic functions ( and ): , so the identity: . There are  different complex numbers  satisfying , and these are called the n-th roots of unity , given by this formula: , where .

Squaring the circle is impossible in normal Euclidean real space, but there have been some serious claims of quadrature over certain non-Euclidean spaces, such as the hyperbolic plane, known as Bolyai-Lobachevskian geometry, but they may be not fully realistic, because there are actually no squares in the hyperbolic plane or any non-Euclidean space, and these quadratures mostly apply for certain situations only.

Therefore, in order to be able to square the circle in Euclidean geometry, we need to redefine space by taking into account the actual geometry of creation, or the genuinely-complex nature of time using the hyperbolic space, or the split-complex numbers. This is the only way we can calculate  with finite number of quadratures, and as a bonus we can discover the reason behind dividing the circle into 360 degrees, and why .

The fundamental reason why it is impossible to square the circle in normal Euclidean space is because we are considering both space and time to be real, or uniform and infinitely divisible. In reality, the dimensions of space-time are hyperbolic, and we need two complementary directions of such hyperbolic time in order to make one real dimension of space. This means that multiplicity and dimensions are inevitably ruled by discrete time, because we can only think of dimensions as a result of our ability to move in time, even if it is only in abstract thought.

When we draw any shape, including the circle, as the simplest and most symmetrical shape, we are making a number of movements in various directions, and even when we try to imagine such objects we need at least two moments to conceive of their relative dimensions. Therefore, it is impossible to make these shapes perfectly smooth, because ultimately at some microscopic level, what looks like a smooth line or curve is actually made of a finite number of discrete points, polygons or angles.

Therefore, in order to calculate the area of a circle from a square, or vice versa, we should take into account the dimension of time as well as space. For example, when we draw a straight line, we are actually drawing a curve, because we must be moving the pen at some speed, so we have to take into account time dilation and length contraction according to Lorentz transformations. We can therefore make the appropriate calculations to find a suitable speed at which the square will appear as a circle when viewed from a reference frame moving at this speed, and therefore the area of the stationary square will be equal to the area of the circle factored according length contraction.

Figure I.4: Obviously, it is impossible to make the curve of the circle perfectly smooth, because ultimately at some microscopic level, what looks like a smooth line or curve is actually made of a finite number of discrete points or squares that also fill the area of the circle. The area of the circle is not exactly equal to all the squares inside it, unless the side of the square is infinitely small, which is impossible in the actual physical world.

However, Lorentz transformations are equivalent to a rotation in the imaginary plane, since: , while , and we have seen above that in the polar coordinates, where a unit circle is defined by: , we could split this circle into two conjugate hyperbolas: , which are symbolically represented in Figure I.5, which means that in reality, when we take the hyperbolic duality nature of time into account, drawing a circle is equivalent to drawing a star that starts from the center to one point on the circumference, and then going back to the center before making the second point, and so on. This actually defines the inner part of the circle which represents the physical world that evolves in the imaginary direction of time, and the other= part is defined by the complementary dashed lines which represents the conjugate of the physical world, that is the psychical world. Each world alone can not define a perfect circle in a finite number of movements, but being complementary, together they do, because each one of them is the inverse of the other and when multiplied they produce the unit: . This is what actually happens in reality when we draw a circle, because the points of space are sequentially created in the inner level of time, which makes space-time hyperbolic. Therefore, drawing a circle is equivalent to drawing a star that starts from the center to one point on the circumference, and then going back to the center before making the second point, and so on. This actually defines the inner part of the circle which represents the physical shape that evolves in the imaginary direction of time, and the other complementary part is defined by the dashed lines which represents the conjugate of the physical world, that is the psychical world. Each world alone can not define a perfect circle in a finite number of movements, but being complementary, together they do, because each one of them is the inverse of the other and when multiplied together they produce the unit as in the equation above.

In reality, therefore, both the physical and psychical worlds have discrete space-time structures, and perception or consciousness happens as a result of their instantaneous coupling that produces an absolute symmetrical and spherical space which is the image that formed by imagination. Physical material objects, as well as psychical objects such as information, are granular, and they are only approximated by imagination into real space or smooth surfaces. For this reason, we can see an image of a circle at once, but we can not draw a real perfect circle in finite time or finite number of steps.

In the actual extreme granular structure of space, the circle is composed of squares, as symbolically represented in Figure I.5, because the minimum shape that can be created, or drawn on a plane space, in one instance of time is a square, and not a circle, and the minimum shape that can be created in three-dimensional space is a cube and not a sphere.

To make such a minimum square in the plane space, the Single Monad, or the  geometrical point makes four physical movements, each one produces an abrupt rotation by , which is what makes space genuinely complex. In reality, however, there areeight fundamental movements; four in the physical plane and four in the psychological plane. For this reason, the ancient Sumerians divided the circle into four quarters that make the 360 degrees.

Figure I.5: Splitting the circle into two complementary stars.

As we described in the previous volume, it is also possible to fill the two-dimensional plane or the three-dimensional volume in one linear flow as it can be achieved through Hilbert curve, which is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890. Although it is a curve, it can be shown that its fractal dimensions are two, which is called the Hausdorff dimension, that is the unit square, and its graph is a compact set homeomorphic to the closed unit interval.