In this post I shall discuss the nature of π as a mathematical constant and reveal its relationship with the fabric of space. As an irrational number π represents the ratio of a circle’s circumference to its diameter. An irrational number is a real number that cannot be expressed as a ratio (a/b), where (a) and (b) are integers and (b≠0).
Returning briefly to the cubical universe, which we considered in a previous post, if the observer there begins to probe his world at the level of the individual cubes defining his space and decides to form different geometries at that level, he could do so only by using those cubes. He would have no other means. Using cubes to define circles, he would soon discover that the geometric properties of his circles vary according to the orientation of the cubes. For example, the number of elements defining the diameter of the same circle could vary depending upon the orientation of the cubes in the circumference. Therefore, in a universe defined by cubical elements π, as the ratio of the units of length of a circle’s circumference to that of its diameter, cannot be constant.
A unit of length in a well-defined and homogenous space is defined by the element defining that space, so that a given length is measured in terms of the number of elements regardless of their orientation. Any space between the elements cannot be included in a measurement, because it is an indefinable quantity and therefore irrelevant to the measurement.
Now, if the space in that universe were defined by spherical elements instead of cubical ones, the problem of a variable π would not arise. In such a universe π and other geometric ratios, such as trigonometric functions, would be constant. In fact, the only type of element that could define any geometry with invariable properties is a spherical element. However, when modelling the geometry of objects in our universe, we tend to assume their boundaries (lines and surfaces) are smooth continuum of infinite points, because a theoretical point is assumed to have no physical extent! If space were defined by elements of a finite size, that assumption would be incorrect. A point would be defined by a single element and any volume of space smaller than an element would be indefinable, because elements defining a space are indivisible. Although different geometries at the level of individual elements could be modelled using theoretical points, such models are not representative of real geometries. For example, in the case of a circle formed by a few spherical elements as shown in Fig. 1, the circumference and diameter would be more accurately defined in terms of the number of elements. If modelled using a series of lines of theoretical points representing the circumference and diameter, its derived geometric properties would be incorrect. Its area would be given by (πD), where (D) is the diameter in some arbitrary units. However, the true area is the sum of the projected areas of the individual elements forming the circle.
Therefore, contrary to our intuition, modelling geometries at a quantum level could reduce the accuracy of the models and thereby affect their predictability of the behaviour of real systems. This is related directly to our understanding, or rather lack of understanding of the phenomenon of chaos. In fact, the adoption of a mathematical model of theoretical points, coupled with the nature of the circle as a deficient quantum geometry, conceal the nature of π as an irrational number, as we shall now explain.
The main problem that a mathematician faces when defining a given geometry using a finite number of discrete elements is incompleteness of a geometry! Consider the case of the circle defined by spherical elements in Fig. 2. Regardless of how many elements are used to define it, in order to maintain the correct curvature of the circumference a gap will always exist in the diameter. Any attempt to close it by bringing the elements closer while maintaining the circular geometry, results in opening up a gap in the circumference, hence the above description of the circle as a deficient quantum geometry. The reference to quantum relates to the manifestation of this problem at the level individual elements, which is concealed by the use of lines of infinite theoretical points.
In order to increase definition of the circle in Fig. 2, we could model it using a greater number of much smaller elements, so that the size of the gap is reduced relative to the diameter. However, a gap will always persist and can never be eliminated, because spaces are indefinable in terms of the elements. Our attempt to use smaller and smaller elements, as we shall explain below, is what lies behind the infinite array of digits that characterise irrational numbers.
Besides π there are infinite other irrational numbers, including those of trigonometric functions. They are all based on deficient quantum geometries, which confirms mathematics’ dependency on the fabric of physical reality. For example, comparing the values of sine and cosine of (30o) angle— Fig. 3, we find the sine of the angle, which is ratio of the opposite side to the hypotenuse, computable exactly and is equal to (1/2). However, when determining the cosine of the angle, there is a gap in the adjacent side. That gap is indeterminable in terms of the elements defining the geometry. So, how does the gap relate to the infinite string of digits in an irrational number?
The infinite array of decimals in an irrational number does not relate to a specific gap in a given geometry. It relates to our continued attempts to compute its size using smaller and smaller scale units to redefine the whole geometry, as we continue our computation. Each decimal computed reduces the scale of the units adopted by a factor of ten, because of our decimal system. When the ratio of elements in the numerator to those in the denominator in a given geometry, such as the circle, are not based on a specific multiplier— i.e., when their ratio varies according to the number of elements, successive computations do not reproduce the same digit in successive computations. Thus, digits fluctuate indefinitely. For example, a circle’s circumference defined by 100 elements has 31 elements in its diameter. When elements defining its circumference are increased ten folds, to 1000 elements, its diameter increases to 318 and not 310, because the multiplier in each case is different. As such, the ratio fluctuates with each successive computation. In the case of inexact fractions, in which a pattern of digits repeats indefinitely, the multiplier of the two parts of the fraction is constant.
From our perspective on physical reality, spherical elements as the three shown in Fig. 4, may represent either an equilateral triangle or a circle, because we can assume either straight or curved lines joining their centres. However, as we stated above, such modelling is inaccurate. In fact it could be unrepresentative of the real geometries and we could test that by checking the resulting properties, areas and volumes of the models developed assuming lines of theoretical points.
Let us follow the logic proposed here step by step and consider gradual betterment of the definition of the three element circle in Fig.4. The next geometric shape would have four elements. The geometry is more of a parallelogram, which is some way away from a circle. In this case too, the geometry produced lacks accurate definition. To produce anything resembling a circle requires at least five elements. Any subsequent increase in the number of elements would improve the definition, as shown in Table 1, below.
Since the elements of space are indivisible, neither π nor any other number could be considered irrational at the level of the individual elements, because decimals at that level are meaningless. In any case, any attempt to introduce such a system would render the models unrepresentative of geometry at that level. Therefore, we must set aside the concept of decimals. Only fractions could be used, which may or may not be reducible to whole numbers. For example, a circle’s circumference defined by (22) elements would have (7) elements defining its diameter. Its value for π cannot be an irrational number. It is expressed as (π = 22/7 = 3), leaving out the fraction or decimal as an indefinable quantity relating to the gap in the diameter. To increase the definition of that circle, it has to be modelled using a larger number of elements.
Decimals become relevant only when we move away from modelling in elements to modelling in groups of elements, in multiples of tens for example. Only then would reducing fractions to decimal becomes meaningful. As more groups of elements are used in tens, hundreds, thousands, etc., ratios acquire greater definition as decimal places. However, despite its apparent infinity, the process is finite. It is infinite only in theory and in as far as we which to increase the level of definition. In reality, the limit is determined by the finite size of the elements of the fabric of space. This means it is impossible to define an absolutely perfect circle or a sphere regardless of the amount of computation we may carry out. Therefore, the ever-increasing decimals in computing π represent continuous increase in the graphic resolution of curvature, which may or may not be representative of the system to which they relate.
We could consider the argument for the source of irrational numbers and their relationship with the geometry of the fabric of space from a different perspective. If we attempt to increase the definition of a circle at the quantum level, we might consider adding one element to its circumference. However, by doing so, we find in most cases we have reduced the accuracy of its definition, because we would have increased the gap in the diameter. Increasing the circumference by a single element does not generate enough space in the diameter to add an element to it. A series of simple sketches starting with a hexagonal shows that for every one element added to the diameter of a circular geometry, three elements must be added to the circumference in every six successive steps, then in the seventh step, for one element added to the diameter four elements, rather than three, have to be added to the circumference. This pattern repeats indefinitely— the first few steps of it are given in Table 1. Note that the case in which four elements are added to the circumference for every one element added to the diameter, are shown in italic.
Given the dependency of the value of π on the number of elements defining the circumference and diameter of a circle, could π be considered a constant? The answer depends on whether one considers physical reality defined by theoretical points of no physical dimensions, in which case π is inherently a constant that imposes itself on physical reality, or considers physical reality defined by elements of finite size, in which case π is a variable parameter the value of which is dictated by the number of elements defining a given circle. The same question applies to all other irrational numbers.