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God, the Universe, & the Value of Pi – Part 1

By Malama Katulwende

Drama in Heaven

Malama Katulwende

Malama Katulwende

Not so long ago, a renowned pastor from Thorn park Assemblies of God sought God’s immediate intervention in a case  in which some men and women of science were trying to reproduce the “first moments of creation”.  Afraid that this would profane the name of God and consequently befoul the land, Pastor Mtembo sought refuge in his church in order to seek the Lord’s guidance over this matter. Thus, bending on his knees and clutching the Holy Bible in both hands, the Man of God cried out to the vaults of heaven, and immediately heard a voice speak back to him – like an echo:

God: Why are you crying out, my son?

Pastor: [Looking up with tears in his eyes] My God, pardon me for saying this, but why do you aid the wicked? Look at what the Satanists recently discovered, something they have been calling the “God particle”. They have tried to simulate the moment of creation, when you, according to your word in the Holy Bible, created the earth and the heavens in six days, from nothing. Now these infidels, blinded by science and the ingenuity of the devil, have conspired against thee by splitting the atom and blasting electrons at almost the speed of light in a particle accelerator in order to prove the Big Bang. Oh, my God, how can this be? How can you allow the Enemy to triumph over the Cross?

God: [Surprised] My son, I have already received a report of these wretched events from Pentecostal churches across the land. The anointed ones want me to take action. I am also aware that these cosmologists have even constructed a space station above the earth, and they are pondering to launch ships to locations beyond the solar system. Now, since you are my devotee for whom I will do anything, let me now hear what you want me to do about this menace.

Pastor: My God, in order to punish the wicked, let us destroy their Large Hadron Collider (LHC), which is their largest and highest energy particle accelerator. We should also target other stations where they attempt to conduct experiments to prove what they call the “Higgs boson”, or the “God particle”.

God: [Looking reflective] The particle accelerator. How do you propose me to destroy it? Can we use a smart bomb like the Americans used in Iraq?

Pastor: No my Lord. Any bomb would be too unpleasant because it would be very devastating in its consequences. Rather, let us use a more subtle method – let us destroy the circle!

God: [Exclaiming] What! The circle? How do we eliminate the particle accelerator by destroying the circle? I can’t quite see the connection here, am afraid.

Pastor:  My God, the particle accelerator is circular, including almost all the tools which the scientists use. By altering the properties of a circle, we will have put an end to all the works of these evil people.

God: [Beaming with contentment] Very good, pastor. I will act as you have suggested. I will, from this very moment, change the value of Pi so that nothing ever approaches it nor will anything geometrical or otherwise be expressed either as Pi, or in terms of Pi ever again.   Have a good day Pastor, and enjoy the fray as we scamper the depraved souls into the valley of Hell! Ha ha ha ha ha ha ha ha ha ha!!

…Now let us take this story at face value, and suppose that indeed God exists. Let us, to wit, also assume that the God whose existence we have so far admitted to be true, has such powers and qualities as the pastor Mtembo has attributed to Him; that is to say: God is uncreated, all-powerful, all-seeing, omnipotent, immortal, and that God alone brought into existence from nothingness all that is in space and time.

Now let us also suppose that God possesses the freedom to do whatever takes to His fancy.  By “freedom” we mean having the ability to act autonomously and without any external influences; that God is a sovereign entity who acts independently and lacks restrictions in any one of His choices. He is, so to speak, absolutely free to do anything He chooses – including, for example, changing the direction of the earth’s rotation on its axis, or suspending gravity to allow his devotees to hop like astronauts in zero-gravity conditions.

Suppose that everything we have so far heard about God’s free-will and His self-determination – from the Churches, especially Pentecostal movements most of whose members are Christian extremists – is held to be true, how would His absolute freedom interact with the structure of the universe?  How would God’s unfettered freedom behave in contexts of the laws of nature as we understand them to be?

In order to test the case of God’s absolute freedom and how that freedom would impact the universe, we have chosen a very curious concept – the value of pi. In particular, we ask two very simple questions: would God really dare intervene in the workings of the physical laws of the universe – whether the universe be one, several or infinite – in order to accommodate all or some of His wishes and fantasies?  Would He, for instance, deploy His absolute freedom and free-will to fiddle with the value of pi?  What would be the metaphysical consequences of playing with pi?

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The concept of pi ( ) refers to the constant ratio of the diameter to the circumference of any circle irrespective of the number of degrees contained within that circle*. In historical retrospect, the idea of pi had baffled mathematicians for a long time. The challenge was how to attempt to divide the diameter of a circle (which is a straight line) into a circumference of a circle (which is a curved line), and arrive at the exact value of pi.  No matter how hard mathematicians tried, the act of bending either the straight line or the curved line altered the nature of the problem and yielded naught.

To put it simply, as soon as one of the line is bent the results are distorted, whereas the very thickness of the line being measured in length – whether one measures the inner part of the curved line of the circumference or the outer edge – makes a great deal of a difference, especially when one is attempting to achieve an exactness in the concept of pi to hundreds or even thousands of decimal places.

Throughout history, though, the value of pi ( ) has taken on many variations. For example, the Babylonians assumed it was 3 1/8; the Egyptians thought it was 4(8/9)²; for the Chinese, the value of pi was 3.1724; and for Archimedes, it was 3.14084 < < 3.142858 (31/7); for the great Ptolemy, it was 3.14167; for Fibonacci, 3.141818; and finally for Viète, 3.141592635 < < 3.1415926537. The computer language of FORTAN estimated pi to be 3.14159265358979324.

Having said this, however, let us now inquire why the ancient people were so fascinated by the geometry of the circle and consequently interested in knowing the value of pi. What was the significance of pi then, and does this importance still hold true today?

According to a story recounted by Simplicius, the Greek philosopher, Plato, posed the following question for Greek astronomers: “By the assumption of what uniform and orderly motions can the apparent motions of the planets be accounted for?”  Like other thinkers before him, Plato had looked at the motions of the celestial bodies and sought a mathematical description to account for observable phenomena. The philosopher’s proposal was that the apparent erratic motions of the planets could be explained by combinations of uniform circular motions centered on a spherical Earth, an idea that was in vogue in the 4th century BC.

This problem was attempted by one Eudoxos of Knidos , who was born approximately  between 395-390 BCE and lived 53 years.  A great man who made significant contributions to geography, metaphysics, and ethics, Eudoxos was best known for his work in geometry, the theory of proportion, and astronomy. In astronomy, in particular, he was the first to present a general, geometrical model of celestial motion, whose five principles are the following: (1) The earth is the center of the universe. (2) All celestial motion is circular. (3) All celestial motion is regular. (4) The center of the path of any celestial motion is the same as the center of its motion and (4) The center of all celestial motion is the center of the universe.

Eudoxus accepted the Platonic notion of the rotation of the planets around the Earth on crystalline spheres, yet noticed discrepancies with observations. He attempted to adjust Plato’s model by postulating that each crystalline sphere had its poles set to the next sphere. In order for his model to preserve the five basic principles and account for the motions of the planets, Eudoxus needed to construct the apparent motions of celestial bodies as combinations of circular motions, a mathematical description which formed the basis for the theory of concentric spheres – the basic idea of which influenced mathematical astronomy from Ptolemy up to Kepler.

[*For the purposes of this article we will take it for granted that perfect circles exist as physical realities in natural phenomenon.]

Today, however, the Eudoxan mathematical description of the motion of celestial bodies may not hold true. His system has been faulted with serious flaws. Yet in spite of these flaws, Euxodus’ underlying assumption that all planets, apart from being spherical themselves, do actually move in “circular” or “elliptical” orbits is still valid today. In essence, therefore, “circularity,” – thus implying the value of pi – has since permeated non-Euclidean or curved geometries, even though the mathematical description of pi does not depend on the circle: π is twice the smallest positive x for which cos(x) equals 0.

Now circles, ellipses and spheres (which are denoted by pi in flat geometry) are not merely ideal concepts which subsist in the mind of the perceiver. On the contrary, they are part of natural objects and the behavior of those very objects.

In the world around us, for example, we abstract circles, ellipses and spheres every day: the egg, when cut at various angles to the horizontal plane, yields fantastic circles and ellipses or a parabola. The cross section of a tree produces circular rings; the firing of a trajectory from point A to B yields a parabolic curve;   droplets of water are sometimes spherical; the ripples in the pool as they move outwards are sometimes circular; a sunflower head is circular, so that in general, nature herself demonstrates  the fact that circularity and, therefore pi,  do exist in nature itself.

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Perhaps for this very reason pi is one of the most known mathematical constants found in many formulae in trigonometry and geometry, particularly those concerning circles, ellipses, or spheres. Pi is also found in formulae from other branches of science, such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism.

Yet perhaps the best demonstration of the application of the concept of circularity (and therefore pi) is the case of the atom. In his planetary model of an atom, Ernest Rutherfold (1911) conceived of the structure of an atom as being made up of constituent parts – electrons, protons, and the nucleus. The central, positively charged core is the nucleus, and the negatively charged particles called electrons are found in orbits around the nucleus.  These orbitals or shells are circular.

Now what is the shape of atoms, you might ask?  Well, a study by Igor Mik­hail­ovskij and his collaborators at the Kharkov Institute of Physics and Technology in Ukraine, imaged the shapes of those orbitals in carbon atoms by improving an old imaging technique called field-emission microscopy. They ‘fashioned a chain of carbon atoms, dangled it from a graphite tip, and then placed it in front of a detection screen. When they applied an electric field of thousands of volts between the graphite and the screen, electrons flowed one by one through the graphite and along the carbon chain, until the electric field pulled them off the last atom in the chain. From the places where the electrons landed on the screen, the investigators could trace back the points where they left their orbital on the last atom. The “denser” parts of the probability clouds had a higher chance of emitting an electron, and the information from many electrons combined into an image of the clouds.’ “We really have an image of single atoms,” Mikhailovskij [said]’. This image of single atoms is sphericalhttp://www.scientificamerican.com/article.cfm?id=the-shape-of-atoms.

The classical atomic model is, however, inadequate to explain the nature of orbitals. In the natural world electrons normally remain in an uncertain and non-deterministic, orbital path around (or through) the nucleus, defying the mathematical description of classical electromagnetism.  That is to say, unlike planets orbiting a sun, electrons are charged particles which, in losing some of their energy, may spiral toward the nucleus and collide with it in a matter of seconds.  This means the orbits are unstable. Secondly, the planetary model could not explain the highly peaked emission and absorption spectra of atoms that were observed.

On the other hand, quantum models, starting with the Borh model  of an atom,  allow for far more dynamic, chaotic possibilities in explaining the nature of a particle. Using a mathematical function called the wave function, they provide information about the probability amplitude of the position, momentum, and other physical properties of a particle such as an electron. The wave function treats the object as a quantum harmonic oscillator, and the mathematics is similar to that describing acoustic resonance. In a quantum mechanical model, the ground state is a non-zero energy state that is the lowest permitted energy state of a system, unlike the “traditional” systems of classical atomic theory in which the ground state is thought of as simply being at rest, with zero kinetic energy.

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 The result of expressing electrons in terms of waveforms is that mathematically it is impossible to simultaneously derive the position and momentum of an electron at any given time. This is called the Heisenberg uncertainty principle – named after the theoretical physicist Werner Heisenberg, who first described it and published it in 1927.   Thus this principle has demolished the Bohr’s model, with its neat, clearly defined circular orbits. The modern atomic model defines the positions of electrons in an atom in terms of probabilities. That is – an electron can potentially be found at any distance from the nucleus, but, depending on its energy level, exists more frequently in certain regions around the nucleus than others; this pattern is referred to as its atomic orbital, which come in a variety of shapes such as sphere, dumbbell, and torus.

Can God erase the value of Pi?

We have stated that for thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Archimedes and Liu Hui, both of whom used geometrical techniques based on polygons to estimate the value of π, appreciated its relation to physical phenomenon.  Others (such as Madhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan, respectively), used new algorithms based on infinite series and thus revolutionized the computation of π – but were no different.

In general, then, we have discovered that pi is here to stay – it is inseparable from natural events.  The question we should now be addressing is this: Can God erase the value of pi? By “erasure” we mean causing pi to disappear or, at the very least, altering its numerical value.

 

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Comment:

One response to “God, the Universe, & the Value of Pi – Part 1”

  1. Thank you for the insightful comment. There is the last part remaining, which will be conclusive and bring out something. On the other hand, this discussion is also meant to stretch our minds far and imagine the “impossible”. I am aware most religious/christian people would not not like this, but the greatest value lies in asking questions and trying to find answers in the dark.

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