Fractal Geometry and the Bible, Part 2--the mathematics:
CHRISTIAN HEADS UP:
1. All math hubs have something about God in their final paragraph. In this hub it is actually the last 3 paragraphs, and they address how God's questions are over our heads( beyond our understanding)
THE COASTLINE OF GREAT BRITAIN:
2. In the Wikipedia article,
we are given a practical real-life example of fractal geometry in action. It demonstrates how the length of what you are measuring may depend on the length of the measuring instrument. The objective in this hub is to give you some logical and mathematical confirmation of why God's question to Job is still a valid question to us today. If you were to ask someone, as God did to Job, the question, How long is Britain's coastline?, obviously(from the link above) the answer would depend on the length of the measuring instrument. We are given 2 answers in this link. It is either 2400Km( 1,491 miles) or 3400Km( 2,113 miles), which give us a difference of 622 miles. The official length is given as 17,820Km( 11,073 miles). If you are selling Great Britain per linear foot, then you want to refer to the last figure, but if you are the buyer then stick to the first figure.
CALCULATING THE VARIABLES:
3. We are not given the values of the variables, M and D, in the above link( equation is repeated at L1); therefore, we will calculate them. The article says M is a constant, but since we do not know what it is, then it is a variable until we determine what it is. So M is a constant and D is the fractal dimension. We plug in the values we do know at L2 and L3. We have 2 equations with 2 unknowns--sounds like simultaneous equations( Pemekwulu shows you 3 ways to solve them). We divide both sides of each equation by their right side factors; thereby isolating M, and since M = M we can equate the left sides at L4. We cross multiply giving us L6 and L7.
LOG IT AND SOLVE IT :
4. When the variable you want to solve is an exponent, then the easiest way to solve for it is with logarithms. We will use common logs( base 10) at L8, and the laws of logarithms tells us we can put the exponent, 1-D, in front of the log at L9. Log is an operator; it tells you what to do to a number or a function, just as +(plus) or --(minus) or X(times--multiply) tells you what to do to a number or function. We isolate the exponent, 1--D at L10 and also take the logarithms at L10 giving us L11, which is the answer after subtracting 1 and multiplying by a --1( negative 1) on both sides of the equation. So we have a fractal dimension, D, of 1.25125.
SOLVING FOR THE CONSTANT, M:
5. Next we simply plug the value of D into one of the equations at L2 or L3; I chose L2 at L12, and arrived at an answer of 9,085.4 for the constant, M, at L14. We arrive at the final equation at L15, which is the same as what the link in paragraph 1 has except we plugged in the values of M and D so that we can put it to use. Remember that the exponent of G is 1-D, or one minus the fractal dimension of 1.25125, which gives us a minus 0.25125.
A NEW MEASURING STICK:
6. The official length of Great Britain's coastline is given as 11,072.76 miles, which is about 17,820Km. We plug that into L16, and then we will solve for G, the length of the measuring stick, at L17 to L23. At L22 we raise 10 to the Log(base 10)G but that cancels out the 10^Log 10 , and we are just left with G = 10^(--1.16444), which is 0.06848Km at L23. This is the same length as 224.7 feet at L23. This represents the length of the measuring stick to get a value of 17,820 Km for the length of Great Britain's coastline.
ACME ANT COMPANY MEASURES THE COASTLINE:
7. We hire Acme Ant Company to measure the coastline with their hardest working ant, Antie. I'm no expert on ants, but it shouldn't be too tough to find one about 0.3 inch long at L24. We plug this value of G into the equation of L15 at L24. We have to keep our units consistent; therefore 0.3 inch must be converted to kilometers(Km). This is done by multiplying by those conversion ratios. This works because each ratio is equal to one as shown at L26, 27 and 28. When doing this you must make sure the unit dimensions cancel so that you are left with Km. You can do this because 1 foot / 12inches equals 12 inches / 1 foot. Antie measures the coastline to be 175,492 Km at L25, which is 109,051 miles at L29; this is a distance of 4.36 times around the earth.
LENGTH OF COASTLINE----->∞(INFINITY) AS G----->0(ZERO):
8. At the length of a virus at L30, the length of Great Britain's coastline is measured as 3,276,916 miles at L32, which is a distance of 13.72 trips to the moon( nearly 7 round trips) at L33. As G approaches zero, the measured distance approaches infinity! This gives mathematical confirmation of why God's question to Job is still valid.
200 KILOMETER MEASURING STICK:
E1. The image below uses the 200km measuring stick mentioned in the text. There are 12 of them giving the length of Great Britain's coastline to be 2,400km. Compare this to the next image.
50 KILOMETER MEASURING STICK:
E2. In this next image the measuring stick is 50km long; therefore, 68 of them can fit on the coastline thereby including more of the curve-linear distances. So 68 times 50 gives a length of 3,400km.
GOD'S QUESTIONS ARE BEYOND OUR COMPREHENSION:
9. Questions as those in Job, chapter 38, are beyond our comprehension. I do not think we can properly articulate what God is asking, let alone answer His questions. This verse of which this hub is about, 38:18, is a good example of what I'm talking about. We have assumed this question is concerning the circumference of earth, or perhaps its diameter. At least that is what some Bible commentaries claim; however, closer scrutiny reveals something much deeper. An argument can be made that God is not referring to circumference but rather surface area. Analogous to an infinite line as described in this hub, it is possible to have an infinite surface area enclosing a finite volume; nevertheless, I think God's question goes much deeper than even a surface area approaching infinity. We cannot answer God's question even if we take a simple--minded approach and assume He is referring to the circumference of the earth. Although it would be very much more complicated if we assume God is referring to fractal geometry and the surface area of earth; nevertheless, I still think this would be a simplistic assumption of what God is actually asking. Just stop for a moment and ponder some of the other questions God asked Job: At Job 38:19 God is implying that all light has already been created and is being stored in some other place or dimension. It is also implied here that darkness has structure because it can't have a "place thereof " if it has no existence. Verse 20 implies there is a specific path by which light, and perhaps darkness, must transverse in order to manifest themselves in our 3 dimensional existence, and that path for light was blocked at Exodus 10:22,23. Verse 36 could be referring to, among other things, the information stored in DNA molecules; however, the "inward parts" could be the structure of space itself. This would explain why God had to stretch space with understanding at Jeremiah 51:15; it would not take any understanding to stretch a rubber band but it would most certainly require some skill to store information in the rubber band through the process of stretching it.
THE BREADTH OF EARTH, AND STRETCHING OF SPACE:
10. Thousands of years before we ever discovered it, God told us that He is stretching space( Isaiah 45:12; Jeremiah 10:12; 51:15), but along with stretching space, God includes spreading out the earth( Isaiah 42:5; 44:24). The original Hebrew word that was translated to, "breadth" at Job 38:18 was also translated to enlarge at Ex. 34:24; Deuteronomy 12:20; 19:8; 33:20, and many other places. It was also translated as, make room, make wide, open wide, etc. Additionally a study of the word seems to indicate it is usually present tense( happening now). The word that was translated to spreading out means to beat and spread out as a blacksmith would do to a piece of metal-- beat and flatten it out. The Bible seems to distinguish between the stretching of space and the breadth, or spreading out, of the earth; however, I believe they both involve space itself. Finally, Isaiah 51:13 mentions the foundations of earth.
COMPLEXITY AND GOD'S QUESTIONS---A SUMMARY:
11. So here is my take( interpretation) on this tiny portion of all God's unfathomable questions to Job: Yes, I believe God was referring to fractal geometry and the circumference of the earth, but also to fractal geometry and the surface area of the earth. They( perimeter and area) can not be known because of dimensional complexity. We do not know the fractal dimension and besides, it probably changes depending on position and time. God implies that the length of the measuring instrument changes length at Job 38:5 to coincide with space being stretched by God. Why else would He say something so strange as, "STRETCHED the line upon it "? The foundations are necessary because the earth would not stay in its orbit around the Sun, nor would it remain a sphere as God stretches space. Yes I know about the triple integral( ∫ ∫ ∫ ) that describes earth as a sphere under the influence of gravity, but the stretching of space changes things, especially under Einstein's General Relativity in which gravity is described as a manifestation of space-time. I can go on and on because God's magnificence is infinite. Simple-minded thinking, as for example that laughable conjecture called Darwinian Evolution, is not going to get us any closer to understanding God's creation. The best we can hope to do is continue to learn but being increasingly more confused the more we look into it. One final note. I mentioned that enlarge is usually present tense; therefore, the circumference and surface area is constantly changing, on top of all the other complexities. So anyway: GOOD LUCK IN ANSWERING GOD'S QUESTIONS!!!
CHECK OUT THIS HUB:
An exhaustive, professionally written and informative hub written by Syzygyastro called Defining the Fractal will give you much to ponder on this subject, and there is a really cool video at the end of the hub that demonstrates how amazing fractals are.