- 2966 - MATH - Invented to Solve Problems? This review discusses the languages of math and how they were invented to solve problems. Mathematics is a language. It can be used to explain observations, to solve problems, to find results and to predict results in the future.
----------------------------------- 2966 - MATH - Invented to Solve Problems?
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- Math’s effectiveness in explaining nature is totally unreasonable. Why does it work so well? It is a wonderful gift that we neither fully understand nor deserve.
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- Isaac Newton invented “Calculus” to explain motion and things that change. He breaks change down into infinitesimally small frame by frame sequences in order to solve problems. Lots of things change so Calculus has wide application in today’s world.
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- When Calculus is the rate of change of quantities it is interpreted as the slope of curves.
Example: velocity is the slope of distance versus time. This is called “Differential Calculus“. When the math is used for lengths, areas, and volumes of objects it is called “Integral Calculus“. Example: Calculus can add up the areas of circles and get the volume of a sphere.
- “Quantum Electrodynamics” is a more recent math invention that explains how light and matter interact. Scientists used this math to calculate the magnetic moment of an electron. Their answer agrees with experiment to an accuracy of a few parts per trillion.
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----------- electron’s magnetic moment = 1.0011565218073
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- “Quantum Mechanics” is a math concept that assumes matter and energy behave as particles as well as waves. It states that measurements at the atomic level are “always uncertain“. Position and velocity, time and energy are pairs can not both be known with precision. There is always a trade-off in play between these pairs.
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- “Group Theory” was a math invented in 1800 to solve polynomial equations . Groups are algebraic structures made up of sets of objects, say a set of integers, united under a specific operation, say addition, that obeys a specific set of rules, say adding zero to any number does not change the number.
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- Now, in the 21st century this same Group Theory is used to develop the Standard Model in Particle Physics. It models the building blocks of matter. The fundamental particles of matter are given a specific group called SU(3) that can be used to describe how atomic nuclei are held together. Example: SU(3) math predicts that there is a Higgs Boson, a force carrier particle that gives matter its mass.
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- The Large Hadron Collider in CERN, Switzerland, is a particle accelerator smashing protons and anti-protons together with 14 trillion watts of energy. Physicists have discovered the Higgs Boson in this debris.
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- “Group Theory” becomes a way to represent symmetry in nature. “Lie Groups” represent symmetries of solutions to differential equations that are infinite. There is a “Monster Group” the defines the structure of the Universe at the subatomic level.
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- Then, there is “Knot Theory” that studies knots, closed curves in 3 dimensions. The math was born out of attempts to model the atom. Today it is used in String Theory, Topology, DNA research and molecular biology.
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- “Topology” is the math that describes objects that remain unaffected by smooth deformations, such as stretching or squeezing, but do not involve tearing or cutting. Example: a doughnut is the same Topology as a coffee cup.
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- “Number Theory” is the math used in cryptography
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- The “General Theory of Relativity” uses the math of non-Euclidian geometry. Non-Euclidian are spaces where parallel lines converge and diverge.
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- “String Theory” is math that describes fundamental particles as musical notes of vibrating strings. The strings are very tiny , 10^-33 centimeters in length, but, they have no thickness. The strings are 1 dimensional, yet the math needed to describe their behavior must use 10 dimensions.
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- Space-time has 4 dimensions, the other 6 dimensions are called Calabi-Yau spaces. All the properties of elementary particles such as mass, spin , charge can be mathematically described in vibrating strings. The faster the vibration the greater the energy. Since matter and energy are the same thing, the faster the vibrations the greater the mass.
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- “Loop Quantum Gravity” is Knot Theory math that is trying to reconcile space -time relativity with quantum mechanics The math for one describes the Universe of big things. The math for the other describes small things, things at the sub-atomic level. But, the two math’s are not compatible. Neither one will work in the other’s domain.
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- “Chaos Theory” uses math whereby the tiniest change in the initial conditions can produce entirely different end results. Example: A butterfly wing flap in the Amazon can cause a tornado in Kansas.
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- Math is a language that just works. We can not really explain why. We just use it because it is reliable in learning stuff about our Observable Universe. There appears to be fundamental symmetries that Nature uses in her own math. There appears to be a grand design that we are discovering, not inventing.
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- Here is and example using Calculus to measure the mass of a comet. NASA spacecraft caught up with “Comet Hartley-2” on November 4, 2010 and took a picture of the icy dirtball that was 1.4 miles in length. Scaling the picture and using Calculus to determine the volume, the mass of the comet was put at 290,000,000 metric tons. This review shows the math involved.
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- News, will the “Comet Elenin” fulfill the 2012 prophecy of the end of the Earth to occur December 21, 2012. Alarmist want to make the most of this modest-sized icy dirtball that will pass within 22,000,000 miles of our planet.
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- The comet is 3 to 5 kilometers wide. Astronomers measure the mass of one object by its gravitational effect on another. But, these comets are far too small and too distant to have any measurable influence on Earth. Except, for alarmist news headlines.
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- Comet Hartley-2 flew by the Earth at its closest 11,000,000 miles away on October 20, 2010. That is between us and the Sun which is 93,000,000 miles away. The Earth-Sun distance is called an Astronomical Unit, an AU.
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- NASA sent a space craft to catch up with the comet. On November 4, 2010 the space craft took pictures from a distance of only 435 miles. The comet was dumbbell shaped and 1.4 miles in length. It was composed of water ice, carbon dioxide ice, and silicate dust.
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- It was rotating a complete cycle every 18 hours. This comet was first discovered in 1986. The picture showed a pitted surface free of large craters but evidence of active ejection of gas plumes. The scale of the picture was 1 millimeter equaled 25 meters.
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- The comet was 2,000 meters long. We can use Calculus to calculate the volume of the comet by measuring the radius at each point along its length then summing up, (integrating) an infinite number of areas of circles with each radius over the entire length.
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---------------------- Length ------------------- Radius
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------------------------ 0 ---------------------------- 0 meters
------------------------ 200 ------------------------- 420 meters
------------------------ 400 ------------------------- 500 meters
------------------------ 600 ------------------------- 410 meters
------------------------ 800 ------------------------- 290 meters
------------------------ 1000 ------------------------ 210 meters
------------------------ 1200 ------------------------ 220 meters
------------------------ 1400 ------------------------ 290 meters
------------------------ 1600 ------------------------ 340 meters
------------------------ 1800 ------------------------ 310 meters
------------------------ 2000 ------------------------ 0 meters
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- This dumbbell shape can be approximated by a function with Volume as a function of “x” length. The plot of the function is the y axis radius versus the x axis length, from 0 to 2 kilometers.
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---------- Volume = pi * radius^2 * length
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----------- y(radius) as a function of length x = -1.22x^4 +5.04x^3-6.78x^2+3.14x +0.03)
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----------- Volume = ( -1.22x^4 +5.04x^3-6.78x^2+3.14x +0.03)^2 * length
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----------- Volume = Integral 0 to 2 * pi * ( -1.22x^4 +5.04x^3-6.78x^2+3.14x +0.03)^2 * dx
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----------- Volume = Integral 0 to 2 * pi * ( -1.22x^4 +5.04x^3-6.78x^2+3.14x +0.03) *
- (1.22x^4 +5.04x^3-6.78x^2+3.14x +0.03) * dx
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----------- Volume = Integral 0 to 2 * pi * ( 1.49x^8 - 12.3x^7 + 41.94x^6-76x^5 + 77.55x^4 - 42.28x^3 + 9.46x^2 + 0.18x + 0.0009) * dx
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- The integral of a function of the form: Integral x^n * dx = x^(n+1) / (n+1)
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- Example : Integral 1.49x^8 = 1.49 / 9 * 2^9 = 0.17* (512) = 87
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------------------------Applying this integration to each term of the above function:
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---------- Volume - pi* (0.17x^9 - 1.54x^8 + 5.99x^7 - 12.67x^6 + 15.51x^5 - 10.57x^4 + 3.15x^3 +0.09x^2 + a constant ) from x equal 0 to 2
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--------- Volume = (3.14) ( 0.1655*2^9 - 1.5375*2^8 +5.9913*2^7 - 12.6667*2^6 + 15.51*2^5 - 10.57*2^3 + 0.09*2^2 +0.0009*2 + a constant
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---------- Volume (3.14) * (0.157)
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---------- Volume = 0.49 kilometers^3
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--------- Volume = 4.9 *10^14 centimeters^3
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- To get the mass of the comet we need to know the density in grams / cubic centimeter. Taking the average density from other comets:
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---------------------- Halley’s Comet ----------------- 0.6 gm / cm^3
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---------------------- Templ-1 Comet ----------------- 0.62 gm / cm^3
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---------------------- Borrells Comet ----------------- 0.3 gm / cm^3
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---------------------- Wild Comet ----------------- 0.6 gm / cm^3
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------------------------- A good estimate would be Hartly-2 density is 0.6 gm / cm^3.
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------------------------- mass = density * volume
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------------------------- mass = (0.6 gm / cm^3) * (4.9*10^14 cm^3)
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------------------------ mass = 2.9*10^14 grams
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------------------------- 1000 kilograms = 1 metric ton
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------------------------ Hartlley-2 mass = 290,000,000 metric tons
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- To double check this calculation another way is to estimate the dumbbell volume as 2 spheres connected by a cylinder. Using the scale of the picture where 1 millimeter = 25 meters. The two spheres are 1000 meter and 750 meter in diameter.
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------------------- 40 mm * 25 m/mm = 1000 meters
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------------------- 30 mm * 25 m/mm = 750 meters
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- The overall length is 2000 meters. Subtracting the diameters of the 2 spheres, 2000- 1750 gives the length of the cylinder to be 250 meters.
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------------------ Volume of a sphere = 4/3 *pi*r^3
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------------------ Volume = 1.33*3.14* (750 / 2)^3 = 2.2 * 10^8 m^3
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------------------ Volume = 1.33*3.14* (1000 / 2)^3 = 5.2 * 10^8 m^3
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----------------- Volume of a cylinder = pi * r^2 * length
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-------------------Volume = 3.14 * (250)^2 * 250 = 0.49 * 10^8 m^3
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----------------- Total Volume = 7.9*10^8 m^3
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----------------- Total Volume = 7.9*10^14 cm^3
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----------------- mass = (0.6 gm / cm^3) * (7.9*10^14 cm^3)
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----------------- mass = 4.7*10^14 gm
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----------------- mass = 470,000,000 metric tons.
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- This rough calculation is in the ballpark, but , about 60% too high. The Calculus calculation is thought to be the more accurate.
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- The NASA mission included a probe impact on Temple-1 Comet on July 3, 2005 . The mission launch was January 12, 2005. The mission was extended to meet up with Hartley-2 Comet on November 4, 2010 traveling 2,900,000,000 miles to get there. The total cost of the mission was $309 million. An expensive picture. But, a lot more than this was learned.
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-------------------------- To learn more math, request these reviews:
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- 1235 Equations are sentences in sort hand
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- 1213 How to calculate odds for poker hands.
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- 1086 Formulas for space time, velocity.
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- 1095 Math is a learned discipline.
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- 1042 Calculate areas by connecting the dots, Pick’s Theorem
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- 1041 The star with the golden ratio
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- 1042 The calculus of a circle.
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- 853 Math, the golden ratio
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- 803 Transcendental numbers, “e” and “pi”.
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- 649 The Greeks invented numbers.
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- 805 Gambling
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- 796 Science and Math, Part I
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- 798 Science and Math, Part II.
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- 799 Science and Math, Part III.
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January 8, 2021 MATH - Invented 1281 1292 2966
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--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
--------------------- --- Friday, January 8, 2021 ---------------------------
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