mechanical Engineering basic questions and answers:-
Sunday, September 14, 2014
Wednesday, September 10, 2014
Maths in nature
Maths in nature
Mathematics might seem an ugly and irrelevant subject at high school, but it's ultimately the study of truth - and truth is beauty! You might be surprised to find that maths is in everything in nature from rabbits to seashells.Many mathematical principles are based on ideals, and apply to an abstract, perfect world. This perfect world of mathematics is reflected in the imperfect physical world.
Sun-Moon Symmetry
Coincidentally, while the sun’s width is about four hundred times larger than that of the moon, the sun is also about four hundred times further away. The symmetry in this ratio makes the sun and the moon appear almost the same size when seen from Earth, and therefore makes it possible for the moon to block the sun when the two are aligned.
The Earth’s distance from the sun can increase during its orbit—and when an eclipse occurs during this time, we see an annular, or ring, eclipse, because the sun isn’t entirely concealed. But every one to two years, everything is in precise alignment, and we can witness the spectacular event known as a total solar eclipse.
Fibonacci numbers:The growth pattern following the logarithmic spiral can be found not only in plants but also in auricle, cochlea, fingers, seahorse’s tail, ram’s horn, and nautilus, etc. There is very important reason for these to grow in logarithmic spiral pattern. If they do, they can keep the same shape when they were young and after they were fully grown. The cochlea in your ear has nice geometric shape for best hearing. If it changing shape while you are growing-up, you will have hearing problem.
Likewise, if bones in your fingers don’t grow in logarithmic spiral pattern, you will have hard time to grasp objects with your fingers when you become a grown-up.
Fibonacci numbers can also be found in sneeze wort. If we draw horizontal lines through axis of the sneeze wort, we can see a growing pattern of the stem and leaves. The main stem produces branch shoots at the beginning of each stage. Branch shoots rest during their first two stages, and then produce new branch shoots at the beginning of each subsequent stage.
The same law applies to all branches. Now, if we count the number of branches in each section, the counted numbers are all Fibonacci numbers. Furthermore, if we count the number of leaves in each stage, they also form Fibonacci numbers.
Nature holds many evidences of God’s creation. One of the evidences of God’s creation inscribed in nature is mathematical principles. The well-known mathematical principles found in nature are golden ratio, golden angle, golden rectangle, Fibonacci sequence, logarithmic spiral, and fractal. Let’s first briefly describe what these mathematical principles are.
When the ratio of the larger length to the smaller one is equal to the ratio of the sum of the larger and smaller lengths to the larger one (a/b = (a + b)/a if expressed algebraically, it is called golden ratio. The golden ratio is approximately 1.618 (1+sqrt(5))/2). The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio. The approximate value of the golden angle is 137.51 degrees. The golden rectangle is one whose side lengths are in the golden ratio.
The Fibonacci sequence is the series of numbers where the next number is found by adding the preceding two numbers (ex: 1, 1, 2, 3, 5, 8, …). The logarithmic spiral is the one within which any line emanating from the origin cuts the curve under a constant angle.
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Leaves, petals and other organs in plants are arranged orderly and follow some mathematical rules. This geometric arrangement of organs is the main topic in phyllotaxis study. In about 80% of plants, leaves are arranged along the stem tracing a helix. Suppose we fix our attention to some leaf on the bottom of a stem on which there is a single leaf at any one point. If we number that leaf "0" and count the leaves upward along the stem until we come to the one which is directly above the starting one, the number we counted becomes one of the Fibonacci sequence.
If we count how many revolutions we made when we count the number of leaves along the stem, the number of revolutions also becomes one of the Fibonacci sequence. The arrangement of leaves can be expressed as a ratio of number of leaves to number of revolutions. For example, if the number of leaves in our sample plant is "5" and the number of revolutions is "2", then our plant is said to have prophylaxis 2/5. Each plant can be characterized by its own prophylaxis.
Almost always the ratios encountered are ratios of consecutive or alternate terms of the Fibonacci sequence (for example, 1/2, 1/3, 2/5, 3/8, 5/13, etc.; notice here that numerators and denominators are also Fibonacci numbers). The angle formed by two adjacent leaves remains approximately constant and approaches to the golden angle (which is 137.5 deg) as the number of leaves increase. If they have golden angle, all of the leaves in the plant receive the same amount of sun light and can collect water effectively and direct it to the center of the plant to pass down to the root.
Not only for the leaves, but also for the buds, seed heads, and shoots follow mathematical rule. If seed head has golden angle, it can have the largest number of seeds in a given area (sunflower seed is a good example). If we look closely to these shoots or seed heads that have golden angle, we can find spiral patterns (logarithmic spiral) as shown in fig. Here, we can find interesting properties in these spirals. The number of spirals is two or three adjacent Fibonacci numbers.
For example, the number of spirals is 21 in clockwise direction and 34 counterclockwise direction in daisy flower, 8 clockwise direction and 13 counterclockwise direction in pine cone, 5 counterclockwise direction, 8 clockwise direction, and 13 counterclockwise direction in pineapple, 5 clockwise direction and 8 counterclockwise direction in cauliflower, and 13 clockwise direction and 21 counterclockwise direction in romanesco broccoli. If the number of spirals form Fibonacci numbers, they preserve the same shape while they are growing. For example, the shape of the daisy flowers is circular when they are small and keeps the same circular shape while they are growing. The mathematical principle behind having invariant shape is that the spirals having Fibonacci numbers form logarithmic spiral and anything growing in logarithmic spiral pattern doesn't change shape.
With the sun having a diameter of 1.4 million kilometers and the Moon having a diameter of a mere 3,474 kilometers, it seems almost impossible that the moon is able to block the sun’s light and give us around five solar eclipses every two years.
Mathematics might seem an ugly and irrelevant subject at high school, but it's ultimately the study of truth - and truth is beauty! You might be surprised to find that maths is in everything in nature from rabbits to seashells.Many mathematical principles are based on ideals, and apply to an abstract, perfect world. This perfect world of mathematics is reflected in the imperfect physical world.
Sun-Moon Symmetry
Coincidentally, while the sun’s width is about four hundred times larger than that of the moon, the sun is also about four hundred times further away. The symmetry in this ratio makes the sun and the moon appear almost the same size when seen from Earth, and therefore makes it possible for the moon to block the sun when the two are aligned.
The Earth’s distance from the sun can increase during its orbit—and when an eclipse occurs during this time, we see an annular, or ring, eclipse, because the sun isn’t entirely concealed. But every one to two years, everything is in precise alignment, and we can witness the spectacular event known as a total solar eclipse.
Fibonacci numbers:The growth pattern following the logarithmic spiral can be found not only in plants but also in auricle, cochlea, fingers, seahorse’s tail, ram’s horn, and nautilus, etc. There is very important reason for these to grow in logarithmic spiral pattern. If they do, they can keep the same shape when they were young and after they were fully grown. The cochlea in your ear has nice geometric shape for best hearing. If it changing shape while you are growing-up, you will have hearing problem.
Likewise, if bones in your fingers don’t grow in logarithmic spiral pattern, you will have hard time to grasp objects with your fingers when you become a grown-up.
Fibonacci numbers can also be found in sneeze wort. If we draw horizontal lines through axis of the sneeze wort, we can see a growing pattern of the stem and leaves. The main stem produces branch shoots at the beginning of each stage. Branch shoots rest during their first two stages, and then produce new branch shoots at the beginning of each subsequent stage.
The same law applies to all branches. Now, if we count the number of branches in each section, the counted numbers are all Fibonacci numbers. Furthermore, if we count the number of leaves in each stage, they also form Fibonacci numbers.
Nature holds many evidences of God’s creation. One of the evidences of God’s creation inscribed in nature is mathematical principles. The well-known mathematical principles found in nature are golden ratio, golden angle, golden rectangle, Fibonacci sequence, logarithmic spiral, and fractal. Let’s first briefly describe what these mathematical principles are.
When the ratio of the larger length to the smaller one is equal to the ratio of the sum of the larger and smaller lengths to the larger one (a/b = (a + b)/a if expressed algebraically, it is called golden ratio. The golden ratio is approximately 1.618 (1+sqrt(5))/2). The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio. The approximate value of the golden angle is 137.51 degrees. The golden rectangle is one whose side lengths are in the golden ratio.
The Fibonacci sequence is the series of numbers where the next number is found by adding the preceding two numbers (ex: 1, 1, 2, 3, 5, 8, …). The logarithmic spiral is the one within which any line emanating from the origin cuts the curve under a constant angle.
.
Leaves, petals and other organs in plants are arranged orderly and follow some mathematical rules. This geometric arrangement of organs is the main topic in phyllotaxis study. In about 80% of plants, leaves are arranged along the stem tracing a helix. Suppose we fix our attention to some leaf on the bottom of a stem on which there is a single leaf at any one point. If we number that leaf "0" and count the leaves upward along the stem until we come to the one which is directly above the starting one, the number we counted becomes one of the Fibonacci sequence.
If we count how many revolutions we made when we count the number of leaves along the stem, the number of revolutions also becomes one of the Fibonacci sequence. The arrangement of leaves can be expressed as a ratio of number of leaves to number of revolutions. For example, if the number of leaves in our sample plant is "5" and the number of revolutions is "2", then our plant is said to have prophylaxis 2/5. Each plant can be characterized by its own prophylaxis.
Almost always the ratios encountered are ratios of consecutive or alternate terms of the Fibonacci sequence (for example, 1/2, 1/3, 2/5, 3/8, 5/13, etc.; notice here that numerators and denominators are also Fibonacci numbers). The angle formed by two adjacent leaves remains approximately constant and approaches to the golden angle (which is 137.5 deg) as the number of leaves increase. If they have golden angle, all of the leaves in the plant receive the same amount of sun light and can collect water effectively and direct it to the center of the plant to pass down to the root.
Not only for the leaves, but also for the buds, seed heads, and shoots follow mathematical rule. If seed head has golden angle, it can have the largest number of seeds in a given area (sunflower seed is a good example). If we look closely to these shoots or seed heads that have golden angle, we can find spiral patterns (logarithmic spiral) as shown in fig. Here, we can find interesting properties in these spirals. The number of spirals is two or three adjacent Fibonacci numbers.
For example, the number of spirals is 21 in clockwise direction and 34 counterclockwise direction in daisy flower, 8 clockwise direction and 13 counterclockwise direction in pine cone, 5 counterclockwise direction, 8 clockwise direction, and 13 counterclockwise direction in pineapple, 5 clockwise direction and 8 counterclockwise direction in cauliflower, and 13 clockwise direction and 21 counterclockwise direction in romanesco broccoli. If the number of spirals form Fibonacci numbers, they preserve the same shape while they are growing. For example, the shape of the daisy flowers is circular when they are small and keeps the same circular shape while they are growing. The mathematical principle behind having invariant shape is that the spirals having Fibonacci numbers form logarithmic spiral and anything growing in logarithmic spiral pattern doesn't change shape.
With the sun having a diameter of 1.4 million kilometers and the Moon having a diameter of a mere 3,474 kilometers, it seems almost impossible that the moon is able to block the sun’s light and give us around five solar eclipses every two years.
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