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MODULE 1 - MATHEMATICS IN OUR MODERN WORLD
WHAT IS MATHEMATICS?
Mathematics is the science that deals with the logic of shape, quantity and arrangement. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.
Further, mathematics was derived from the contributions of thinkers throughout the ages and across the globe. It gives us a way to understand patterns, to quantify relationships, and to predict the future. Math helps us understand the world — and we use the world to understand math.
"Mathematics is universal, it is everywhere."
Mathematics is the science that deals with the logic of shape, quantity and arrangement. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.
Further, mathematics was derived from the contributions of thinkers throughout the ages and across the globe. It gives us a way to understand patterns, to quantify relationships, and to predict the future. Math helps us understand the world — and we use the world to understand math.
"Mathematics is universal, it is everywhere."
PATTERNS IN NATURE
Patterns in nature are observable regularities of form that may be discovered in the natural world. These patterns repeat in several situations and can be quantitatively described in some cases. Symmetries, trees, spirals, meanders, waves, foams, tessellations, fissures, and stripes are examples of natural patterns. Plato, Pythagoras, and Empedocles were among the first Greek philosophers to study pattern, striving to explain order in nature. Over time, the contemporary notion of visual patterns evolved.
Mathematics, physics and chemistry can explain patterns in nature at different levels and scales. Patterns in living things are explained by the biological processes of natural selection and sexual selection. Studies of pattern formation make use of computer models to simulate a wide range of patterns.
Patterns in nature are observable regularities of form that may be discovered in the natural world. These patterns repeat in several situations and can be quantitatively described in some cases. Symmetries, trees, spirals, meanders, waves, foams, tessellations, fissures, and stripes are examples of natural patterns. Plato, Pythagoras, and Empedocles were among the first Greek philosophers to study pattern, striving to explain order in nature. Over time, the contemporary notion of visual patterns evolved.
Mathematics, physics and chemistry can explain patterns in nature at different levels and scales. Patterns in living things are explained by the biological processes of natural selection and sexual selection. Studies of pattern formation make use of computer models to simulate a wide range of patterns.
SYMMETRY
Symmetry is variously defined as "proportion," "perfect, or harmonious proportions," and "a structure that allows an object to be divided into parts of an equal shape and size."
There are three types of symmetry observed in living beings, according to biology. Any straight incision across the center point separates the organism into mirrored halves, according to point symmetry (a type of reflecting symmetry). This group includes certain floating organisms with radiating portions, as well as some tiny protozoa. All of the animals in this pattern are incredibly little. A cone or disk form with radial symmetry (a kind of rotational symmetry) is symmetrical around a central axis. Radial symmetry may be seen in starfish, sea anemones, jellyfish, and certain flowers. Finally, plane or bilateral symmetry (also known as reflected symmetry) describes how a body can be split into two equal halves by a central (sagittal) plane. Bilateral symmetry is seen in humans, insects, and animals.
Symmetry is variously defined as "proportion," "perfect, or harmonious proportions," and "a structure that allows an object to be divided into parts of an equal shape and size."
There are three types of symmetry observed in living beings, according to biology. Any straight incision across the center point separates the organism into mirrored halves, according to point symmetry (a type of reflecting symmetry). This group includes certain floating organisms with radiating portions, as well as some tiny protozoa. All of the animals in this pattern are incredibly little. A cone or disk form with radial symmetry (a kind of rotational symmetry) is symmetrical around a central axis. Radial symmetry may be seen in starfish, sea anemones, jellyfish, and certain flowers. Finally, plane or bilateral symmetry (also known as reflected symmetry) describes how a body can be split into two equal halves by a central (sagittal) plane. Bilateral symmetry is seen in humans, insects, and animals.
SPIRAL
A spiral is a curved design that revolves around a central point and a sequence of circular forms. Pine cones, pineapples, and hurricanes are examples of spirals. Plants employ a spiral structure, such as the leaf in the image above, since they are always seeking to develop while remaining secure. Plants with a spiral form compact themselves and take up less space, making them stronger and more resistant to the weather.
A spiral is a curved design that revolves around a central point and a sequence of circular forms. Pine cones, pineapples, and hurricanes are examples of spirals. Plants employ a spiral structure, such as the leaf in the image above, since they are always seeking to develop while remaining secure. Plants with a spiral form compact themselves and take up less space, making them stronger and more resistant to the weather.
FRACTAL
A fractal is a complex pattern that seems the same at every scale and repeats itself throughout time. The pattern of a fractal becomes increasingly complicated as you study it at bigger scales. Simple forms increase over time while preserving the same pattern in this fractal sample. Snowflakes, tree branches, lightning, and ferns are all examples of fractals in nature.
A fractal is a complex pattern that seems the same at every scale and repeats itself throughout time. The pattern of a fractal becomes increasingly complicated as you study it at bigger scales. Simple forms increase over time while preserving the same pattern in this fractal sample. Snowflakes, tree branches, lightning, and ferns are all examples of fractals in nature.
TESSELLATION
A tessellation is a pattern of shapes that fit together perfectly, without any gaps. Tessellations are a type of pattern that can be found in nature. Tessellation patterns can be seen in nature, such as the hexagonal cells in a honeycomb or the diamond-shaped scales that decorate snake skin. Several geometric units (tiles) are used to create distinct forms that fit together without gaps or overlaps to generate an appealing and unified pattern. Tessellating patterns are abstract and non-representational, allowing anybody to interpret them whatever they like.
A tessellation is a pattern of shapes that fit together perfectly, without any gaps. Tessellations are a type of pattern that can be found in nature. Tessellation patterns can be seen in nature, such as the hexagonal cells in a honeycomb or the diamond-shaped scales that decorate snake skin. Several geometric units (tiles) are used to create distinct forms that fit together without gaps or overlaps to generate an appealing and unified pattern. Tessellating patterns are abstract and non-representational, allowing anybody to interpret them whatever they like.
FIBONACCI SEQUENCE
Leonardo Pisano Bogollo (Fibonacci), an Italian, was the first to discover the Fibonacci sequence. The Fibonacci sequence is made up of entire numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The Fibonacci sequence is the name given to an endless series. Each word, starting at 0 and going up to 1, is the total of the two preceding ones. This is referred to as "nature's hidden code." Sunflowers, daisies, broccoli, cauliflowers, and seashells all have spiral designs that follow the Fibonacci sequence.
Leonardo Pisano Bogollo (Fibonacci), an Italian, was the first to discover the Fibonacci sequence. The Fibonacci sequence is made up of entire numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The Fibonacci sequence is the name given to an endless series. Each word, starting at 0 and going up to 1, is the total of the two preceding ones. This is referred to as "nature's hidden code." Sunflowers, daisies, broccoli, cauliflowers, and seashells all have spiral designs that follow the Fibonacci sequence.
BINET'S FORMULA
As stated, the Fibonacci numbers can be obtained by adding the two preceding integers. But how about when you are asked to get the nth Fibonacci number? For example, find the 40th Fibonacci number. It would be so time consuming to use the traditional method, that is why, we have the Binet's Formula. Binet's formula is an explicit formula used to find the nth term of the Fibonacci sequence. It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre. Shown on the right is the formula.
As stated, the Fibonacci numbers can be obtained by adding the two preceding integers. But how about when you are asked to get the nth Fibonacci number? For example, find the 40th Fibonacci number. It would be so time consuming to use the traditional method, that is why, we have the Binet's Formula. Binet's formula is an explicit formula used to find the nth term of the Fibonacci sequence. It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre. Shown on the right is the formula.
FIBONACCI NUMBERS IN NATURE
When looking at the geometry of plants, flowers, or fruit, it's simple to spot recurring patterns and forms. The Fibonacci sequence is important in phyllotaxis, which is the study of the arrangement of leaves, branches, flowers, or seeds in plants with the goal of showing the occurrence of regular patterns. D'arcy Thompson noticed that the plant world has an odd affinity for specific numbers and spiral shapes, and that these numbers and geometries are tightly connected. In the spirals made by individual flowers in the composite inflorescences of daisies, sunflowers, cauliflowers, and broccoli, we may readily locate the Fibonacci numbers. The number of petals on flowers is another basic illustration of where the Fibonacci sequence may be seen in nature. Most flowers have three petals (like lilies and irises), five (parnassia, rose hips) or eight (cosmea), 13 (certain daisies), 21 (chicory), 34, 55, or 89 petals (like lilies and irises) (asteraceae).
When looking at the geometry of plants, flowers, or fruit, it's simple to spot recurring patterns and forms. The Fibonacci sequence is important in phyllotaxis, which is the study of the arrangement of leaves, branches, flowers, or seeds in plants with the goal of showing the occurrence of regular patterns. D'arcy Thompson noticed that the plant world has an odd affinity for specific numbers and spiral shapes, and that these numbers and geometries are tightly connected. In the spirals made by individual flowers in the composite inflorescences of daisies, sunflowers, cauliflowers, and broccoli, we may readily locate the Fibonacci numbers. The number of petals on flowers is another basic illustration of where the Fibonacci sequence may be seen in nature. Most flowers have three petals (like lilies and irises), five (parnassia, rose hips) or eight (cosmea), 13 (certain daisies), 21 (chicory), 34, 55, or 89 petals (like lilies and irises) (asteraceae).
FIBONACCI SPIRAL
Fibonacci spiral is formed If we construct a series of squares with lengths equal to the Fibonacci numbers (1, 1, 2, 3, 5,8 etc) and trace a line through the diagonals of each square. This Fibonacci spiral is a close approximation of the spirals found in nautilus shells, snail shells, and chameleon tails, but it does not perfectly match these natural spirals.
Fibonacci spiral is formed If we construct a series of squares with lengths equal to the Fibonacci numbers (1, 1, 2, 3, 5,8 etc) and trace a line through the diagonals of each square. This Fibonacci spiral is a close approximation of the spirals found in nautilus shells, snail shells, and chameleon tails, but it does not perfectly match these natural spirals.
THE GOLDEN RECTANGLE
The Golden Rectangle , also called the perfect rectangle by some, is a rectangle in which the ratio of its length to its width is the Golden Ratio . Many believe that this is one of the most visually pleasing of all geometric shapes. It appears in many works of art and architecture. The Parthenon of ancient Greece is the most famous example of the use of the golden rectangle. People find the golden rectangle in the Mona Lisa, and other Renaissance art works.
The Golden Rectangle , also called the perfect rectangle by some, is a rectangle in which the ratio of its length to its width is the Golden Ratio . Many believe that this is one of the most visually pleasing of all geometric shapes. It appears in many works of art and architecture. The Parthenon of ancient Greece is the most famous example of the use of the golden rectangle. People find the golden rectangle in the Mona Lisa, and other Renaissance art works.
GOLDEN RATIO
Golden ratio is simply the ratio of the golden rectangle's long side to its short side. It is approximately equal to 1.618 which all starts with the fact that a golden rectangle can be broken up into a square and another smaller rectangle, where the smaller rectangle is another golden rectangle. It is an irrational number like pi and e, meaning that its terms go on forever after the decimal point without repeating.
Golden ratio is simply the ratio of the golden rectangle's long side to its short side. It is approximately equal to 1.618 which all starts with the fact that a golden rectangle can be broken up into a square and another smaller rectangle, where the smaller rectangle is another golden rectangle. It is an irrational number like pi and e, meaning that its terms go on forever after the decimal point without repeating.
DERIVING THE GOLDEN RATIO
Salazar BSCE 1B ( Review Exercises 1).pdfREVIEW EXERCISES #1
Salazar-Activity1-BSCE1B.mp4ACTIVITY #1
Salazar BSCE 1B (Quiz 1).pdfQUIZ #1 FIBONACCI SEQUENCE
REFLECTION
In this module, I was able to learn and appreciate the different patterns in nature, the Fibonacci numbers, the golden rectangle and the golden ratio. By doing the activity 1 or the video presentation above, I was able to observe the different patterns in nature especially those just within our vicinity. Most of them are leaves and flowers with different patterns either tessellation, symmetry or fractals. I also appreciated the Binet's formula in the review exercises knowing how important the formula is especially when dealing with large nth Fibonacci number. Further, checking Fibonacci equations if valid or not in quiz 1 was a bit challenging for me because it's a bit confusing which nth Fibonacci number should be substituted to a certain equation. Nevertheless, I was satisfied with my works given that I have checked them multiple times to ensure precision and accuracy. When it comes to the learning material provided by our teacher, I would say that it was comprehensive and understandable enough for everybody. I would also like to praise our teacher for being responsible and always prompt when it comes to discussion.
In this module, I was able to learn and appreciate the different patterns in nature, the Fibonacci numbers, the golden rectangle and the golden ratio. By doing the activity 1 or the video presentation above, I was able to observe the different patterns in nature especially those just within our vicinity. Most of them are leaves and flowers with different patterns either tessellation, symmetry or fractals. I also appreciated the Binet's formula in the review exercises knowing how important the formula is especially when dealing with large nth Fibonacci number. Further, checking Fibonacci equations if valid or not in quiz 1 was a bit challenging for me because it's a bit confusing which nth Fibonacci number should be substituted to a certain equation. Nevertheless, I was satisfied with my works given that I have checked them multiple times to ensure precision and accuracy. When it comes to the learning material provided by our teacher, I would say that it was comprehensive and understandable enough for everybody. I would also like to praise our teacher for being responsible and always prompt when it comes to discussion.
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