Chapter 1 - Patterns and Numbers in the World PDF

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MATHEMATICS IN THE MODERN WORLD...

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PART

1 THE NATURE OF MATHEMATICS Science of Patterns and Relationships A way of Thinking

F

A Discipline

MATHEMATICS A Language

An Art

Mathematics is a science of patterns and relationships, patterns that help us understand and solve problems that originate in the world around us. It is also a way of thinking, as it relies on logic and creativity. It is an art, as numerous patterns can be found in numbers and in geometric figures. Its essence lies not only its beauty and intellectual challenge. It has a language of its own and it is a broad discipline that continues to grow as this is applied in daily situations and in the world of work. Let us explore this beautiful world of patterns and numbers … the world of MATHEMATICS!

1

Mathematics in Our

World Mathematics plays a central role in the modern world, and it helps us to understand the world, thru patterns, relationships and possibilities. The world is interconnected and math shows these connections on a daily basis. Math helps us understand the world — and we use the world to understand math. As such, college students need to perceive it as part of any scientific endeavor, understand the nature of mathematical inquiry and become acquainted with key mathematical concepts and abilities.

Your Target Outcomes At the end of this lesson, you are expected to: 1. identify patterns and regularities in the world 2. articulate the importance of mathematics in the patterns, 3. argue about the nature of mathematics, what it is and how it is expressed, represented and used, and 4. express appreciation for mathematics as a human endeavor.

What if … - there is no pattern and regularity in everything you find around you? - there is no pattern and regularity in the stairs of your house, no pattern in the room where you sleep, no pattern in the tiles of the floor, and no pattern and regularity in the classrooms where you attend classes? - there is no pattern and regularity in the skin of animals, like the zebra, in insects like the butterfly, no pattern in plants’ leaves, trunks and other parts? - no pattern on how cars, buses and planes travel? - and most importantly, what if there is no pattern and regularity in your body? The answer is that there would be chaos, no beauty, no strength, no comfort and most of all, everything is disorganized! Things were all built following a pattern and regularity that makes it beautiful, sturdy and comfortable to stay.

Therefore, we find harmony in the world because of the patterns, regularities and numbers that we find in nature.

Let us proceed to the first lesson is order to understand how patterns and numbers in nature provide us with an almost perfect harmony.

Understand the Concept Patterns and Numbers in Nature and the World A pattern is an arrangement of lines or shapes, especially a design in which the same shape is repeated at regular intervals over a surface. Patterns in nature are visible regular forms found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks, and stripes. Symmetries There is symmetry when a original shape after being folded, is a mirror image. It allows an equal shape and size.

shape looks identical to its flipped or turned. Symmetry too, object to be divided into parts of

Symmetry can be reflective and rotational. Reflective, or line, symmetry has one half of an image is the mirror image of the other half, as in the wings of a butterfly. When an object or image can be turned around a center point and match itself some number of times (as in a five pointed star), there is rotational symmetry. There is perfect symmetry in our human body. The human body has two matching hands, feet, eyes, ears, our smile etc. The left side and the right side of the nose can also be divided into two symmetrical parts. Fractals Fractals are a never-ending pattern. These are extremely intricate patterns that are similar across different levels, created by doing a simple process repeatedly in an ongoing feedback loop. Fractals are images of dynamic systems – the pictures of Chaos. Fractal patterns are very familiar, since there are lots of them in nature like the coastlines, rivers, hurricanes, clouds, mountains, seashells, and trees etc. The leaves of ferns are self-similar (pinnate) to 2, 3 or 4 levels.

Spirals

Spirals are ordinary in some plants and animals. A spiral is a coiled or curved pattern that focuses on a center point and a series of circular shapes that revolve around it. Some examples of spirals are pine cones, pineapples, and hurricanes. Plants use a spiral form to constantly grow while staying secure.

Waves Waves are disturbances travel. Mechanical waves spread swing back and forth as they

that transmit energy as they by means of air or water, making it pass by.

Waves create patterns of sand. Dunes are created as sand, sometimes in extensive

ripples as water or wind pass over winds blow over large bodies of dune fields

Dunes may form a range of patterns including crescents, very long straight lines, stars, domes, parabolas, and longitudinal or Seif (‘sword’) shapes.

Tessellations A tessellation is a pattern of shapes that fit perfectly together. When a surface is covered with a pattern of flat shapes so that there are no overlaps or gaps, then tessellation is done. This is more popularly known as tiling. These have appeared throughout art history and are also found in nature. Specific examples include oriental carpets, quilts, origami, and Islamic architecture. Examples of synthetic tessellations include nets and chain-link fences.

One of the most common tessellation as found in nature is a segmented covering of a surface such as the scales covering a reptile, the spider’s web or in a pineapple.

Cracks Cracks are openings that develop in materials to release stress. When a flexible material stretches or shrinks evenly, it finally attains its breaking strength and then suddenly fails in all directions, creating cracks with 120-degree joints, so three cracks meet at a node. On the other hand, when an inelastic material fails, straight cracks develop to let go of the stress. More stress in the same path would then simply open the remaining cracks; pressure at right angles can create new cracks, at 90 degrees to the old ones. Thus the pattern of cracks indicates whether the material is elastic or not.

Spots, stripes Leopards are spotted, zebras are striped. These patterns may be explained in terms of their functions that increases the chances that the offspring will survive and reproduce. One function of animal patterns is disguise; for instance, a leopard that is more difficult to see catches more prey. Another function is signaling — for instance, a ladybird is less expected to be attacked by predatory birds, if it has bold warning colors, and is also distastefully bitter or poisonous, or mimics other distasteful insects. A young bird may see a warning patterned insect like a ladybird and try to eat it, but it will only do this once; very soon it will spit out the bitter insect; the other ladybirds in the area will remain unmolested. The young leopards and ladybirds, inherit genes that somehow create spottiness and survive. But while these evolutionary and functional arguments explain why these animals need their patterns, they do not explain how the patterns are formed. Mathematics Helps Organize Patterns and Regularities in the World Mathematics is the study of patterns; it is not just confined to numbers and arithmetic. When you use patterns to analyze and solve problems, things are made easier. Nature can be thought of as the glue or the substrate that binds art and mathematics, and observing nature teaches us about both art and math. The Fibonacci Sequence Leonardo of Pisa, more popularly known as Fibonacci, is best known for his introduction of

a particular number sequence, which as the Fibonacci Numbers or the discovered the sequence while involving the growth of a hypothetical idealized assumptions. He noted that, the number of pairs of rabbits 8 to 13, etc. He found a pattern, noting 1… and calculated the next numbers Thus, he arrived at a sequence of 0, 1, 144, 233, 377, 610, 987, and so on.

later became known to the world Fibonacci sequence. He considering a practical problem population of rabbits based on after each monthly generation, increased from 1 to 2 to 3 to 5 to that the sequence begins with 0, from the sum of the previous two. 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,

As defined, the first four sequence are 0, 1, 1, 2, and each the previous two, therefore, the the nth Fibonacci number, the Binet may be used:

numbers of the Fibonacci subsequent number is the sum of sequence Fn = Fn-1 + Fn-2. To find form of the nth Fibonacci number

10

Illustration: To solve for F10 = ((1+√5) 2 −(1−√5) 2

)

10

(√5) = 55 You may also use the online Fibonacci sequence calculator to solve for the F n :

https://keisan.casio.com/exec/system/1180573404

When you make squares with the numbers in the Fibonacci sequence, you can develop a nice spiral. Look at the Figure at the left.

The Fibonacci sequence in nature The Fibonacci numbers are interesting in that they occur throughout both nature and art. When you observe plants, flowers or fruit, you can recognize some recurrent forms and structures. For example, the Fibonacci sequence has a

vital role to play in phyllotaxis, which is the study of the arrangement of flowers, leaves, seeds, branches, or flowers in plants, to determine if there exists a regular pattern. Arrangements of elements in nature elements follow surprising mathematical forms and regularities. The plant kingdom has a leaning for particular numbers. An example is the number of petals of flowers. Most have three (like lilies and irises), five (parnassia, rose hips) or eight (cosmea), 13 (some daisies), 21 (chicory), 34, 55 or 89 (asteraceae). The number of leaves, and for certain spiral shapes, like in a growing fern, daisies, cauliflowers, sunflowers, and broccoli and that these numbers and shapes are closely related. Focusing on the sunflower, look at the arrangement of the seeds in its head and you will notice that they form spirals. Depending on the species, you can count 34 and 55 or 55 and 89 or 89 and 144. This arrangement keeps the seeds uniformly packed no matter how large the seed head is. Mathematics Helps Predict the Behavior of Nature and Phenomena in the World Mathematics is a science of pattern and order. A lot of observed natural phenomena can be measured and patterns can be perceived, and from those observations and patterns, predictions about nature can be made. Predicting the size, location and timing of some natural hazards and calamities such as typhoons, floods, volcanic eruption and even landslides, aftershocks after an earthquake, can now be predicted with the help of mathematics. For example, weather forecasting is a real-life application of mathematics side by side with meteorology. This is done by working with a set of equations that describe the atmosphere, considering temperature, pressure and humidity. In forecasting the weather, powerful computers are used to predict the weather by solving mathematical equations that model the atmosphere and oceans. On a daily basis, these supercomputers collect and organize billions of observations about the Earth like temperature, air pressure, moisture, water levels, and wind speed, which are very important in order to prepare all numerical weather prediction models. All these observations are represented by numbers. Mathematics Helps Control Nature and Occurrences in the World for our Own Ends Mathematics cannot predict nor control natural disasters like earthquakes. However, Mathematics has been helpful in understanding that the “laws of nature” and that nature is such a powerful force and how things in nature form and develop like the Fibonacci sequence. Mathematics can be used to explain how something recurs in nature. It gives us a way to understand patterns, to quantify relationships, and to predict the future. It is used to explain why the Sun “sets”, “where” it went, and why it “returned” because it is easier to count these events in numbers than to put them into words. Similarly, formulas became a way of using numbers to show how things in nature happen together or oppose one another. Since Mathematics has been most helpful in tracking typhoons with precision, it has also been helpful in forecasting floods, volcanic eruptions and landslides, thus it has been helpful in reducing the effects of these disasters by calculating hazards and building safer structures. We can prepare for untoward consequences, or better yet, maybe we can stop them from happening. Needless to say, Mathematics has numerous applications in our everyday life making it indispensable.

How Far Have You Understood #1 Name: __________________________________ Total Score: ____

A. Answer the following questions concisely: 1. Discuss how do each of the patterns and regularities in nature help in making the world a better place to live in: 1.1 symmetries _______________________________________________ ________________________________________________________________________________________________ __________________________ 1.2 spirals ____________________________________________________ ________________________________________________________________________________________________ __________________________ 1.3 tessellations ______________________________________________ ________________________________________________________________________________________________ __________________________ 1.4 stripes_____________________________________________________ ________________________________________________________________________________________________ __________________________ 1.5 cracks ____________________________________________________ ________________________________________________________________________________________________ __________________________

2. How does Mathematics organize patterns and regularities in the world? _____________________________________________________________________________________________________ _____________________________________________________________________________________________________ _____________________________________________________________________________________________________ ___________________________ 3. Explain: Mathematics helps predict the behavior of nature and phenomena in the world. _____________________________________________________________________________________________________ _____________________________________________________________________________________________________ _____________________________________________________________________________________________________ ___________________________ 4. Does Mathematics really help control nature and occurrences in the world for our own ends? Support your answer.

_____________________________________________________________________________________________________ _________________________________________________________________________________________________ __________________________________________________________________ B. Calculate the following: 1. The first three Fibonacci numbers are 0, 1, and 1. What is the 7th? _________________ 2. What would be the next value for the Fibonacci number 233? ______________ 3. If two numbers in the Fibonacci series are 34 and 55, what would be the next? ____________ 4. If two numbers in the Fibonacci series are 34 and 55, what would be before 34? 5. If two numbers in the Fibonacci series are 34 and 55, what would be the 4th number? 6. Solve for the nth Fibonacci sequence using the Binet form or the online Fibonacci sequence calc: F25, F32, F36, F40 and F62...