Mathematics In The Modern World PDF

1 1 | Mathematics in the Modern World UNIT 1. THE NATURE OF MATHEMATICS 1.0 Learning Outcomes On completion of the module, you should be able to: 1. identify the patterns in nature and articulate the importance of mathematics...

1 1 | Mathematics in the Modern World UNIT 1. THE NATURE OF MATHEMATICS 1.0 Learning Outcomes On completion of the module, you should be able to: 1. identify the patterns in nature and articulate the importance of mathematics in one’s life; and 2. solve problems involving Fibonacci Sequence. 1.1 Introduction Have you gone into the woods? Have you experienced walking along the beaches? Run or jogged around a public park or took a walk with your favourite pet in a garden? Will you agree with me that these places reveal an unending variation of forms in nature? They not only delight your imagination but also challenge your understanding. How did these patterns develop? What are the rules and guidelines used to create the patterns of the things that surround you? Some patterns are molded with strict regularity while others are with unexplained irregularity. All these may increase your curiosity to find the answers for the existence of these patterns in nature and in fact many theories have been proposed as an attempt to do so. As students in mathematics in the modern world, you need to look deeply into the geometry and mechanism of the patterns in living and non-living things in your environment. Let us start our lesson with these types of patterns in nature. The “You try this” section of the module should be answered. This will be submitted to me for checking. It will serve as your grade in the participation. So, you must answer all of them. 1.2 The Nature of Mathematics 1.2.1 Types of Pattern in Nature Patterns in nature can be seen in our environment. These patterns occur in different forms and can be modelled mathematically. Natural patterns include the following: 1. symmetry, 2. trees & fractals, 3. spirals, 4. chaos, flow & meanders, 5. waves and dunes, 6. bubbles & foam, 7. tessellations, 8. cracks, 9. spots & stripes, and 10. pattern formation. Let us discuss them one by one.  2 1 | Mathematics in the Modern World 1. Symmetry Symmetry is pervasive in living things. Animals mainly have bilateral or mirror symmetry, as do the leaves of plants and some flowers such as orchids. Plants often have radial or rotational symmetry, as do many flowers and some groups of animals such as sea anemones. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies. Among non-living things, snowflakes have striking sixfold symmetry; each flake's structure forms a record of the varying conditions during its crystallization, with nearly the same pattern of growth on each of its six arms. Crystals in general have a variety of symmetries and crystal habits; they can be cubic or octahedral, but true crystals cannot have fivefold symmetry (unlike quasicrystals). Rotational symmetry is found at different scales among non-living things, including the crown-shaped splash pattern formed when a drop falls into a pond, and both the spheroidal shape and rings of a planet like Saturn. Symmetry has a variety of causes. Radial symmetry suits organisms like sea anemones whose adults do not move: food and threats may arrive from any direction. But animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The head becomes specialised with a mouth and sense organs (cephalisation), and the body becomes bilaterally symmetric (though internal organs need not be). More puzzling is the reason for the fivefold (pentaradiate) symmetry of the echinoderms. Early echinoderms were bilaterally symmetrical, as their larvae still are. Sumrall and Wray argue  3 1 | Mathematics in the Modern World that the loss of the old symmetry had both developmental and ecological causes. Below are examples of symmetry. 2. Trees, fractals Fractals are infinitely self-similar, iterated mathematical constructs having fractal dimension. Infinite iteration is not possible in nature so all 'fractal' patterns are only approximate. For example, the leaves of ferns and umbellifers (Apiaceae) are only selfsimilar (pinnate) to 2, 3 or 4 levels. Fern-like growth patterns occur in plants and in animals including bryozoa, corals, hydrozoa like the air fern, Sertularia argentea, and in non-living things, notably electrical discharges. Lindenmayer system fractals can model different patterns of tree growth by varying a small number of parameters including branching angle, distance between nodes or branch points (internode length), and number of branches per branch point. Animals Sea Water splash Echinoderms Snowflakes often show anemones approximates like this have sixfold mirror or have radial starfish have symmetry. bilateral rotational symmetry. fivefold symmetry, symmetry. symmetry. like this tiger. Fivefold Garnet symmetry can Volvox has showing Fluorite showing be seen in spherical symmetry. rhombic cubic crystal habit. many flowers dodecahedral and some fruits crystal habit. like this medlar.  4 1 | Mathematics in the Modern World Fractal-like patterns occur widely in nature, in phenomena as diverse as clouds, river networks, geologic fault lines, mountains, coastlines, animal coloration, snow flakes, crystals, blood vessel branching, actin cytoskeleton, and ocean waves. Leaf of cow parsley, Anthriscus sylvestris, is 2 - or 3 -pinnate, not Angelica flowerhead, a sphere made infinite. of spheres (self-similar) Here are some examples of trees and fractals 3. Spirals Spirals are common in plants and in some animals, notably molluscs. For example, in the nautilus, a cephalopod mollusc, each chamber of its shell is an approximate copy of the next one, scaled by a constant factor and arranged in a logarithmic spiral. Given a modern understanding of fractals, a growth spiral can be seen as a special case of self-similarity. Plant spirals can be seen in phyllotaxis, the arrangement of leaves on a stem, and in the arrangement (parastichy) of other parts as in composite flower heads and seed heads like the sunflower or fruit structures like the pineapple and snake fruit, as well as in the pattern of scales in pine cones, where multiple spirals run both clockwise and anticlockwise. These arrangements have explanations at different levels – mathematics, physics, chemistry, biology – each individually correct, but all necessary together. Phyllotaxis spirals can be generated mathematically from Fibonacci ratios: the Fibonacci sequence runs 1, 1, 2, 3, 5, 8, 13... (each subsequent number being the sum of the two preceding ones). For example, when leaves alternate up a stem, one rotation of the spiral touches two leaves, so the pattern or ratio is 1/2. In hazel the ratio is 1/3; in apricot it is 2/5; in pear it is 3/8; in almond it is 5/13. In disc phyllotaxis as in the sunflower and daisy, the florets are arranged in Fermat's spiral with Fibonacci numbering, at least when the flowerhead is mature so all the elements are the same size. Fibonacci ratios approximate the golden angle, 137.508°, which governs the curvature of Fermat's spiral.  5 1 | Mathematics in the Modern World From the point of view of physics, spirals are lowest-energy configurations which emerge spontaneously through self-organizing processes in dynamic systems. From the point of view of chemistry, a spiral can be generated by a reaction-diffusion process, involving both activation and inhibition. Phyllotaxis is controlled by proteins that manipulate the concentration of the plant hormone auxin, which activates meristem growth, alongside other mechanisms to control the relative angle of buds around the stem. From a biological perspective, arranging leaves as far apart as possible in any given space is favoured by natural selection as it maximises access to resources, especially sunlight for photosynthesis. Take a look at these examples of spirals! Fibonacci Bighorn sheep, Spirals: Nautilus shell's Fermat's spiral Ovis phyllotaxis of logarithmic spiral: seed canadensis spiral aloe, growth spiral head of Aloe sunflower, polyphylla Helianthus annuus Multiple Spiralling Water Fibonacci shell of spirals: red droplets fly Trochoidea off a wet, cabbage in liebetruti cross section spinning ball 4. Chaos, flow, meanders In mathematics, a dynamical system is chaotic if it is (highly) sensitive to initial conditions (the so-called "butterfly effect", which requires the mathematical properties of topological mixing and dense periodic orbits. Alongside fractals, chaos theory ranks as an essentially  6 1 | Mathematics in the Modern World universal influence on patterns in nature. There is a relationship between chaos and fractals—the strange attractors in chaotic systems have a fractal dimension. Some cellular automata, simple sets of mathematical rules that generate patterns, have chaotic behaviour, notably Stephen Wolfram's Rule 30. Vortex streets are zigzagging patterns of whirling vortices created by the unsteady separation of flow of a fluid, most often air or water, over obstructing objects. Smooth (laminar) flow starts to break up when the size of the obstruction or the velocity of the flow become large enough compared to the viscosity of the fluid. Meanders are sinuous bends in rivers or other channels, which form as a fluid, most often water, flows around bends. As soon as the path is slightly curved, the size and curvature of each loop increases as helical flow drags material like sand and gravel across the river to the inside of the bend. The outside of the loop is left clean and unprotected, so erosion accelerates, further increasing the meandering in a powerful positive feedback loop. Examples of chaos, flow, and meanders are found below. Chaos: shell of Meanders: dramatic meander Chaos: vortex gastropod mollusc the scars and oxbow lakes in the cloth of gold cone, street of clouds broad flood plain of the Rio Conus textile, resembles Rule 30 Negro, seen from space cellular automaton. Meanders: sinuous Meanders: sinuous Meanders: symmetrical path of Rio Cauto, snake crawling. brain coral, Diploria Cuba strigosa 5. Waves, dunes Waves are disturbances that carry energy as they move. Mechanical waves propagate through a medium – air or water, making it oscillate as they pass by. Wind waves are sea surface waves that create the characteristic chaotic pattern of any large body of water, though their statistical behaviour can be predicted with wind wave models. As waves in water or  7 1 | Mathematics in the Modern World wind pass over sand, they create patterns of ripples. When winds blow over large bodies of sand, they create dunes, sometimes in extensive dune fields as in the Taklamakan desert. Dunes may form a range of patterns including crescents, very long straight lines, stars, domes, parabolas, and longitudinal or seif ('sword') shapes. Here are some examples: Waves: Dunes: sand Dunes: barchan Wind ripples with breaking wave dunes in crescent sand dislocations Taklamakan dune In Sistan, in a ship's desert, from Afghanistan wake space 6. Bubbles, foam A soap bubble forms a sphere, a surface with minimal area — the smallest possible surface area for the volume enclosed. Two bubbles together form a more complex shape: the outer surfaces of both bubbles are spherical; these surfaces are joined by a third spherical surface as the smaller bubble bulges slightly into the larger one. A foam is a mass of bubbles; foams of different materials occur in nature. Foams composed of soap films obey Plateau's laws, which require three soap films to meet at each edge at 120° and four soap edges to meet at each vertex at the tetrahedral angle of about 109.5°. Plateau's laws further require films to be smooth and continuous, and to have a constant average curvature at every point. For example, a film may remain nearly flat on average by being curved up in one direction (say, left to right) while being curved downwards in another direction (say, front to back). Structures with minimal surfaces can be used as tents.  8 1 | Mathematics in the Modern World Examples of these bubbles and foams are found below: Foam of soap Haeckel's bubbles: four edges Radiolaria Spumellaria; the Buckminsterfullerene: meet at each vertex, drawn by skeletons of Richard Smalley and at angles close to Haeckel in his these Radiolaria colleagues synthesised 109.5°, as in two C- Kunstformen der have foam-like the fullerene molecule H bonds in methane. Natur (1904). forms. in 1985. Brochosomes (secretory microparticles produced Equal spheres by leafhoppers) often (gas bubbles) in Beijing's National Aquatics approximate fullerene a surface foam. Center for the 2008 Olympic geometry. games has a Weaire–Phelan structure. 7. Tessellations Tessellations are patterns formed by repeating tiles all over a flat surface. There are 17 wallpaper groups of tilings. While common in art and design, exactly repeating tilings are less easy to find in living things. The cells in the paper nests of social wasps, and the wax cells in honeycomb built by honey bees are well-known examples. Among animals, bony fish, reptiles or the pangolin, or fruits like the salak are protected by overlapping scales or osteoderms, these form more-or-less exactly repeating units, though often the scales in fact vary continuously in size. Among flowers, the snake's head fritillary, Fritillaria meleagris, have a tessellated chequerboard pattern on their petals. The structures of minerals provide good examples of regularly repeating three-dimensional arrays. Despite the hundreds of thousands of known minerals, there are rather few possible types of arrangement of atoms in a crystal, defined by crystal structure, crystal system, and point group; for example, there are exactly 14 Bravais lattices for the 7 lattice systems in three-dimensional space.  9 1 | Mathematics in the Modern World Crystals: cube-shaped Tilings: tessellated Arrays: Bismuth hopper crystals of halite (rock flower of snake's honeycomb crystal illustrating salt); cubic crystal head fritillary, is a natural the stairstep system, isometric Fritillaria meleagris tessellation crystal habit. hexoctahedral crystal symmetry. Tilings: overlapping scales Tessellated pavement: a of common roach, Rutilus rare rock formation on the rutilus Tasman Peninsula Examples of tessellations are as follow; 8. Cracks Cracks are linear openings that form in materials to relieve stress. When an elastic material stretches or shrinks uniformly, it eventually reaches its breaking strength and then fails suddenly in all directions, creating cracks with 120 degree joints, so three cracks meet at a node. Conversely, when an inelastic material fails, straight cracks form to relieve the stress. Further stress in the same direction would then simply open the existing cracks; stress 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. In a tough fibrous material like oak tree bark, cracks form to relieve stress as usual, but they do not grow long as their growth is interrupted by bundles of strong elastic fibres. Since each species of tree has its own structure at the levels of cell and of molecules, each has its own pattern of splitting in its bark. Take a look at these examples of cracks:  10 1 | Mathematics in the Modern World Old pottery surface, white Drying inelastic glaze with mud in the Rann mainly 90° of Kutch with Veined gabbro cracks mainly 90° cracks with 90° cracks, near Sgurr na Stri, Skye Cooled basalt at Drying elastic Giant's Causeway. mud in Sicily with Vertical mainly mainly 120° 120° cracks Palm trunk with cracks giving hexagonal branching vertical columns cracks (and horizontal leaf 9. Spots, stripes Leopards and ladybirds are spotted; angelfish and zebras are stripe. These patterns have an evolutionary explanation: they have functions which increase the chances that the offspring of the patterned animal will survive to reproduce. One function of animal patterns is camouflage; for instance, a leopard that is harder to see catches more prey. Another function is signalling— for instance, a ladybird is less likely to be attacked by predatory birds that hunt by sight, if it has bold warning colours, 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 undisturbed. The young leopards and ladybirds, inheriting genes that somehow create spottedness, survive. But while these evolutionary and functional arguments explain why these animals need their patterns, they do not explain how the patterns are formed. Here are some examples:  11 1 | Mathematics in the Modern World Grevy's zebr a, Array of Breeding pattern Equus grevyi ladybirds by of cuttlefish, G.G. Sepia officinalis Jacobson 10. Pattern formation Alan Turin, and later the mathematical biologist James Murray, described a mechanism that spontaneously creates spotted or striped patterns: a reaction-diffusion system. The cells of a young organism have genes that can be switched on by a chemical signal, a morphogen, resulting in the growth of a certain type of structure, say a darkly pigmented patch of skin. If the morphogen is present everywhere, the result is an even pigmentation, as in a black leopard. But if it is unevenly distributed, spots or stripes can result. Turing suggested that there could be feedback control of the production of the morphogen itself. This could cause continuous fluctuations in the amount of morphogen as it diffused around the body. A second mechanism is needed to create standing wave patterns (to result in spots or stripes): an inhibitor chemical that switches off production of the morphogen, and that itself diffuses through the body more quickly than the morphogen, resulting in an activator-inhibitor scheme. The Belousov–Zhabotinsky reaction is a non-biological example of this kind of scheme, a chemical oscillator. Later research has managed to create convincing models of patterns as diverse as zebra stripes, giraffe blotches, jaguar spots (medium-dark patches surrounded by dark broken rings) and ladybird shell patterns (different geometrical layouts of spots and stripes, see illustrations. Richard Prum's activation-inhibition models, developed from Turing's work, use six variables to account for the observed range of nine basic within-feather pigmentation patterns, from the simplest, a central pigment patch, via concentric patches, bars, chevrons, eye spot, pair of central spots, rows of paired spots and an array of dot. More elaborate models simulate complex feather patterns in the guineafowl Numida meleagris in which the individual feathers feature transitions from bars at the base to an array of dots at the far (distal) end. These require an oscillation created by two inhibiting signals, with interactions in both space and time. Patterns can form for other reasons in the vegetated landscape of tiger bush and fir waves. Tiger bush stripes occur on arid slopes where plant growth is limited by rainfall. Each roughly horizontal stripe of vegetation effectively collects the rainwater from the bare zone immediately above it. Fir waves occur in forests on mountain slopes after wind disturbance, during regeneration. When trees fall, the trees that they had sheltered become exposed and are in turn more likely to be damaged, so gaps tend to expand downwind. Meanwhile, on the windward side, young trees grow,  12 1 | Mathematics in the Modern World protected by the wind shadow of the remaining tall trees. Natural patterns are sometimes formed by animals, as in the Mima mounds of the Northwestern United States and some other areas, which appear to be created over many years by the burrowing activities of pocket gophers, while the so-called fairy circles of Namibia appear to be created by the interaction of competing groups of sand termites, along with competition for water among the desert plant. In permafrost soils with an active upper layer subject to annual freeze and thaw, patterned ground can form, creating circles, nets, ice wedge polygons, steps, and stripes. Thermal contraction causes shrinkage cracks to form; in a thaw, water fills the cracks, expanding to form ice when next frozen, and widening the cracks into wedges. These cracks may join up to form polygons and other shapes. The fissured pattern that develops on vertebrate brains are caused by a physical process of constrained expansion dependent on two geometric parameters: relative tangential cortical expansion and relative thickness of the cortex. Similar patterns of gyri (peaks) and sulci (troughs) have been demonstrated in models of the brain starting from smooth, layered gels, with the patterns caused by compressive mechanical forces resulting from the expansion of the outer layer (representing the cortex) after the addition of a solvent. Numerical models in computer simulations support natural  13 1 | Mathematics in the Modern World and experimental observations that the surface folding patterns increase in larger brains. Here are examples of pattern formation. You try this! (number 1) Helmeted guineafowl, Snapshot of Aerial view Giant Detail of giant of a tiger pufferfish skin Numida meleagris, simulation of pufferfish, bush pattern feathers transition from Belousov plateau in Tetraodon mbu barred to spotted, both in Zhabotinsky Niger feather and across the bird reaction. Fir waves in Patterned ground: a Human brain White melting pingo with Fairy circles in the (superior view) Mountains, surrounding ice wedge Marienflusstal area exhibiting patterns New polygons near in Namibia of gyri and sulci Hampshire Tuktoyaktuk, Canada  14 1 | Mathematics in the Modern World 1. Identify the type of pattern whether 1.symmetry, 2. trees & fractals, 3. spirals, 4. chaos, flow & meanders, 5. waves and dunes, 6. bubbles & foam, 7. tessellations, 8. cracks, 9. spots & stripes, and 10. pattern formation. 1. 2. 3. 4. 5. 6. 7. 1. trees & fractals 2.trees & fractals 3. Tessellation 4.trees & fractals 5.trees & fractals 6.waves, dunes 7. Trees & fractals 2. Take a picture of an object in your surroundings and discuss how mathematics is embedded in your chosen object. it shows a pattern that is related to mathematics it shows the arrangement of leaves on a plant called phyllotaxis, the position of the these whorled leaves on the stem can be often be predicted by mathematical formula called fibonacci series.  15 1 | Mathematics in the Modern World 1.2.2 The Fibonacci Sequence and the Golden Ratio The Fibonacci sequence is one of the most famous formulas in mathematics. It is significant because of the so called golden ratio of 1.618, or its inverse 0.618. Each number in the sequence is the sum of the two numbers that precede it. The sequence goes this way: 0, 1, 1, 2, 3, 5, …, and so on. In the Fibonacci sequence, any given number is approximately 1.618 times the preceding number, ignoring the first few numbers. Fibonacci sequence is found in nature by counting the number of petals of flowers. Most have three petals like lillies and irises, five like parnassia and rose hips, or eight like cosmea, 13 like daisies, 21 like chicory, and 34, 55, or even 89 like asteraceae. Fibonacci spirals and Golden spirals appear in nature, but not every spiral in nature is related to Fibonacci numbers or Phi. Most spirals in nature are equiangular spirals, and Fibonacci and Golden spirals are special cases of the broader class of equiangular spirals. Spirals and the Golden Ratio. Fibonacci numbers and Phi are related to spiral growth in nature. If you sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number. This property results in the Fibonacci spiral, based on the following series: 12 + 12 + 22+ 32+ 52 = 5 x 8  16 1 | Mathematics in the Modern World 12 + 12 + … + F(n)2 = F(n) x F(n+1) The Fibonacci spirals and Golden spirals appear in nature, but not every spiral in nature is related to Fibonacci numbers or Phi. Most spirals in nature are equiangular spirals, and Fibonacci and Golden spirals are special cases of the broader class of Equiangular spirals. An equiangular spiral itself is a special type of spiral with unique mathematical properties in which the size of the spiral increases but its shape remains the same with each successive rotation of its curve. The curve of an equiangular spiral has a constant angle between a line from origin to any point on the curve and the tangent at that point, hence its name. In nature, equiangular spirals occur simply because the forces that create the spiral are in equilibrium, and are often seen in non-living examples such as spiral arms of galaxies and the spirals of hurricanes. Fibonacci spirals, Golden spirals and golden ration-based spirals often appear in living organisms. Previously, you learned that numbers are everywhere. Numerical sequences are also patterns of numbers known as terms. Let us try to know some of the usefulness of these patterns in our lives. Look at these sequences: a. 2, 4, 8, , , b. 3, 5, 7, , , c. 1, 4, 9, , , What are the next three terms of these sequences? Let also consider the sequence below: 0, 1, 1, 2, 3, 5, … What is the sixth term of this sequence? The seventh? Can you tell how the terms are generated? This is called the Fibonnaci Sequence. You try this! (number 2) A. Use a tape measure to find the ratio of the following measures in your body:  17 1 | Mathematics in the Modern World 20 1. foot and hand length = 𝑓𝑜𝑜𝑡 = =.5714 ℎ𝑎𝑛𝑑 𝑙𝑒𝑛𝑔𝑡ℎ 35 19 2. the length of your forearm and the length of your hand = 𝑓𝑜𝑟𝑒𝑎𝑟𝑚 = =.5428 ℎ𝑎𝑛𝑑 𝑙𝑒𝑛𝑔𝑡ℎ 35 43 3. shoulder length and waistline = 𝑠ℎ𝑜𝑢𝑙𝑑𝑒𝑟 𝑙𝑒𝑛𝑔𝑡ℎ = =.6515 𝑤𝑎𝑖𝑠𝑡𝑙𝑖𝑛𝑒 66 80 4. bust and right palm = 𝑏𝑢𝑠𝑡 = = 5.33 𝑟𝑖𝑔ℎ𝑡 𝑝𝑎𝑙𝑚 15 78 5. hips and armpit = ℎ𝑖𝑝𝑠 = = 2.516 𝑎𝑟𝑚𝑝𝑖𝑡 31 B. Determine whether the ratio give the value of (phi). 3.1416 1. 2. 3. 4. 5. 1.3 References Livio, M. (2003). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.). Broadway Books. ISBN 978-0-7679-0816-0. Padovan, Richard (2002). "Proportion: Science, Philosophy, Architecture". Nexus Network Journal. 4 (1). https://doi:10.1007/s00004-001-0008-7. Knott, Ron. "Fibonacci's Rabbits". University of SurreyFaculty of Engineering and Physical Sciences. Hannavy, J. (2007). Encyclopedia of Nineteenth-Century Photography. 1. CRC Press. ISBN 978-0-415-97235-2. Forbes, P. (2012). All that useless beauty. The Guardian. Review: Non-fiction. Koning, R. (2012). "Plant Physiology Information Website". Pollination Adaptations. Minamino, Ryoko; Tateno, Masaki (2014). "Tree Branching: Leonardo da Vinci's Rule versus Biomechanical Models". https://doi:10.1371/journal.pone.0093535. Sadegh, Sanaz (2017). "Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork". Physical Review https://doi:10.1103/PhysRevX.7.011031. PMC 5500227. PMID 28690919.  18 1 | Mathematics in the Modern World Prusinkiewicz, P., Lindenmayer, A. (1990). The Algorithmic Beauty of Plants. Springer-Verlag. ISBN 978-0-387-97297-8. Lewalle, J. (2006). "Flow Separation and Secondary Flow: Section 9.1" (PDF). Lecture Notes in Incompressible Fluid Dynamics: Phenomenology, Concepts and Analytical Tools. Syracuse, NY: Syracuse University. Archived from the original (PDF). Tolman, H.L. (2008), "Practical wind wave modeling", in Mahmood, M.F. (ed.), CBMS Conference Proceedings on Water Waves: Theory and Experiment (PDF), World Scientific Publishing. Strahler, A. & Archibold, O.W. Physical Geography: Science and Systems of the Human Environment. John Wiley, 4th edition 2008. Brodie, C. (2005). "Geometry and Pattern in Nature 3: The holes in radiolarian and diatom tests". Microscopy-UK. Hook, J.R.; Hall, H.E. (2010). Solid State Physics (2nd Edition). Manchester Physics Series, John Wiley & Sons. ISBN 978-0-471-92804-1 Murray, James D. (2013). Mathematical Biology. Springer Science & Business Media. ISBN 978-3-662-08539-4. Prum, R. O.; Williamson, S. (2002). "Reaction–diffusion models of within-feather pigmentation patterning"(PDF). Proceedings of the Royal Society of London B. 269(1493). https://doi:10.1098/rspb.2001.1896. PMC 1690965. PMID 11958709. D'Avanzo, C. (2004). "Fir Waves: Regeneration in New England Conifer Forests". TIEE. Retrieved 26 May 2012. Morelle, Rebecca (2013). "'Digital gophers' solve Mima mound mystery". BBC News. Sample, I. (2017). "The secret of Namibia's 'fairy circles' may be explained at last". The Guardian. Retrieved 18 January 2017. Ghose, T. (2018). "Human Brain's Bizarre Folding Pattern Re-Created in a Vat". Scientific American. Tallinen, T.; Chung, J; Biggins, J. S.; Mahadevan, L. (2014). https://youtu.be/me6Dn12DOtM  19 1 | Mathematics in the Modern World 1.4 Acknowledgment The images, tables, videos, figures and information contained in this module were taken from the references cited above. Finding patterns is the essence of wisdom. -Dennis Prager Course, Year and Section Exercise No. 1(Mathematics in Our World) symmetry, trees, fractals, spirals, chaos, flow, meanders, waves, dunes, bubbles, foam, tessellations, cracks, spots, stripes, pattern formation. I. Choose from the box above the type of pattern that is seen in the picture. 1. Pattern formation 2. Pattern formation _ _3. Pattern formation _4. Wave  20 1 | Mathematics in the Modern World _5.cracks 6_trees _7. Spots, _ _8. Trees 9 10 11 12 9.symmetry 10. stripe 11. Cracks 12.symmetry 13. spirals 14. Pattern formation 15. tessellation 16. Bubbles, 17.chaos, _ 18 tessellation _19. tessellation 20. symmetry II. Let F(n) be the nth term of the Fibonacci sequence, with F(1) = 1, F(2) = 1, F(3) = 2, so on. 1. Find F(8)= 21 2. Find (19)= 4181 3. If F(22) = 17,711 and F(24) = 46,368, what is F(23)? F(23)= 28657 4. What is F(1) + F(2)? F(3)=2  21 1 | Mathematics in the Modern World 5. What is F(1) + F(2) + F(3) + F(4) + F(5) + F(6)? 20 III. Watch the video “Nature by Numbers/The Golden Ratio and Fibonacci Numbers” in youtube (use this link: https://youtu.be/me6Dn12DOtM) and answer the following questions as briefly and concisely as can be: (I will also send this video to the group chat of your section.) 1. What new ideas about mathematics did you learn from the lesson? -the new ideas about mathematics that I learned from the lesson is that the golden ratio describes pattern on everything from atoms to huge stars in the sky, and nature uses this ratio to maintain balance, and financial market seems to as well and the fibonacci sequence can be applied to finance by using four main techniques retracements, fans, time zones and arcs. 2. What is it about mathematics that might have changed your thoughts after learning the patterns in nature in both living and non-living things? -that mathematics is not just about number and solving it is also a study of patterns and studying patterns can help us to observe, experiment and create. By learning the patterns in nature in both living and non- living things we can discover and explain abstract patterns or regularities of all kinds. And it is an important skills in math that help us to learn complex number concepts and mathematical operation, patterns allow us to see relationships and develop generalization. 3. How useful is mathematics to your daily lives? - It is very helpful it helps us understand the words and help us in many ways important things like government budget, business financial problems, school, sport, making building and houses, cooking and many more it is a huge part of our life, many people may say why study math we can’t even use it in real that is not true cause we use math everyday some people may not know but its true without math our world today will be lost.Example number is a math without numbers people could not read clocks, measure anything, call anybody and much more. Everyone has been put into position where they were presented with different forms of data. without math the world will be pure chaos.