Beautiful Examples of Mathematics in                              Nature

                                         

  1. Sunflowers

                                    

Bright, bold and beloved by bees, sunflowers boast radial symmetry and a type of numerical symmetry known as the Fibonacci sequence, which is a sequence where each number is determined by adding together the two numbers that preceded it. For example: 1, 2, 3, 5, 8, 13, 21, 24, 55, and so forth.

 2. Nautilus Shell

     

                                        

• A nautilus is a cephalopod mollusk with a spiral shell and numerous short tentacles around its mouth.
• Although more common in plants, some animals, like the nautilus, showcase Fibonacci numbers. A nautilus shell is grown in a Fibonacci spiral. The spiral occurs as the shell grows outwards and tries to maintain its proportional shape.

3. Romanesco Broccoli

  

Romanesco broccoli has an unusual appearance, and many assume it’s another food that’s fallen victim to genetic modification. However, it’s actually one of many instances of fractal symmetry in nature.n geometric terms, fractals are complex patterns where each individual component has the same pattern as the whole object. In the case of romanseco broccoli, each floret is a miniaturised version of the whole head’s logarithmic spiral. This means the entire veggie is one big spiral composed of smaller, cone-like mini-spirals.

4. Pinecones

                                        

Pinecones have seed pods that arrange in a spiral pattern. They consist of a pair of spirals, each one twisting upwards in opposing directions.

• The number of steps will almost always match a pair of consecutive Fibonacci numbers. For example, a three–to–five cone meets at the back after three steps along the left spiral and five steps along the right.

5. Honeycombs

                                               

• Honeycombs are an example of wallpaper symmetry. This is where a pattern is repeated until it covers a plane. Other examples include mosaics and tiled floors.
• Mathematicians believe bees build these hexagonal constructions because it is the shape most efficient for storing the largest possible amount of honey while using the least amount of wax. Shapes like circles would leave gaps between the cells because they don’t fit perfectly together.

6. Milky Way Galaxy

                                                                                                                                                                                 

Recently, a new section on the edges of the Milky Way Galaxy was discovered, and, by studying this, astronomers now believe the galaxy is a near-perfect mirror image of itself.

• Using this new information, scientists have become more confident in their theory that the galaxy has only two major arms: the Scutum-Centaurus and the Perseus.
• As well as having mirror symmetry, the Milky Way has another amazing design. Like nautilus shells and sunflowers, each ‘arm’ of the galaxy symbolises a logarithmic spiral that begins at the galaxy’s centre and expands outwards.
• 
  

7. Peacocks

                                

• Most animals have bilateral symmetry, which means drawing an even centre line would create two matching halves.
• The peacock takes the earlier principle of using symmetry to attract a mate to the nth degree. In fact, Charles Darwin, who famously conceived the survival of the fittest theory, detested peacocks. 
• Darwin thought the peacock’s tail was a burden that made no evolutionary sense. He remained furious until coming up with the theory of sexual selection, which asserts that animals develop certain features to increase their chances of mating. Male peacocks utilise their variety of adaptations to seduce sultry peahens. These include bright colours, a large size, a symmetrical body shape and repeated patterns in their feathers.

 8. Sun-Moon Symmetry

                              

                                         

• The sun has a diameter of 1.4 million kilometres, while his sister, the Moon, has a meagre diameter of 3,474 kilometres. With these figures, it seems near impossible that the moon can block the sun’s light and give us around five solar eclipses every two years.                                                          By sheer coincidence, the sun’s width is roughly four hundred times larger than that of the moon, while the sun is about four hundred times further away. The symmetry in this ratio causes the moon and sun to appear almost the same size when seen from Earth, and, therefore, it becomes possible for the moon to block the sun when the two align.
• Earth’s distance from the sun can increase during its orbit. If an eclipse occurs during this time, we see what’s known as an annular or ‘ring’ eclipse. This is because the sun isn’t completely hidden.
• Every one to two years, though, the sun and moon become perfectly aligned, and we can witness a rare event called a total solar eclipse. 
• Astronomers don’t know how common this symmetry is between other planets, suns, and moons, but theorise that it’s quite rare. Every year, though, our moon drifts roughly four centimetres further from Earth. This means that, billions of years ago, every solar eclipse would have been a total eclipse.
• If things continue as they are, total eclipses will eventually cease entirely – as will annular eclipses, assuming the planet lasts that long. With this in mind, it’s easy to conclude that we’re simply in the right place at the right time to witness this phenomenon. Some have theorised that this sun-moon symmetry is the special factor which makes life on Earth possible.

                                          

  9.  Starfish

                                          

• Starfish or sea stars belong to a phylum of marine creatures called echinoderm. Other notable echinoderm include sea urchins, brittle stars, sea cucumbers and sand dollars.
• The larvae of echinoderms have bilateral symmetry, meaning the organism’s left and ride side form a mirror image. However, during metamorphosis, this is replaced with a superficial radial symmetry, where the organism can be divided into similar halves by passing a plane at any angle along a central axis.
• Sea stars or starfish are invertebrates that typically have five or more ‘arms’. These radiate from an indistinct disk and form something known as pentaradial symmetry.
• Their evolutionary ancestors are believed to have had bilateral symmetry, and sea stars do exhibit some superficial remnant of this body structure.

                     


10.  Orb Web Spiders

                             

• There are approximately 5,000 types of orb web spiders. All of them create near-perfect circular webs that have near-equal-distanced radial supports coming out of the middle and a spiral that is woven to catch prey.
• It’s not clear why orb spiders are so geometrically inclined. Tests have shown that orbed webs are no better at catching prey than irregularly shaped webs.
• Some scientists theorise that orb webs are built for strength, with radial symmetry helping to evenly distribute the force of impact when a spider’s prey makes contact with the web. This would mean there’d be less rips in the thread.

  

Mathematics Laboratory

The mathematics laboratory is a place where anybody can experiment and explore patterns and ideas. It is a place where one can find a collection of games, puzzles, and other teaching and learning material. The materials are meant to be used both by the students on their own and with their teacher to explore the world of mathematics, to discover, to learn and to develop an interest in mathematics. The activities create interest among students or in anybody who wants to explore, and test some of their ideas, beliefs about mathematics.

The activities in the maths lab should be appealing to a wide range of people, of different ages and varying mathematical proficiency. While the initial appeal is broad-based, the level of engagement of different individuals may vary. The maths lab activities listed here have been done with students and teachers of different grade levels. The activities are intended to give children an experience of doing mathematics and not merely for the purpose of demonstration.

The maths lab provides an opportunity for the students to discover mathematics through doing. Many of the activities present a problem or a challenge, with the possibility of generating further challenges and problems. The activities help students to visualize, manipulate and reason. They provide opportunity to make conjectures and test them, and to generalize observed patterns. They create a context for students to attempt to prove their conjectures.

It is important to note that while in science experiments provide evidence for hypotheses or theories, this is not so in mathematics. Observed patterns can only suggest mathematical hypotheses and conjectures, not provide evidence to support them. Mathematical truths are accepted only on the basis of proofs, and not through experiment.

Mathematics laboratory is a place to enjoy mathematics through informal exploration. It is a place where anyone can generate problems and struggle to get a answer. It is a space to explore and design new mathematical activities. So, the maths lab should not be used to assess students’ knowledge of mathematics. Often mathematics lab takes students knowledge beyond the curriculum.

Mathematics laboratory is a self-explanatory lab with activities, in which students could come any time and engage in the work, continue working on the problems/tasks, and use teachers as and when they are stuck. In this way, the role of the teacher is not to teach how to progress in the activity but to facilitate inquiry with the mathematics in it. The facilitation could be done either by probing questions, giving an extra resource or asking to follow or discuss with peers.

                                                               MATHEMATICS

                       Mathematics is a very important subject. The term mathematics may be defined in a number of ways. It is a exact science which is related to measurements, calculations, discovering relationships and dealing with the problems of space.

                                           DIFFERENT QUOTES RELATED TO MATHEMATICS

1.   " Mathematics is the indispensable  instrument of all physical researches". - Kant
2.   "Mathematics is the queen of sciences and arithmetic is the queen of all mathematics". - Gauss
3.   "Mathematics is the gateway and key to all sciences". - Bacon
4.  "Mathematics is the science of indirect measurement". - Comets
5.  "Mathematics is the language in which god has written the universe".- Galileo 
6.  "Mathematics is the study of quantity". - Aristotle 

TORUS


An  torus is a surface having genus one, and therefore possessing a single "hole" . The single-holed "ring" torus is known in older literature as an "anchor ring." It can be constructed from a rectangle by gluing both pairs of opposite edges together with no twists . The usual torus embedded in three-dimensional space is shaped like a donut, but the concept of the torus is extremely useful in higher dimensional space as well.

                     Pierre de Fermat

Born	17 August 1601 or 1607 Beaumont-de-Lomagne, France
Died	12 January 1665 (aged 63 or 57) Castres, France
Residence	France
Nationality	French
Fields	Mathematics and Law
Known for	Number theory Analytic geometry Fermat's principle Probability Fermat's Last Theorem Adequality
Influences	François Viète, Gerolamo Cardano, Diophantus

 

Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus' Arithmetica.

Life and work

Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne (present-day Tarn-et-Garonne), France; the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a wealthy leather merchant, and served three one-year terms as one of the four consuls of Beaumont-de-Lomagne. His mother was either Françoise Cazeneuve or Claire de Long. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth. There is little evidence concerning his school education, but it was probably at the Collège de Navarre in MontaubanHe attended the University of Orleans from 1623 and received a bachelor in civil law in 1626, before moving to Bordeaux. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius's De Locis Planis to one of the mathematicians there. Certainly in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Etienne d'Espagnet who clearly shared mathematical interests with Fermat. There he became much influenced by the work of Francois Viete.
In 1630 he bought the office of a councillor at the Parlement de Toulouse, one of the High Courts of Judicature in France, and was sworn in by the Grand Chambre in May 1631. He held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat. Fluent in six languages: French, Latin, Occitan, classical Greek, Italian, and Spanish, Fermat was praised for his written verse in several languages, and his advice was eagerly sought regarding the emendation of Greek texts.
He communicated most of his work in letters to friends, often with little or no proof of his theorems. Secrecy was common in European mathematical circles at the time. This naturally led to priority disputes with contemporaries such as Descartes and Wallis.
Anders Hald writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebraic methods."

Work

Fermat's pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes' famous La géométrie. This manuscript was published posthumously in 1679 in "Varia opera mathematica", as Ad Locos Planos et Solidos Isagoge, ("Introduction to Plane and Solid Loci").
In Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed a method (adequality) for determining maxima, minima, and tangents to various curves that was equivalent to differential calculus. In these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature.
Fermat was the first person known to have evaluated the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series.The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus.
In number theory, Fermat studied Pell's equation, perfect numbers, amicable numbers and what would later become Fermat numbers. It was while researching perfect numbers that he discovered the little theorem. He invented a factorization method—Fermat's factorization method—as well as the proof technique of infinite descent, which he used to prove Fermat's right triangle theorem which includes as a corollary Fermat's Last Theorem for the case n = 4. Fermat developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, and so on.
Although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss, doubted several of his claims, especially given the difficulty of some of the problems and the limited mathematical methods available to Fermat. His famous Last Theorem was first discovered by his son in the margin on his father's copy of an edition of Diophantus, and included the statement that the margin was too small to include the proof. He had not bothered to inform even Marin Mersenne of it. It was not proved until 1994 by Sir Andrew Wiles, using techniques unavailable to Fermat.
Although he carefully studied, and drew inspiration from Diophantus, Fermat began a different tradition. Diophantus was content to find a single solution to his equations, even if it were an undesired fractional one. Fermat was interested only in integer solutions to his Diophantine equations, and he looked for all possible general solutions. He often proved that certain equations had no solution, which usually baffled his contemporaries.
Through their correspondence in 1654, Fermat and Blaise Pascal helped lay the fundamental groundwork for the theory of probability. From this brief but productive collaboration on the problem of points, they are now regarded as joint founders of probability theory.Fermat is credited with carrying out the first ever rigorous probability calculation. In it, he was asked by a professional gambler why if he bet on rolling at least one six in four throws of a die he won in the long term, whereas betting on throwing at least one double-six in 24 throws of two dice resulted in his losing. Fermat subsequently proved why this was the case mathematically.
Fermat's principle of least time (which he used to derive Snell's law in 1657) was the first variational principle enunciated in physics since Hero of Alexandria described a principle of least distance in the 1st century CE. In this way, Fermat is recognized as a key figure in the historical development of the fundamental principle of least action in physics. The terms Fermat's principle and Fermat functional were named in recognition of this role.