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# Maths in nature (complete)

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### Maths in nature (complete)

1. 1. Symmetry in nature Made by:-Abhay goyal X-B 754
2. 2. INTRODUCTION Mathematics is all around us. As we discover more and more about our environment and our surroundings we see that nature can be described mathematically. The beauty of a flower, the majesty of a tree, even the rocks upon which we walk can exhibit natures sense of symmetry. Although there are other examples to be found in crystallography or even at a microscopic level of nature, we havechosen representations within objects in our field of view that exhibit many different types of symmetry. 1.Bilateral symmetry 2. Radial symmetry 3. Strip patterns 4.Wallpaper patterns
3. 3. Radial symmetryRadial symmetry is rotational symmetry around a fixed point known as thecenter. Radial symmetry can be classified as either cyclic or dihedral.Cyclic symmetries are represented with the notation Cn, where n is the numberof rotations. Each rotation will have an angle of 360/n. For example, an objecthaving C3 symmetry would have three rotations of 120 degrees.Dihedral symmetries differ from cyclic ones in that they have reflectionsymmetries in addition to rotational symmetry. Dihedral symmetries arerepresented with the notation Dn where n represents the number of rotations,as well as the number of reflection mirrors present. Each rotation angle will beequal to 360/n degrees and the angle between each mirror will be 180/ndegrees. An object with D4 symmetry would have four rotations, each of 90degrees, and four reflection mirrors, with each angle
4. 4. 1.A starfish provides us with a Dihedral 5 symmetry. Not only do wehave five rotations of 72 degrees each, but we also have five linesof reflection.2.Another example of a starfish - as we can see, starfish can beembedded in a pentagon, which can then be connected to theGolden Ratio ...
5. 5. 3.Jellyfish have D4 symmetry - four rotations of 90 degrees each. It alsohas four lines of symmetry, and in the middle you have a four-leafedclover for good luck 4.Hibiscus - C5 symmetry. The petals overlap, so the symmetry might not be readily seen. It will be upon closer examination though
6. 6. Strip pattern symmetry can be classified in sevendistinct patterns. Each pattern contains all or some ofthe following types of symmetry: Translationsymmetry, Horizontal mirror symmetry, Vertical mirrorsymmetry, Rotational symmetry, or Glide reflectionsymmetry.The seven types are T, TR, TV, TG, TRVG, TGH, andTRGHV.
7. 7. 1.An Eastern White Pine has interesting symmetry on its trunk.Each year, as the tree grows, it develops a new ring of branches(most of which have been broken off in the picture above). Therings move up by similar translation vectors, but some variationoccurs due to the conditions for that year.2.Another picture of the white pine - this time with branchesshowing. The white pine exhibits T symmetry
8. 8. 3.The copperhead is one of the four poisonous snakes in the UnitedStates. Can you name the other three? Highlight the text between thearrows for the answer:>> The Cottonmouth (Water Mocassin), Rattle Snake, Coral Snake <<As with most snakes, it has TRGHV symmetry.The black rat snake is a non-poisonous snake, and like thecopperhead (and most other snakes with patterns), it has TRGHVsymmetry.
9. 9. Wallpaper patterns are patterns of symmetry thattessellate the plane from a given fundamental region.There are seventeen different types of wallpaperpatterns. In the examples below, you will see thefundamental regions highlighted, as well as thetranslation vector generators that can be used tocomplete the pattern by translation, after the otherisometries of the pattern are completed
10. 10. 1.The Giants Causeway, located in Ireland, is an fascinating *632formation found in nature. It is a collection of hexagons tesselatingthe ground - even in 3D at some points.2.Bees form their honeycombs in a *632 pattern as well. There seemsto be a lot of hexagonal symmetry in nature. Any conjectures on whythats the case? The answer lies with steiner points and minimalnetworks.
11. 11. Bilateral symmetry is symmetry across a line of reflection.Are people symmetric? We think we are, but upon closeranalysis, we are less symmetric than we think. The moresimple the creature (ants --> elephants), the more likeley itis that it will be perfectly symmetric.We took two professors, cut and pasted half of their headin Photoshop, and flipped that half horizontally. We thenaligned the two halves so that it came closest roresembling a human head. You be the judge on how goodof a job we did and how symmetric people around us are ingeneral ...
12. 12. Many mathematical principles are based on ideals, and apply to anabstract, perfect world. This perfect world of mathematics is reflectedin the imperfect physical world, such as in the approximate symmetryof a face divided by an axis along the nose. More symmetrical facesare generally regarded as more aesthetically pleasing
13. 13. conclusionSymmetry is the ordering principle in nature thatrepresents the center of balance between two or moreopposing sides. As a fundamental design principle, itpermeates everything: from man-made architecture tonatural crystalline formations. In nature, symmetryexists with such precision and beauty that we canthelp but attribute it to intelligence-such equalproportions and organization would seem to becreated only on purpose. Consequently, humans haveborrowed this principle for its most iconic creationsand symbols.