MATHEMATICS

1. MATHEMATICS SYMMETRY

2. INDEX Symmetry Picture1 Symmetry in the field of mathematics Picture2 Reflection symmetry Symmetry in history, religion, and culture picture3 Symmetry in religious symbols Symmetry in Social Interactions Symmetry in architecture Picture4 Symmetry in pottery and metal vessels Picture5 Symmetry in carpets and rugs Picture6 Symmetry in other arts and crafts Picture7 Questions related- Search Engines THANKS GIVING Thank you

3. Symmetry Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise. This article describes these notions of symmetry from three perspectives. The first is that of mathematics, in which symmetries are defined and categorized precisely. The second perspective describes symmetry as it relates to science and technology. In this context, symmetries underlie some of the most profound results found in modern physics, including aspects of space and time. Finally, a third perspective discusses symmetry in the humanities, covering its rich and varied use in history, architecture, art, and religion. The opposite of symmetry is asymmetry.

5. Symmetry in the field of mathematics In formal terms, we say that an object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation does not change the object or its appearance. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa). Symmetries may also be found in living organisms including humans and other animals. In 2D geometry the main kinds of symmetry of interest are with respect to the basic Euclidean plane isometrics: translations, rotations, reflections, and glide reflections.

7. Reflection symmetry Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. In 1D, there is a point of symmetry. In 2D there is an axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror Symmetric. The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror image. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry, for the same reason.

8. Symmetry in history, religion, and culture In any human endeavor for which an impressive visual result is part of the desired objective, symmetries play a profound role. The innate appeal of symmetry can be found in our reactions to happening across highly symmetrical natural objects, such as precisely formed crystals or beautifully spiraled seashells. Our first reaction in finding such an object often is to wonder whether we have found an object created by a fellow human, followed quickly by surprise that the symmetries that caught our attention are derived from nature itself. In both reactions we give away our inclination to view symmetries both as beautiful and, in some fashion, informative of the world around us.

10. Symmetry in religious symbols The tendency of people to see purpose in symmetry suggests at least one reason why symmetries are often an integral part of the symbols of world religions. Just a few of many examples include the six fold rotational symmetry of Judaism's Star of David, the twofold point symmetry of Taoism's Taijitu or Yin-Yang, the bilateral symmetry of Christianity's cross and Sikhism's Khanda, or the fourfold point symmetry of Jain's ancient (and peacefully intended) version of the swastika. With its strong prohibitions against the use of representational images, Islam, and in particular the Sunni branch of Islam, has developed intricate and visually impressive use of symmetries

11. Symmetry in Social Interactions People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of reciprocity, empathy, apology, dialog, respect, justice, and revenge. Symmetrical interactions send the message "we are all the same" while asymmetrical interactions send the message "I am special; better than you." Peer relationships are based on symmetry, power relationships are based on asymmetry.

12. Symmetry in architecture Another human endeavor in which the visual result plays a vital part in the overall result is architecture. Both in ancient times, the ability of a large structure to impress or even intimidate its viewers has often been a major part of its purpose, and the use of symmetry is an inescapable aspect of how to accomplish such goals. Just a few examples of ancient examples of architectures that made powerful use of symmetry to impress those around them included the Egyptian Pyramids, the Greek Parthenon, the first and second Temple of Jerusalem, China's Forbidden City, Cambodia's Angkor Wat complex, and the many temples and pyramids of ancient Pre-Columbian civilizations. More recent historical examples of architectures emphasizing symmetries include Gothic architecture cathedrals, and American President Thomas Jefferson's Monticello home. India's unparalleled Taj Mahal is in a category by itself, as it may arguably be one of the most impressive and beautiful uses of symmetry in architecture that the world has ever seen.

14. Symmetry in pottery and metal vessels Since the earliest uses of pottery wheels to help shape clay vessels, pottery has had a strong relationship to symmetry. As a minimum, pottery created using a wheel necessarily begins with full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point cultures from ancient times have tended to add further patterns that tend to exploit or in many cases reduce the original full rotational symmetry to a point where some specific visual objective is achieved. For example, Persian pottery dating from the fourth millennium B.C. and earlier used symmetric zigzags, squares, cross-hatchings, and repetitions of figures to produce more complex and visually striking overall designs.

16. Symmetry in carpets and rugs A long tradition of the use of symmetry in carpet and rug patterns spans a variety of cultures. American Navajo Indians used bold diagonals and rectangular motifs. Many Oriental rugs have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs typically use quadrilateral symmetry—that is, motifs that are reflected across both the horizontal and vertical axes .

18. Symmetry in other arts and crafts The concept of symmetry is applied to the design of objects of all shapes and sizes. Other examples include beadwork, furniture, sand paintings, knot work, masks, musical instruments, and many other endeavors.

20. Questions related- • What is the opposite of symmetry? • Define symmetry. • Symmetry is used in what other topics?

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22. THANKS GIVING We thank our teacher Vijay kumari ma’am to have given us an opportunity to be doing this project. Unnathi and I Saloni read the lesson and also gained external knowledge from the net. It was a lot of fun while doing this project!