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Ananthakrishanan X ‘B’ Introduction : Introduction Geometry is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere, served as an important source of geometric problems during the next one and a half millennia. A mathematician who works in the field of geometry is called a geometer. Slide 3: Geometry is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Slide 4: (a) Computer graphics is based on geometry - how images are transformed when viewed in various ways. Graphics used in Mirror’s Edge, the videogame. Geometry being used in : Slide 5: (b) Computer-aided design, computer-aided geometric design. Representing shapes in computers, and using these descriptions to create images, to instruct people or machines to build the shapes, etc. (e.g. the hood of a car, the overlay of parts in a building construction, even parts of computer animation). Geometry being used in : Graphics used for industrial purposes. Slide 6: (c) Robotics. Robotic vision, planning how to grasp a shape with a robot arm, or how to move a large shape without collission. Geometry being used in : A Korean Robot, D2E. Slide 7: (d) Medical imaging - how to reconstruct the shape of a tumor from CAT scans, and other medical measurements. Lots of new geometry and other math was (and still is being) developed for this. Geometry being used in : CAT Scan Chamber. Slide 8: (e) Structural engineering. What shapes are rigid or flexible, how they respond to forces and stresses. Statics (resolution of forces) is essentially geometry. This goes over into all levels of design, form, and function of many things. Geometry being used in : Rigid Buckminster Structure. Slide 9: (f) Protein modeling. Much of the function of a protein is determined by its shape and how the pieces move. Mad Cow Disease is caused by the introduction of a 'shape' into the brain (a shape carried by a protein). Many drugs are designed to change the shape or motions of a protein - something that we are just now working to model, even approximately, in computers, using geometry and related areas (combinatorics, topology). Geometry being used in : Protein Structure. Slide 10: (g) Physics, chemistry, biology, .... . Symmetry is a central concept of many studies in science - and also the central concept of modern studies of geometry. Students struggle in university science if they are not able to detect symmetries of an object (molecule in stereo chemistry, systems of laws in physics, ... ). the study of transformations and related symmetries has been, since 1870s the defining characteristic of geometric studies. Geometry being used in : Symmetry of a Leaf. Slide 11: Geometry is used everywhere. Everywhere in the world there is geometry, mostly made by man. Most man made structures today are in a form of Geometric. How, you ask? Well some examples would the a CD, that is a 3-D circle and the case would be a rectangular prism. Buildings, cars, rockets, planes, maps are all great examples. Here's some examples on how the world uses Geometry in buildings and structure….. Pictures of Geometry used in Real Life Slide 12: 1. This a pictures with some basic geometric structures. This is a modern reconstruction of the English Wigwam. As you can there the door way is a rectangle, and the wooden panels on the side of the house are made up of planes and lines. Except for really planes can go on forever. The panels are also shaped in the shape of squares. The house itself is half a cylinder. Slide 13: 2. Here is another modern reconstruction if of a English Wigwam. This house is much similar to the one before. It used a rectangle as a doorway, which is marked with the right angles. The house was made with sticks which was straight lines at one point. With the sticks in place they form squares when they intercepts. This English Wigwam is also half a cylinder. Slide 14: 3. This is a modern day skyscraper at MIT. The openings and windows are all made up of parallelograms. Much of them are rectangles and squares. This is a parallelogram kind of building. Slide 15: 4. This is the Hancock Tower, in Chicago. With this image, we can show you more 3D shapes. As you can see the tower is formed by a large cube. The windows are parallelogram. The other structure is made up of a cone. There is a point at the top where all the sides meet, and There is a base for it also which makes it a cone. Slide 16: 5. This is another building at MIT. this building is made up of cubes, squares and a sphere. The cube is the main building and the squares are the windows. The doorways are rectangle, like always. On this building There is a structure on the room that is made up of a sphere. Slide 17: 6. This is the Pyramids, in Indianapolis. The pyramids are made up of pyramids, of course, and squares. There are also many 3D geometric shapes in these pyramids. The building itself is made up of a pyramid, the windows a made up of tinted squares, and the borders of the outside walls and windows are made up of 3D geometric shapes. Slide 18: 7. This is a Chevrolet SSR Roadster Pickup. This car is built with geometry. The wheels and lights are circles, the doors are rectangular prisms, the main area for a person to drive and sit in it a half a sphere with the sides chopped off which makes it 1/4 of a sphere. If a person would look very closely the person would see a lot more shapes in the car. Too many to list. Conclusion : Conclusion Two developments in geometry in the nineteenth century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Lobachevsky, Bolyai and Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann, working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics. The traditional type of geometry was recognized as that of homogeneous spaces, those spaces which have a sufficient supply of symmetry, so that from point to point they look just the same Slide 20: Thank You You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.