CONTENTS 1. Introduction what is geometry? 02 where is geometry? 03 2. Body geometry in structures 04 The golden rectangle 07 Fractal 10 Parabola 12 Geometry in astronomy 14 Optical illusions 17 Geometry in fashion 19 Sacred geometry 23 3. conclusion 26 4. bibliography 27 What is geometry?
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The word geometry derives from the Ancient Greek geo (earth) and metron (measurement). “Earth – measure” is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Where is geometry? Geometry is everywhere. Everywhere in the world you see geometry . It is happening in the world all around us. It is found in art, architecture, engineering, robotics, land surveys, astronomy, sculptures, space, nature, sports, machines, cars and much more.
Geometry is used and found everywhere in the world in some way or the other. Man-made structures that include geometric structures would be almost everything. If a person looks closely, they would see geometry in the structure. Nature even has its own geometric structures. The world is a big sphere, so is the moon and the other 8 planets in the Solar System. The whole Universe is a geometric structure, which proves that not only man-made structures are geometric but that even nature has geometry. Geometry exists even in things a human cannot see, just know it’s there.
Geometry in buildings and structures All the structures in the world are geometric. This is a structure with basic geometric shapes found everywhere. This is a modern reconstruction of the English Wigwam. As you can see 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. 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. 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. 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. 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. These were all basic geometry shapes in the world today. There are a lot more structure and things that contain geometry but that would be too many to list. The golden rectangle
A golden rectangle is one whose side lengths are in the golden ratio, or approximately 1:1. 618o34. A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a , will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship The golden rectangle R, constructed by the Greeks, has the property that when a square section is removed, the remainder is another golden rectangle; that is, with the same proportions as the first.
Square removal can be repeated infinitely in which case corresponding corners of the square form an infinite sequence of points on the Golden Spiral, the unique logarithmic spiral with this property. The sides are in the “golden proportion” (1:1. 618o34). The Golden Rectangle is found everywhere in nature and as well as in the work of men. It somehow appeals to our aesthetic sense of beauty. The Golden Rectangle was considered by the Greeks to be of the most pleasing proportions, and it was used in ancient architecture.
The ancient Parthenon Temple in Greece is the most famous example of the use of Golden Rectangle. The Architects of the Parthenon used the Golden Proportion (Rectangle) in a number of areas in its design. For example, on the gable side of the building, the width and height of the building combine to create a Golden Rectangle. The total height of the building is approximately 1. 618o34 times the height to the top of the columns, and the frieze sculptures (columns) and metopes (sculptures) on the entablature mimic those proportions as well. The Great Pyramid of Giza is also built on these proportions.
The Golden Rectangle is prevalent in the construction of trees, architecture, music, human and animal bodies and even in art ( the Golden Rectangle is found in the Mona Lisa). FRACTAL A fractal is “a rough or fragmented geometric shape that can be split into parts, each of which is (at least) a reduced-size copy of the whole,” a property called self-similarity. While fractals are a mathematical construct, they are found in nature, which has led to their inclusion in artwork. They are useful in medicine, soil mechanics, seismology, and technical analysis. Approximate fractals are easily found in nature.
These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, river networks, mountain ranges, snow- flakes, cauliflower, and systems of blood vessels and pulmonary vessels and ocean waves. Trees and ferns are fractal in nature and can be modeled on a computer by using recursive algorithm (an effective method expressed as an finite list of well-defined instructions for calculating a function). The connection between fractals and leaves is currently being used to determine how much carbon is contained in trees.
Fractal patterns have also been found in paintings, which appear to be composed of chaotic dripping and splattering, computer analysis has found fractal patterns in some work of art. Fractals are also used in the classification of histopathology (refers to the microscopic examination of tissue) slides in medicine, signal and image compression, seismology and even in the generation of new music and T-shirts and fashion. Parabola A parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.
Given a point (the focus) and a corresponding line (the directix) on the plane, the locus of points in that plane that are equidistant (equal distance) from them is a parabola. The line perpendicular to the directix and passing through the focus (that is the line that splits the parabola through the middle) is called the “axis of symmetry”. Parabolas can open up, down, left, right, or in some arbitrary direction. The parabola has many important applications, from automobile headlight reflectors to the design of ballistic missiles. They are frequently used in Physics, Engineering, and many other areas.
In nature, approximations of parabolae and paraboloids (such as catenary curves) are found in many diverse situatuons. The best known instance of the parabola in the history of Physics is the trajectory (the path a moving object follows through space as a function of time) of a particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a baseball flying through the air, neglecting air friction). Another situation in which parabolae may arise in nature is in two-body orbiys, for example, of a small planetoid or other object under the influence of the gravitation of the Sun.
The best known paraboloids is the parabolic reflector, which is a minor or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point. Aircraft used to create a weightless state for purposes of experimentation, such as NASA’s “Vomit Comet”, follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall. Which produces the same effect as zero gravity for most purposes. Astronomy Astromomy is the scientific study of celestial objects such as stars, planets, comets and galaxies.
Being one of the oldest sciences. Astronomy is concerned with evolution and the formation nad development of the universe. Our galaxy is filled with many geometric shapes, from the stars forming certain angles to form a constellation, to the shape of the many planets found in it. The Greeks used geometry to understand the stars and the way that our galaxy works. Geometry helped them to determine how the planets orbited the Sun, and also to measure the planets, Sun and moons. Shapes The Sun, moon, and all the planets in the solar system are all shaped as a sphere.
This earth is actually an “oblate spheriod”, but the reason why these all share a common shape is because of the force of gravoty. The spheroid shape allows the Earth to rotate, and all points of mass are within a certain radius of the cebter so it is a low energy shape. Angles forming constellations The stars in the sky paint certain pictures that are known as constellations. Perharps one of the most natural and earliest sightings of certain shapes, constellations have been noted for ages. Many shapes are contained in constellations. For example, “Sagittarius” is composed of two triangles, one quadilateral, and one trapezoid.
Angles can be seen in the images that these constellations make. Geometry in orbits Orbits are the path of a celestial body as it revolves around another body. Early settlers thought that the Earth was at the center of the Universe, and that Sun, moon and stars and naked eye planets revolved around the Earth. Scientist such as Ptomely and Aristotle embraced this theory, but geometry helped to prove it wrong. When Nicholaus Copernicus formualted helicentrit cosmology, it was discovered that the Sun was the center of the the Solar System and that planets and stars revolved around it.
The orbits that are made have three distinct shapes – ellipses, hyperbolas, and parabolas. Eccentricity= parameter of the orbit that defines its absolute shape Our Solar System is a map of many geometric features, and geometry is very useful in understanding and figuring out astronomical aspects. Optical illusions Optical illusions are deceptive or misleading visuals that trick your mind. They make it appear as though something is different than it really is. There are many kinds of optical illusions in the world. There are illusions that involve: ? Movement ?Luminance ( Contrast ?Colour ?3D and space ?Angles This illusion is called the “Cafe Wall Illusion”. It appears that the horizontal lines slope upwards and downward. However, the horizontal rows are actually parallel. If we were to measure all of the angles of the square blocks, they would be 90 degrees. This proves that the lines are parallel because if they really did slope, the angles of the blocks would change as they got to the smaller end. Here is another example so that you can further see the geometry in optical illusions: This type of illusion is called the “Hering illusion”.
It was discovered by the German physiologist Ewald Hering in 1861. Because of the thick lines of the circle, the sides of the shape in the middle appear to be bent inwards. In reality, the shape is a perfect square with straight sides. Just like the Cafe Wall example, if we were to measure the angles of the square they would be perfect right angles, proving that the curvature is just an illusion. Geometry in fashion If you were to look around your home, or mall, or anywhere else you would notice that most of the people around you were wearing clothing and other fashion pieces with geometrical patterns all over them.
The geometrical patterns can be as simple as a triangle, circle, square, etc, or as intricate ( having many interrelated parts or facets) or tessellation (a pattern of shapes that fit perfectly together) of the already mentioned shapes. Now, more than ever fashion runaways of New York, Milan and Paris are flooded with geometrical shapes present on designers clothing. Geometrical shapes are not just being used by department store designers but by high fashion designers as well. Prada, Gucci. Versace, and Louis Vuitton are only a few of the high fashion designers that are using intricate geometrical patterns in their collections.
One commonly well-known designer took geometrical shapes to another level. Calvin klein designed his entire Spring 2008 collection using geometrical shapes. However he did not just use fabrics that had geometrucal shapes printed on it, he actually made the clothes the models were wearing look geometrical. As you can see in the picture below the modles dress looks as if it is a bunch of triangular pieces placed together. Other types of fashion which contains geometry in them are habdbags and jewelery. Handbags, like clothing, usually have intricate geometrical shapes and or tessellations on them which create the pattern of a handbag.
A handbag also comes in many different sizes and the way the sizes are computed and then constructed is done using geometrical equations, such as the area of a shape, the shape of the bag, or the measurements of the lenghts of the different shapes which make up the handbag. Jewelry also has many geometrical aspects to it as well. Not only is the look of the jewelry usually geometrical, such as the repition of shapes, or just the shapes used, but the way jewelry is constructed also has geometrical features. A bracelet or necklace has to be made a certain length and the jewelry pieces have to fit nicely onto a chain.
For example, to figure out how many silver beads belong on a Tiffany’s necklace the manufacturer must first measure the length of each bead and compute the amount of beads he/ahe will need to fill the chain. Geometry used to manufacture fashion Obviously when clothes are manufactures and produced for mass sale the clothes have to have a size run, of several different sizes. These sizes and size differences are achieved by using geometry. Each size had to follow certainmeasurements which are crucial to be precise. Clothes of all types, shirts, pants, dresses, skirts, etc can be measured using various geometrical strategies.
Sacred geometry “Go down deep enough into anything and you will gind mathematics. ” ~Dean Schlicter They say that the mysteries of the Universe are as vast and numerous as the stars in the sky. While many find answers to their questions in religion or mysticism, others find answers in math and science. But when you combine the two you get a new kind of answer… you get Sacred Geometry. Sacred Geometry is geometyr that involves the sacred universal patterns used in the design of everything in our reality; it is a worldview of pattern recognition, a complex system of religious symbols, and structures involving space, time and form.
Sacred Geometry is most often seen in architecture and in sacred art. The basic patterns of exixtence are perceived as sacred. Sacred Geometry is the basis some believe that all creation is formed from. In Christianity, the major aymbol for generations has been the Cross. The Cross had become such a religious symbol that people often forget that it is also a geometric shape. The cross is nothing more than an unfolded cube. The Cross was and still is such and influnential shape that many Churches have been and are still being built based on the cross shape.
The Cross Many forms observed in nature can be related to geometry. For example, the chambered nautilus grows at a constant rate and so its shell forms a logarithmic spiral to accomadate that growth without changing shape. Also, honeybess construct hexagonal cells to hold their honey. These and other correspondence are seen by believers in sacred geometry to be furthe proof of the cosmic significance of geometric forms. Honeycomb Sunflower The Golden Ratio, geometric ratios, and geometric figures were often employed in the design of Egyptian, ancient Indian, Greek and Roman architecture.
Medieval European cathedrals also incorporated symbolic geometry. Indian and Himalayan spiritual communities often constructed temples and fortifications on design plans of mandala and yantra. Yantra Yin and Yang The Flower of Life is the modern name given to a geometrical figure composed of multiple evenly –spaced, overlapping circles. They are arranged to form a flower-like pattern with a sixfold symmetry, similar to a hexagon. The centre of each circle is on the circumference of six surrounding circles of the same diameter.
It is considered by some to be a symbol of sacred geometry, said to conatian ancient, religious value depicting the fundamental forms of space and time. It is believed to contain a type of Akashic Records of basic information of all living things. There are many apiritual beliefs associated with the Flower of Life; for example, depiction of the five Platonic Solids are found within the symbol of Metatrons Cube, which may be derived from the Flower of Life pattern. These platonic solids are geometrical forms which are said to act as a template from which all life springs.
According to Drunvalo melchizedek, in the Judeo-Christian tradition, the stages which construct the Seed of Life are said to represent the six days of Creation, in which Elohim created life. conclusion In conclusion for my topic, Geometry in the World, I would just like to say that geometry is found everywhere in the world. No matter where you look; up, down, to the right or the left; or what you look at; a car, a building, the sky or the ocean; whether it is a man-made structure or it occurs naturally, you will find geometry in it no matter what.
The world is full of Geometry, one of the oldest mathematical sciences, which has been through the ancient eras to the modern ages and is still here to stay for the future without even giving a sign that it is going to finish anytime sooner… The strands of our DNA, the corners of our eye, snow-flakes, pine cones, flower petals, diamond crystals, the branching of trees, a nautilus shell, the star we spin around, the galaxy we spiral within, the air we breathe, and all life forms as we know them emerge out of timeless geometric codes.
The designs of exalted holy spaces from the prehistoric monuments at Stonehenge and the Pyramid of Khufu at Giza, to the world’s great cathedrals, mosques, and temples are based on these same principles of geometry. bibliography links 1. http://en. wikipedia. org/wiki/Parabola 2. http://faculty. fullerton. edu/crenne/Geometry/geometry. htm 3. http://www. ics. uci. edu/~eppstein/geom. html 4. http://library. thinkquest. org/C006354/pictures. html 5. http://en. wikipedia. org/wiki/Golden_rectangle 6. http://en. wikipedia. org/wiki/Golden_ratio 7. http://www. geom. uiuc. du/~demo5337/s97b/art. htm 8. http://www. britannica. com/EBchecked/topic/229851/geometry/217484/Astronomy-and-trigonometry 9. http://wiki. answers. com/Q/How_is_geometry_used_in_astronomy 10. http://biology. wsc. ma. edu/Math251/node/24 11. http://en. wikipedia. org/wiki/Fractal 12. http://en. wikipedia. org/wiki/Geometry references 1. Lockwood, E. H. (1961): A Book of Curves, Cambridge University Press 2. Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W. H. Freeman and Company.. ISBN 0-7167-1186-9. 3. Briggs, John (1992). Fractals:The Patterns of Chaos.
London : Thames and Hudson, 1992.. p. 148. ISBN 0500276935, 0500276935 4. Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice. 5. Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 320. Taylor & Francis. ISBN 0-419-22780-6: “Both the paintings and the architectural designs make use of the golden section”. 6. Mlodinow, M. ; Euclid’s window (the story of geometry from parallel lines to hyperspace), UK edn. Allen Lane, 1992