Posts Tagged uses of geometry
The Five Most Important Concepts In Geometry
Posted by admin in Mathematics on June 18, 2011
Having just written an article about everyday uses of Geometry and another article about real world applications of the principles of Geometry, my head is spinning with all I found. Being asked what I consider the five most important concepts in the subject is “giving me pause.” I spent almost my entire teaching career teaching Algebra and avoiding Geometry like the plague, because I didn’t have the appreciation for its importance that I have now. Teachers who specialize in this subject may not totally agree with my choices; but I have managed to settle on just 5 and I did so by considering those everyday uses and real world applications. Certain concepts kept repeating, so they are obviously important to real life.
5 Most Important Concepts In Geometry:
(1) Measurement. This concept encompasses a lot of territory. We measure distances both large, like across a lake, and small, like the diagonal of a small square. For linear (straight line) measurement, we use appropriate units of measure: inches, feet, miles, meters, etc. We also measure the size of angles and we use a protractor to measure in degrees or we use formulas and measure angles in radians. (Don’t worry if you don’t know what a radian is. You obviously haven’t needed that piece of knowledge, and now you aren’t likely to need it. If you must know, send me an email.) We measure weight–in ounces, pounds,or grams; and we measure capacity: either liquid, like quarts and gallons or liters, or dry with measuring cups. For each of these I have just given a few common units of measure. There are many others, but you get the concept.
(2) Polygons. Here, I am referring to shapes made with straight lines, The actual definition is more complicated but not necessary for our purposes. Triangles, quadrilaterals, and hexagons are primary examples; and with each figure there are properties to learn and additional things to measure: the length of individual sides, perimeter, medians, etc. Again, these are straight line measures but we use formulas and relationships to determine the measures. With polygons, we can also measure the space INSIDE the figure. This is called “area,” is measured actually with little squares inside, although the actual measure is, again, found with formulas and labeled as square inches, or ft^2 (feet squared).
The study of polygons gets expanded into three dimensions, so that we have length, width, and thickness. Boxes and books are good examples of 2-dimensional rectangles given the third dimension. While the “inside” of a 2-dimensional figure is called “area,” the inside of a 3-dimensional figure is called volume and there are, of course, formulas for that as well.
(3) Circles. Because circles are not made with straight lines, our ability to measure the distance around the space inside is limited and requires the introduction of a new number: pi. The “perimeter” is actually called circumference, and both circumference and area have formulas involving the number pi. With circles, we can talk about a radius, a diameter, a tangent line, and various angles. Read the rest of this entry »
Geometry: Understanding the Rules of Logic
Posted by admin in Mathematics on June 18, 2011
Geometry is the field of mathematics that deals with spatial relationships. It also relates to the deduction of properties, relationships of points, lines and angles. Like algebra, the earliest recorded uses of geometry can be traced back to ancient Babylonia, around 3000 BC. Early geometry was essentially a composite of empirical principals and discoveries concerning lengths, angles, areas, and volumes. Babylonians used it for surveying, construction, astronomy and various projects.
Though the Babylonians invented geometry, the Greeks perfected it. In mid 300 BC Euclidean geometry was developed through Euclid of Alexandria. Euclid, considered ‘the father of geometry,’ is a suspected student of the philosopher Plato. Euclid’s influence began with the release of his book, The Elements of Geometry. In The Elements of Geometry Euclid described geometry in a more fundamental form, later called Euclidean geometry.
Euclidean geometry defined fundamental geometric principles called axioms or postulates, and general quantitative principles, called common notions. Euclidean geometry sought to satisfy all of Euclid’s axioms. This form of math followed five main rules: any two points can be joined by a straight line, any fixed straight line can be extended in a straight line, a circle can be drawn with any center and any radius, all right angles are equal and the parallel postulate.
Geometry was also very important in ancient India and China. In 179 AD, Liu Hui, a 3rd century mathematician, wrote The Nine Chapters on the Mathematical Art. Hui’s book illustrated many geometric problems and solutions including surface areas for squares and circles, volumes of solids and three-dimensional shapes, and the Pythagorean theorem.
Modern geometry began in the 17th century with the development of analytic geometry and projective geometry. Analytical geometry was created by French philosopher, René Descartes and Pierre de Fermat. Analytical geometry refers to geometry with coordinates and equations. This form of math was not only necessary for the progression of geometry, but also formed the foundations for calculus and physics.
In the mid 1600s, French mathematician, Girard Desargues introduced projective geometry. Projective geometry is the study of geometry with the absence of measurement, focusing on the ways in which points align. This type of geometry is typically used to study geometric properties that are consistent under projective transformations. Read the rest of this entry »