Geometria archaic : Geometry: Earth measure Journal Entry 1/26/2000 written by Diana.  Constructions The Greeks, when they got hold of geometry, made a game of it. The rules were simple enough: Only a Straightedge and a compass! What can you do with these?These basic seven constructions include: copying a segment, bisecting a segment, copying an angle, bisecting an angle, constructing a perpendicular to a point on a line, constructing a perpendicular to a point not on the line, and constructing a parallel line through a point not on the line. The Introduction to ruler and compass constructions Construct a line segment equal in length to a given line segment Construct an angle equal in measure to a given angle Bisect a line segment . The constructions the students explore include constructing with compass and straightedge the following: rhombus, right triangle, length of root 2, length of root 3, length of root 5, various angle measures (i.e. 75 degrees), golden ratio, golden rectangle, golden triangle, pentagon  Geometry Student
The Measurement of the Circle The exploration was to try to find some relationship between a) the diameter and the circumference of a circle and b) the diameter to the area of a circle without using complicated instruments or looking it up, forgetting (if possible) all previous knowledge. Most of the students thought the best way was to try to find the area of triangles and squares within the circle in order that the areas of the shapes with straight lines could be measured and added together in order to provide a close approximation. One student thought the diameter was equal to three times the radius (see my explanation of my exploration, following) My Exploration I thought about the above problem, and this is what I came up with. I decided to find a round object like a pie plate, and fasten a string around it. Then I could cut the string, and I would have a string the length of the circumference of the circle. Then I could play with the length of string by seeing how many times it would go back and forth across the circle.So I bought some paper plates and some string and I used tape to hold the string in place and I measured it and I cut it and it went across the plate three times. Student # 10 Carmen Is there really such a thing as a perfect circle? In other words, how do I know that the pie plate is round for sure? Well, one way to answer this question is to take the philosophical position that we can never know what a thing is except by comparison, that is to say we can never say that it is round, we can only say that it has no corners. The ancient Greeks believed there were such things as perfect lines and perfect circles. If, by some circumstance we come to a conclusion regarding a formula, such as: circumference = 31/2 times the diameter diameter = C X 3.5 can we assume the same formula will work for a larger circle? If we find a single truth, is it always true? One way to answer this questions is to do many experiments. Why do certain shapes appeal to us? A t a certain point mathematics becomes the art of the solving of problems. For example, it is theorized that the science of geometry may have started because after the annual flooding of the Nile river people needed a way to re-demarcate boundaries of the fields. Supposedly they used ropes in order to do this. A society must achieve a fair amount of economic success in order for there to be a sufficient amount of surplus food to allow people to sit around and think all day; the birth of the scholar. First mathematics was observational. Then, it became experimental. This is the next step in the historical evolution of mathematics. Questions arose out of practical necessity "How do I know my walls are straight?" Archimedes constructed a polygon with 96 sides in order to approximate a shape close to that of a circle which he could accurately measure the area of. In the "carpenters approximation" Pi = 22/7,which is all the more impressive because the Greeks did not have fractions, but only whole numbers and ratios of whole numbers. Patterns become important in terms of cultural identity. For example, American Indian, Arab, Byzantine, Celtic design and also tattoo are forms of cultural identity. The next area of math we will learn about is that of the classical Greeks, and move away from Babylonian math. This is the era of the first drafting i.e. drawings of how to construct buildings. The Muse of discovery needs to inspire us. In the spirit of designing a city we will be doing drawings and sketches. Compass and straight edge construction T he next set of experiments will require the use of a 12 inch straight edge ruler and a compass (the compass allows us to draw circles) Interesting Websites:
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