For centuries, mathematicians around the world have struggled in the chase for an increasingly accurate approximation of pi. The first recorded avenue of approximating the value of pi by theoretical means was the polygon approximation method, employed by Archimedes around 250 B.C., which was accurate only to about two decimal places when he performed it (Groleau). This was partly due to the fact that Greek mathematicians at the time had no concept of 0, thus only used fractions, not decimals, and the fact that they had no access to algebra (McKeeman). Now, over 2000 years later, pi has been calculated and verified to more than 12.1 trillion digits thanks, in part, to the Bailey-Borwein-Plouffe (BBP) formula, published in 1996, which allows us to quickly and easily calculate a specific digit of pi (Yee and Kondo; Seife). Though the polygon method and the BBP formula each work in entirely …show more content…
The polygon approximation method uses the perimeters of polygons to approximate pi and “was the first theoretical, rather than measured, calculation of pi,” but while it worked great for Archimedes in his time, it has limited practicality today (Groleau). In order to estimate pi, Archimedes used a circle that had a polygon inscribed in and circumscribed about it, then found the perimeter of each polygon, and used those values as the upper and lower limits of pi (Groleau). He started with a hexagon as the inner polygon of a circle with a diameter of 1 whose perimeter is equal to 6r when r is the radius of the circle (McKeeman). This meant that since the circumference of the circle was 2πr, 6r < 2πr and the lower limit of pi was 3 (McKeeman). He then found the perimeter of the outer hexagon to calculate the upper limit and proceeded to double the number of sides and repeat the
The Mayans used tons and tons of geometry throughout their creations. Which is obviously figured out just by thinking of the one thing that most of everybody knows and relates to the Mayans, the calendar, and the Aztec’s then took the Mayan calendar and adapted it to come up with their own calendar. They probably used trial and error, I’m sure of. They created many drawings that all involved geometry in one form or another.
Euclidean Geometry has been around for over thousands of years, and is studied the most in high school as well as college courses. In it's simplest form, Euclidean geometry, is concerned with problems such as determining the areas and diameters of two-dimensional figures and the surface areas and volumes of solids. Euclidean Geometry is based off of the parallel postulate, Postulate V in Euclid's elements, which states that, "If a straight line meets two other straight lines so as to make the two interior angles on one side of it together less than two right angles, the other straight lines, if extended indefinitely, will meet on that side on which the angles are less than two right angles."
A little info: Archimedes was a Greek Mathematician who was born in 287 BC and died in 212 BC. He was born in Syracuse, Sicily; during this time, the city was an independent Greek city-state which held a 500-year history. At the Siege of Syracuse Romans at the siege were specifically ordered not to harm Archimedes but he later was pronounced dead by being stabbed by a Roman soldier. His father (Phidias) was an astronomer and is believed to be related to the King of Syracuse. This information was found in his work “The Sand Reckoner.” Archimedes was labeled as one of the top scientists in classical antiquity. In those times, when blackboards and paper were not yet around, Archimedes constructed ashes, dust or all any available surface to help sketch his geometric figures. It’s been told that he used to get so intrigued with all of the work he did that sometimes he forgot to eat, skipped a meal or two just to finish on the project. He was considered the greatest mathematician in antiquity and possibly the greatest of all time.
The game reveals many mathematical concepts even though it is rather simple. My aim for this mathematical exploration is to put the Tower of Hanoi to the test and find out (according to the legend) how long we have until the end of the world.
Another early attempt at the area of a circle is found in the Bible. In the old testament within the book of Kings Vii.23 and also in Chronicles iv.2 a statement is made that says; “And he made a molten sea, ten cubits from one brim to the other; it was round all about and his height was five cubits: and a line of thirty cubits did compass it round about.” From this verse, we come to the conclusion that Pi is 30/10 or simply 3. The book of Kings was edited around the time of 550 BCE. Mu...
Chudnovky (2014) wonders if it might allude to the ancient problem of expressing pi in algebraic form. Terrance Lynch (1982) speculates that the polyhedron poses the mathematical problem of squaring a circle. Remarkably, Hideko (2009) believes that the solution to the Delian Problem of ancient Greece is hidden in the form of the polyhedron. The Delian Problem is, as described by Dürer: Using only a compass and straightedge, how does one double the volume of a cube? To date, it remains
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
The Greeks were able to a lot of things with only a compass and a straight edge (although these were not their sole tools, the Greeks in fact had access to a wide variety of tools as they were a fairly modern society). For example, they found means to construct parallel lines, to bisect angles, to construct various polygons, and to construct squares of equal or twice the area of a given polygon. However, three constructions that they failed to achieve with only those two tools were trisecting the angle, doubling the cube, and squaring the circle.
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
Historically speaking, ancient inventors of Greek origin, mathematicians such as Archimedes of Syracuse, and Antiphon the Sophist, were the first to discover the basic elements that translated into what we now understand and have formed into the mathematical branch called calculus. Archimedes used infinite sequences of triangular areas to calculate the area of a parabolic segment, as an example of summation of an infinite series. He also used the Method of Exhaustion, invented by Antiphon, to approximate the area of a circle, as an example of early integration.
There are many people that contributed to the discovery of irrational numbers. Some of these people include Hippasus of Metapontum, Leonard Euler, Archimedes, and Phidias. Hippasus found the √2. Leonard Euler found the number e. Archimedes found Π. Phidias found the golden ratio. Hippasus found the first irrational number of √2. In the 5th century, he was trying to find the length of the sides of a pentagon. He successfully found the irrational number when he found the hypotenuse of an isosceles right triangle. He is thought to have found this magnificent finding at sea. However, his work is often discounted or not recognized because he was supposedly thrown overboard by fellow shipmates. His work contradicted the Pythagorean mathematics that was already in place. The fundamentals of the Pythagorean mathematics was that number and geometry were not able to be separated (Irrational Number, 2014).
By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the numbers, which is entirely his own independent discovery.
from his tables, which showed powers of 10 with a fixed number used as a base.
is convergent and ends up converging to φ, the golden ratio [2]. This curious quantity is just a ratio, so what makes it so special? Why is it so mystifying? While the other major constant in mathematics, pi, is a ratio between a circle's circumference and its diameter, phi (φ) considers a rectangle with height, h, and width, w, and forms the following ratio:
However, between 1850 and 1900 there were great advances in mathematics and physics that began to rekindle the interest (Osborne, 45). Many of these new advances involved complex calculations and formulas that were very time consuming for human calculation.