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This is the most boring thread ever created.
Now that I have posted in it, it is no longer boring.
This is the most boring thread ever created.
Now that I have posted in it, it is no longer boring.
By that argument, YOUR thread was never boring.
This is the most boring thread ever created.
Now that I have posted in it, it is no longer boring.
By that argument, YOUR thread was never boring.
I can be stale and tedious at times.
Whenever it suits my purpose.![]()
Wait a minute now, are you trying to make this thread UN-boring.. This is supposed to be a BORING thread
Archimedes' Approximation of Pi
One of the major contributions Archimedes made to mathematics was his method for approximating the value of pi. It had long been recognized that the ratio of the circumference of a circle to its diameter was constant, and a number of approximations had been given up to that point in time by the Babylonians, Egyptians, and even the Chinese. There are some authors who claim that a biblical passage1 also implies an approximate value of 3 (and in fact there is an interesting story2 associated with that).
At any rate, the method used by Archimedes differs from earlier approximations in a fundamental way. Earlier schemes for approximating pi simply gave an approximate value, usually based on comparing the area or perimeter of a certain polygon with that of a circle. Archimedes' method is new in that it is an iterative process, whereby one can get as accurate an approximation as desired by repeating the process, using the previous estimate of pi to obtain a new one. This is a new feature of Greek mathematics, although it has an ancient tradition among the Chinese in their methods for approximating square roots.
Archimedes' method, as he did it originally, skips over a lot of computational steps, and is not fully explained, so authors of history of math books have often presented slight variations on his method to make it easier to follow. Here we will try to stick to the original as much as possible, following essentially Heath's translation3.
The Approximation of Pi
The method of Archimedes involves approximating pi by the perimeters of polygons inscribed and circumscribed about a given circle. Rather than trying to measure the polygons one at a time, Archimedes uses a theorem of Euclid to develop a numerical procedure for calculating the perimeter of a circumscribing polygon of 2n sides, once the perimeter of the polygon of n sides is known. Then, beginning with a circumscribing hexagon, he uses his formula to calculate the perimeters of circumscribing polygons of 12, 24, 48, and finally 96 sides. He then repeats the process using inscribing polygons (after developing the corresponding formula). The truly unique aspect of Archimedes' procedure is that he has eliminated the geometry and reduced it to a completely arithmetical procedure, something that probably would have horrified Plato but was actually common practice in Eastern cultures, particularly among the Chinese scholars.
The Key Theorem
The key result used by Archimedes is Proposition 3 of Book VI of Euclid's Elements. The full statement of the theorem is as follows:
If an angle of a triangle be bisected and the straight line cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and, if the segments of the base have the same ratio as the remaining sides of the triangle, the straight line joined from the vertex to the point of section will bisect the angle of the triangle(http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html)
Please don't post images of pastries or desserts of any kind as they are too exciting. Scones are an exception.
Schrödinger Equation -- from Eric Weisstein's World of Physics
The Schrödinger equation is the fundamental equation of physics for describing quantum mechanical behavior. It is also often called the Schrödinger wave equation, and is a partial differential equation that describes how the wavefunction of a physical system evolves over time. Viewing quantum mechanical systems as solutions to the Schrödinger equation is sometimes known as the Schrödinger picture, as distinguished from the matrix mechanical viewpoint, sometimes known as the Heisenberg picture.
The time-dependent one-dimensional Schrödinger equation is given by
View attachment 7264
Click on the link above for more fun!
Schrödinger Equation -- from Eric Weisstein's World of Physics
The Schrödinger equation is the fundamental equation of physics for describing quantum mechanical behavior. It is also often called the Schrödinger wave equation, and is a partial differential equation that describes how the wavefunction of a physical system evolves over time. Viewing quantum mechanical systems as solutions to the Schrödinger equation is sometimes known as the Schrödinger picture, as distinguished from the matrix mechanical viewpoint, sometimes known as the Heisenberg picture.
The time-dependent one-dimensional Schrödinger equation is given by
View attachment 7264
Click on the link above for more fun!