This is a drawing of Pi as it expands forever closer to a circle. This is a snapshot of an n-sided polygon with n=360 (or 360 right triangles that when you draw secant lines around the edge gives you an area equal to an n sided polygon with n=360). As n gets larger and approaches infinity the value approaches Pi forever because you are getting closer and closer to a circle for ever and as you fill in the edge of the circle (or it gets smoother as n gets larger). The area gets a little larger and the circumference get larger also as you add sides (as n grows larger), but the diameter stays the same.When you use secant lines (a line through two points on the edge of the \'circle\' every one degree in this drawing) you are approaching Pi from the inside of the circle. This is the inner boundary of Pi. If you use tangent lines around the drawing (a line through only one point around the \'circle\') then as you add sides the value you get is larger than Pi but begins to get smaller and it approaches a Pi from the outside of the perimeter. This is the outer boundary of Pi. Then as the secant lines and tangent lines from the inner and outer boundary of Pi approach each other they trap Pi, or a shape forever getting smoother and smoother (a circle), forever between them. But the coolest part is that perfect circles don\'t exist.
The easy way to picture it though is to look at the three drawings I have of Pi next to each other on your screen at the same time. The one with 180 sides has big empty spaces on the edge of the circle, then when you look at this drawing with 360 sides you see that some af that empty space has been filled in so it is closer to a circle and then look at the drawing of Pi with 720 sides and you see that it fills in a little more of the space (area) as it is even closer to a circle. So as you keep adding and adding sides and you get closer and closer to a circle forever but you never get all the way there. Just closer and closer forever. That is the beauty of Pi. The exact equation for the area of this shape is 360sin(180/360)r^2in degree mode on a scientifitc calculator (if you do Pir^2 you get a value that is slightly larger becase Pi is being used as a limit in our calculators) and the circumference=2(sin(180/360))r in degree mode or 2(sin(Pi/360))r in radians.
The area of Pi with 180 sides is 3.141433159.... When you have 360 sides like this drawing the area is 3.141552779... just a little larger....The area of the drawing of Pi with 720 sides is 3.141582685....So a reason Pi can never repeat itself is that each time you add sides to the \'circle\' you get a new and unique area and circumference. The can never find the \'end\' to Pi mathematically because you can add sides to a circle forever and get a larger and unique value as you forever approach an infinite number of sides. They way Pi is calculated now is that they ssay let the number of sides to a n-sided polygon forever approach infinity and it is that idameter divided by its circumference that we will call Pi. The problem with this is that it is describing a shape that is forever approaching a circle as you add more and more sides and it gets smoother and smoother forever towards a circle. But when you try to take a measurement from a shape in motion you cannot do it. The reason Pi can never end is becasue you can mathematically makes the sides to a 'circle smaller and smaller to infinity and the smaller the sides get the further the circumference gets. It is the same as the "Yardstick" or "Coastline" problem in fractal geometry. If you want to measure the circumference of a country and you use a stick a mile long you can't get into all the nook and crannies of the outline of the country. But if you use a yardstick you get a better measurement and you can keep using a smaller yardstick to infinity and you will continually get a better measurement and the circumference will get longer. The problem is that this says that there is an infinitely large perimeter. What ever "circle" it is that you are actually measuring in real life has a million sides, then you enter 1,000,000 for x in the equation f(x)=xsinPi/x). Your calculater says that x goes to infinity so no matterhow many side the polygon has Pi (as it is currently being calculated) will always give you a value that is slightly to large.
As a side note for those into physics. The only way you can avoid this problem with infinity is to apply the Planck length. The Planck length is the smallest observable distance. Once you have a circle where the sides are one planck length the that may be the closest you can get to observing a perfect circle in our universe.
December 1st, 2011
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