Perfect Circles Don't Exist Pi 180

This is what Pi looks like at a value of 180sin(Pi/180).
This is a drawing of Pi as it expands forever closer to a circle. This is a snapshot of... more

by Jason Padgett

Buy the Original Drawing

Price

$5,000

Dimensions

18.000 x 18.000 inches

Title

Perfect Circles Don't Exist Pi 180

Artist

Jason Padgett

Medium

Drawing - Pencil On Paper, Color Added With Computer. Original Is Pencil On Paper With No Color.

Description

This is what Pi looks like at a value of 180sin(Pi/180).
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=180 (or 180 right triangles that when you draw secant lines around the edge gives you an area equal to an n sided polygon with n=180). 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 (and the edge of the 'circle' 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 two degrees 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 bigger empty spaces on the edge of the circle, then when you look at the drawing with 360 sides you see that some of 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 area of Pi with 180 sides is 3.141433159.... When you have 360 sides like the one drawing with 360 sides 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 say let the number of sides to a n-sided polygon forever approach infinity and it is that diameter 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 because 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 calculator says that x goes to infinity so no matter how many sides 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.

Uploaded

January 3rd, 2015