Is it true that π=4? What is the thorough rebuttal?

You are right, if the circumference of a circle is defined as following little vertical and horizontal lines, then you are going to get π=4. Another scenario where π can be equal to 4 is when it is defined as the ratio of the circumference of a square to its diameter (the diameter is the shortest straight line distance from one side, through the middle, to the opposite side. I think that may not be the perfect definition but I hope that much is clear.)

I don't find it to be shocking at all, I'm perfectly secure in my understanding that the arc length of an arc with radius 1 and making 1 complete resolution is less than 4.

A similar issue comes up when you do surface area. Do you know about those telescoping cups that can flatten down and then a series of rings rises up to form a cup, and they lock together to keep in liquids, such as apple juice? If you do your integral like that you can get a wrong answer for the surface area too.

I don't know how many people answered after you said nobody was getting your point.

It seems like you want a definition of arc length that works.

The points on the circle around the origin all satisfy x^2 + y^2 exactly, and that is not true about the pixels on your screen. Although the differences are small between the location of the pixels and the location of the points on the circle, what is important in arc length is the small changes in x and y as you move from point to point. A small line segment that is a secant line to the circle is a good approximation of the arc of the circle. On the other hand a pair of line segments, one vertical and one horizontal (when a secant line to the circle does not make a very small angle with the horizontal or vertical) does not make a good approximation of the arc to the circle. It causes a person to take a sharp corner when going between two points A and B on the circle, when the person would rather walk straight from A to B for a shorter distance. Therefore taking short corners overestimates the distance.

Hopefully I have answered your point although I know I did not put it into very exact mathematical language yet.

Consider a circle and a set of points A1 ... An spaced evenly, consecutively around the circle.

Then consider two paths around the circle, one that visits the points by making straight lines A1 A2, A2 A3, .. A(n-1) A(n), A(n) A(1). (It is not good mathematical language though because I can not make subscripts, sorry!)

Then another one that stops also at points B1 vertical from A1 and horizontal from A2, B2 vertical from B2 and horizontal from A3, etc. and Bn vertical from An and horizontal from A1

In either case the set of points converges to the circle.

So the reason the secant line path is used to define arc length is because the path with vertical and horizontal lines would be silly to use.

Here is why it would be silly. You could take an arc length say from pi/8 to 3pi/8 and measure it with vertical and horizontal steps. You would get the Manhattan distance for (cos pi/8, sin pi/8) to (cos 3 pi/8, sin 3 pi/8).

Then take the same arc length and rotate it so now goes from 3 pi/8 to 5 pi/8. Now it is going to have a different arc length when you measure it with little vertical and horizontal steps (basically it fits in a different shape of rectangle now that you have rotated it). That would be silly. Since mathematicians are all never silly, this is not the definition of arc length that anybody chooses to use. There was once a silly mathematician who tried to define arc length the way you are suggesting, but that is why it took mankind very many millenia before the Wheel was invented.

The linear distance between connected head to tail pixels forming a closed geometry of a two dimensional 360 degree figure will approach 0 as the pixel count approached infinity and the ratio of circumference to diameter of the closed figure will approach Pi.

If you follow the square edges of the the pixels, you are not going around a circle. You are going "stair-case" around a shape that is larger than the circle. Try going "stair-case" around the edges of the pixels on the *inside* of the circle, and see what you get.

Quote: "Is it true that π=4?"

Answer: No.

Your erroneous measurement is being brought about by 2 factors:

(a) insufficient resolution,

(b) inherent measurement errors.

Quote: "Even when you increase the resolution (theoretically even infinitely much) this holds good".

You need to provide evidence of that. A bland statement such as that without any validation is worthless and proves nothing.

Moreover, you have not explained the technique you used to determine your measurements. Again, with this information lacking, your claim has no foundation.

it is just a limitation of your device. pi is an irrational number, which means that it cannot be expressed as a ratio of two numbers (aka a fraction). pi is usually approximated as 3.14, but its decimal places extend to infinity. you can google pi to a ridiculous amount of decimal places.

Pi is equal to 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609

It has to be 3.1416.................. or no wonder my pants don't fit!