“the area of a circle is pi r squared uh and so if the circle had bumps in it or if it was topologically different in some way pi wouldn’t be pi pi would sort of be more variable in such circumstances something that you thought was a constant turns out to be variable”
Seems to me that the variable then in not Pi but rather R.
Not all radii are created equal.
Not true. Pi has the value it does only in Euclidean geometry, that is geometry in a flat space. Pi can have different, non constant values in a curved space. As an example think of the surface of the earth. A circle is defined to be the set of all points equidistant from a center point. Pi is defined to be the ratio of the circumference of a circle to its diameter (double the constant distance that defined the circle).
With those definitions in mind let’s define the North Pole as our center and all the points on the equator (which strictly speaking may not be equidistant on the earth since it is only approximately spherical, but I’m talking about an ideal true sphere here) are our circle. It should be clear that the circumference of this circle is equal to the circumference of the earth and that the radius is one fourth of the earth’s circumference. The diameter is thus half the circumference, so pi equals 2 for this circle.
Next consider the circle formed by the 60 degree north line of latitude centered again at the North Pole. Trigonometry can be used to show that the circumference of this circle is half that of the equator, and therefore half that of the earth. The distance from the North Pole to this circle is one twelfth of the earth’s circumference since we must traverse 30 degrees to get from 90N to 60N latitude and 360 degrees is the full circumference. Therefore the diameter of this circle is one sixth of the earth’s circumference. Pi is then the ratio of one half the earth’s circumference to one sixth of it, which means pi equals 3 for this circle.
In general a different value of pi will be found for any given circle on a spherical surface. The smaller the radius the closer the value will be to the value of pi we learn about in school. The larger the radius the bigger the deviation will be. It’s even possible to define a “circle” for which pi is zero. Just consider the circle consisting of all points at a distance of one half the earth’s circumference centered on the North Pole. Obviously that “circle has only one point - the South Pole. The “circumference” can be said to be zero based on a limit process (consider what happens as you take circles to be the 89 degree south parallel, then the 89.9 degree S, then 89.99 degree S, etc).
“Seems to me that the variable then in not Pi but rather R.”
Well, if “r” is variable, then it’s not a circle because it wouldn’t meet the definition. There might be some circumstances with exotic geometry where you could create a shape that satisfies the definition of a circle, but where the ratio of radius to circumference doesn’t match Pi, or isn’t constant. But I’m no expert on exotic geometry.