The discussion centers on the behavior of the mathematical constant pi (π) in curved space, revealing that π is not constant in non-Euclidean geometries. It highlights that while π remains the ratio of circumference to diameter in Euclidean geometry, this relationship alters in curved spaces, affecting sine waves and trigonometric functions. The validity of the sine function, sin(theta) = opposite/hypotenuse, is questioned in the context of curved spaces, necessitating the use of calculus and differential geometry for accurate analysis. The conversation underscores the need for a deeper understanding of geometric principles beyond traditional Euclidean definitions.
PREREQUISITESMathematicians, physics students, and anyone interested in the implications of geometry on trigonometric functions and the nature of mathematical constants in different spatial contexts.
You may be talking about something other than "\pi". \pi has a very specific value. There are a number of different ways to define \pi, one of them being the ratio between the circumference of a circle divided by the diameter of that circle in Euclidean Geometry. In a variety of forms of non-Euclidean geometries, that ratio might be something other than \pi or the ratio might not be a constant. That has nothing to do with the number \pi.newjerseyrunner said:I was going to ask a question about whether or not pi was constant or changed with curved space. I found the answer on here that it does indeed change. Then I started thinking about the ramifications of that. sine waves are dependent on pi, so they should change too. Does sin(theta) = opposite / hypotenuse still hold true for all spaces?