If you can make a weird non-euclidean mathematical space where parallels lines actually *do* touch eventually, can you make a weird mathematical space where the ratio between a circle’s circumference and its diameter is something other than 3.14159…..etc.? Is there a geometry I don’t know where “pi” is just 3, for instance?
Following from that, the universe as we know it is based on certain fundamental properties, and I’ve heard speculation that there may be other universes (within a multiverse) that have different fundamental properties. In theory, could there some other universe where pi is 3?
In: Mathematics
Yes, in alternate geometries you can have circles where the ratio of circumference to radius is something other than 2pi. For example, in hyperbolic geometry (a plane is “saddle shaped” rather than flat) it’s always more than 2pi (and the value depends on the size of the circle compared to the size of the curvature too).
This isn’t exactly the same as a different value of pi…it depends on how you define pi. There are lots of definitions that don’t depend on circles and wouldn’t change with geometry.
I’m not sure exactly what geometry gives you a ratio of three.
You can invent any geometry you want. If you make a plane be a sphere and a radius be an arch on its surface then that would increase the radius but not the diameter thus changing the ratio.
As an example of the weird stuff that can happen when pi != 3.1414…
A fun related fact is there is no simple formula for calculating the circumference of an ellipse (oval). You don’t just take pi and then multiple that by how much wider it is than tall. For each unique ellipse you have to calculate a special version of pi that isn’t 3.14159. So just in regular geometry there is already infinite versions of pi.
yes and no. There are geometries where the area of a “circle” is not πr^2, but the actual mathmatical constant π has a lot of other uses and meanings that do not depend on geometry. these dont change, and the value π IS doesnt change with geometry.
Surprisingly, pi is also an important constant in a field unrelated to geometry. For example, in Normal Distribution, the PDF is calculated via [this formula](https://wikimedia.org/api/rest_v1/media/math/render/svg/a45cef4ca1e2fcd4d367ecff5806d8a2878d3821). This instance of pi will be the same regardless of the geometry of our world.
Mathematically, yes. Whether that actually corresponds to anything existing in reality is a very different question.
Latest Answers