3 ms·
π is derived from the circle, which is defined by distance from a single point. ϖ is derived from the lemniscate of Bernoulli, which is defined by distances fr
by divbzero 2y ago
π is derived from the circle, which is defined by distance from a single point.
ϖ is derived from the lemniscate of Bernoulli, which is defined by distances from two points.
Is there an analogous constant that is derived from a shape defined by distances from three points?
- clort 2y agoit sounds like you are suggesting it might be turtles all the way down?
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- VHRanger 2y agoI mean the concept of distance from 3 points introduces a mess of metrics or even measure theory. 2 points always have a shortest path between each other, so the constant is about this fact. For 3 points you have the whole universe of possible triangle shapes to contend with.
- jovial_cavalier 2y agoIt's easy to generalize this to more points. https://www.desmos.com/calculator/fo7tqlfjgo https://www.desmos.com/calculator/fo7tqlfjgo
- vitus 2y agoShortest path between two points still depends on your metric. For instance, if you're constrained to travel along the surface of Earth, your shortest path is going to travel along a great circle, rather than pass through the interior of the sphere. That said, you could, for instance, pick the three vertices of an equilateral triangle (using the Euclidean distance as your metric of choice, as we do in order to derive the lemniscate and the circle), and again deal with the product of the distances from each vertex. You again start with small circles around each vertex, which eventually expand to a single looping curve, and then into ovals encircling the entire triangle. https://en.wikipedia.org/wiki/Cassini_oval#Generalizations https://en.wikipedia.org/wiki/Cassini_oval#Generalizations https://en.wikipedia.org/wiki/Polynomial_lemniscate#Erd%C5%91s_lemniscate https://en.wikipedia.org/wiki/Polynomial_lemniscate#Erd%C5%9...
- dahart 2y agoYes, definitely. Pi is just the perimeter of the circle, and varpi is the perimeter of the lemniscate. If you use three points, you get three tear-drops, and you can compute the perimeter of that. Let’s call it a trilemniscate. ;) Here’s a 3d plot of it. If you rotate to view it from +Z downward, then you’ll see the trilemniscate, which is where the volume intersects with the XY plane. Note I subtracted 1 from the product in order to visualize the plane intersection. (And you can turn off the 3 points version and turn on the 2 points version to compare.) https://www.desmos.com/3d/dl9v2vqbqb https://www.desmos.com/3d/dl9v2vqbqb One interesting note about 2 points vs 3 points. The area inside the lemniscate and trilemniscate is the same! (True for more points, as long as they’re evenly space on a circle). The perimeter, of course, goes to infinity as you add more points.
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