Summary:

• Pi's Evil Twin: There is a mathematical constant, often referred to as the lemniscate constant, which acts similarly to pi in certain equations but is not equal to it. It is denoted by ϖ (varpi) and has a value of approximately 2.622057554291118.
• Lemniscate Functions: Corresponding to the sine and cosine functions, there are lemniscatic sine (sl) and lemniscatic cosine (cl) functions, which are associated with the lemniscate curve rather than the circle.
• Gauss's Constant: The arithmetic-geometric mean of 1 and √2 is equal to π/ϖ, known as Gauss's constant.
• Higher Dimensions: There are generalizations of these constants and functions (ϖn) related to hyperelliptic curves with n branch points.
• Book Reference: "Squigonometry: The Study of Imperfect Circles" by Poodiack and Wood discusses these topics in more detail, connecting them to the modular group and concrete mathematics.