In mathematics, the lemniscate constant ϖ is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of π for the circle. Equivalently, the perimeter of the lemniscate ${\displaystyle (x^{2}+y^{2})^{2}=x^{2}-y^{2}}$ is ${\displaystyle 2\varpi }$. The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755. It also appears in evaluation of the gamma and beta function at certain rational values. The symbol ϖ is a cursive variant of π known as variant pi represented in Unicode by the character U+03D6 ϖ GREEK PI SYMBOL.

Sometimes the quantities 2ϖ or ϖ/2 are referred to as the lemniscate constant.

Gauss's constant, denoted by G, is equal to ϖ /π ≈ 0.8346268 and named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as ${\displaystyle 1/M{\bigl (}1,{\sqrt {2}}{\bigr )}}$. By 1799, Gauss had two proofs of the theorem that ${\displaystyle M{\bigl (}1,{\sqrt {2}}{\bigr )}=\pi /\varpi }$ where ${\displaystyle \varpi }$ is the lemniscate constant.

John Todd named two more lemniscate constants, the first lemniscate constant A = ϖ/2 ≈ 1.3110287771 and the second lemniscate constant B = π/(2ϖ) ≈ 0.5990701173.

The lemniscate constant ${\displaystyle \varpi }$ and Todd's first lemniscate constant ${\displaystyle A}$ were proven transcendental by Carl Ludwig Siegel in 1932 and later by Theodor Schneider in 1937 and Todd's second lemniscate constant ${\displaystyle B}$ and Gauss's constant ${\displaystyle G}$ were proven transcendental by Theodor Schneider in 1941. In 1975, Gregory Chudnovsky proved that the set ${\displaystyle \{\pi ,\varpi \}}$ is algebraically independent over ${\displaystyle \mathbb {Q} }$, which implies that ${\displaystyle A}$ and ${\displaystyle B}$ are algebraically independent as well. But the set ${\displaystyle {\bigl \{}\pi ,M{\bigl (}1,1/{\sqrt {2}}{\bigr )},M'{\bigl (}1,1/{\sqrt {2}}{\bigr )}{\bigr \}}}$ (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over ${\displaystyle \mathbb {Q} }$. In 1996, Yuri Nesterenko proved that the set ${\displaystyle \{\pi ,\varpi ,e^{\pi }\}}$ is algebraically independent over ${\displaystyle \mathbb {Q} }$.

As of 2025 over 2 trillion digits of this constant have been calculated using y-cruncher.

Usually, ${\displaystyle \varpi }$ is defined by the first equality below, but it has many equivalent forms:

$${\displaystyle {\begin{aligned}\varpi &=2\int _{0}^{1}{\frac {dt}{\sqrt {1-t^{4}}}}={\sqrt {2}}\int _{0}^{\infty }{\frac {dt}{\sqrt {1+t^{4}}}}=\int _{0}^{1}{\frac {dt}{\sqrt {t-t^{3}}}}=\int _{1}^{\infty }{\frac {dt}{\sqrt {t^{3}-t}}}\\[6mu]&=4\int _{0}^{\infty }{\Bigl (}{\sqrt[{4}]{1+t^{4}}}-t{\Bigr )}dt=2{\sqrt {2}}\int _{0}^{1}{\sqrt[{4}]{1-t^{4}}}\mathop {dt} =3\int _{0}^{1}{\sqrt {1-t^{4}}}dt\\[2mu]&=2K(i)={\tfrac {1}{2}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )}={\tfrac {1}{2{\sqrt {2}}}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{4}}{\bigr )}={\frac {\Gamma (1/4)^{2}}{2{\sqrt {2\pi }}}}={\frac {2-{\sqrt {2}}}{4}}{\frac {\zeta (3/4)^{2}}{\zeta (1/4)^{2}}}\\[5mu]&=2.62205\;75542\;92119\;81046\;48395\;89891\;11941\ldots ,\end{aligned}}}$$