Let \(\varpi=\dfrac{\Gamma^2\left(\frac14\right)}{2\sqrt{2\pi}}=2.62205755\ldots\) be the lemniscate constant. Then,
\[\Large\displaystyle\sum_{n=1}^\infty\dfrac{1}{\sinh^4(\pi n)}=\dfrac{\varpi^4}{30\pi^4}+\dfrac{1}{3\pi}-\dfrac{11}{90}\]

#Series #Sum #InfiniteSum #LemniscateConstant #GammaFunction #Lemniscate #LemniscateOfBernoulli #Bernoulli #Math #Maths #InfiniteSeries #HyperbolicSines

"Lemniscating"

Another math-meditation, this time over lemniscate spirals, bells and an ambient sound...

Inspired by @fractalkitty 🙂

In my series of Tooll3 point generators, today I created first a lemniscate, but then extended it to be able to create lemnispirals by varying the radius over time.

https://youtu.be/l7UrvBH2b7w

#Tooll3 #realtime #particles #livingpainting #creativecoding #hlsl #gpu #shader #generative #procedural #mastoart #audioreactive #sounddesign #math #lemniscate #opensource

An interesting infinite product!

\[\displaystyle\prod_{n=1}^\infty\coth^{(-1)^n}\left(\dfrac{\pi n}{2}\right)=\dfrac{\sqrt\pi}{\sqrt[4]2\sqrt{\varpi}}\]

where \(\varpi=2.62205755\ldots\) is the lemniscate constant (the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of \(\pi\) for the circle). Both \(\varpi\) and \(\pi\) are proven to be transcendental.

#lemniscate #lemniscateconstant #product #infiniteproduct #HyperbolicFunction #HyperbolicCotangent #Function #pi #maths #math

Mathober Day 31 - Unity (Last day!!! - posted early)

from gas to solid

and back again to the stars

you are – lemniscate

#mathober2022 #mathart #haiku
#math #lemniscate #unity #mindfulness #poetry

Infinity dragon...XD
https://www.thingiverse.com/thing:5152665

#OpenSCAD
#dotSCAD
#lemniscate
#infinity
#dragon