Some Cool Looking Surfaces

Some cool looking surfaces. Manim: https://github.com/3b1b/manim Blender: https://www.blender.org/ Here are the parameterizations: Breather Surface https://youtu.be/KWbhhAmlofg x(u,v) = -u + ( 2*(1-4/25)*cosh(2/5 *u)*sinh(2/5 *u) )/( 2/5*( (1-4/25)*cosh(2/5 *u)**2 + 4/25 *( sin(sqrt(1-4/25)*v) ) ) ) y(u,v) = (2*sqrt(1-4/25)*cosh(2/5 *u)*(-sqrt(1-4/25)*cos(v)*cos(sqrt(1-4/25)*v)-sin(v)*sin(sqrt(1-4/25)*v)))/( 2/5*( (1-4/25)*cosh(2/5 *u)**2 + 4/25 *( sin(sqrt(1-4/25)*v) ) ) ) z(u,v) = (2*sqrt(1-4/25)*cosh(2/5 *u)*(-sqrt(1-4/25)*sin(v)*cos(sqrt(1-4/25)*v)+cos(v)*sin(sqrt(1-4/25)*v)))/( 2/5*( (1-4/25)*cosh(2/5 *u)**2 + 4/25 *( sin(sqrt(1-4/25)*v) ) ) ) -14 ≤ u ≤ 14 -37.4 ≤ v ≤ 37.4 Twisted Torus https://youtu.be/57cqix6Cgg0 x(u,v) = cos(u)*(6-(5./4. + sin(3*v))*sin(v-3*u)) y(u,v) = (6-(5./4. + sin(3*v))*sin(v-3*u))*sin(u) z(u,v) = -cos(v-3*u)*(5./4.+sin(3*v)) 0≤u≤2pi 0≤v≤2pi Sea Shell https://youtu.be/_CFqkdVlXHE x(u,v) = cos(v)*(1+cos(u))*sin(v/8) y(u,v) = sin(u)*sin(v/8)+cos(v/8)*1.5 z(u,v) = sin(v)*(1+cos(u))*sin(v/8) 0≤u≤2pi 0≤v≤4pi Dini's Surface https://youtube.com/shorts/aqrsRAcFgbA?feature=share x(u,v) = a*cos(u)*sin(v) y(u,v) = a*sin(u)*sin(v) z(u,v) = a*(cos(v)+ln(tan(v/2)))+b*u a = 1, b = 2 0≤u≤4pi 0.01≤v≤1 Snake https://youtu.be/L79WoEljX8I x(u,v) = 1.2*(1 -v/(2*pi))*cos(3*v)*(1 + cos(u)) + 3*cos(3*v) y(u,v) = 9*v/(2*pi) + 1.2*(1 - v/(2*pi))*sin(u) z(u,v) = 1.2*(1 -v/(2*pi))*sin(3*v)*(1 + cos(u)) + 3*sin(3*v) 0≤u≤2pi 0≤v≤2pi Klein Bottle https://youtu.be/u0gG36Q-qyY x(u,v) = (3*(1+sin(v)) + 2*(1-cos(v)/2)*cos(u))*cos(v) y(u,v) = (4+2*(1-cos(v)/2)*cos(u))*sin(v) z(u,v) = -2*(1-cos(v)/2)*sin(u) 0≤u≤2pi 0≤v≤2pi Boy's Surface https://youtu.be/F66LuPtnXno x(u,v) = 2/3* (cos(u)* cos(2*v) + sqrt(2)* sin(u)* cos(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v)) y(u,v) = sqrt(2)* cos(u)* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v)) z(u,v) = 2/3* (cos(u)* sin(2*v) - sqrt(2)* sin(u)* sin(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v)) 0≤u≤pi 0≤v≤pi