Free, open-source online mathematics for students, teachers and workers

We all know what a torus is: a (hollow) 'donut'

Topologically, the torus is the product of two circumferences, $T^2=S^1\times S^1$. To understand it, one must imagine the decomposition in meridians and parallels

u, v = var('u,v'); t = parametric_plot3d((cos(u)*(2+cos(v)), sin(u)*(2+cos(v)), sin(v)), (u,0,2*pi), (v,0,2*pi), rgbcolor=(1,1,0), opacity=0.7, viewer='threejs', axes_labels=False); t+=sum([parametric_plot3d((cos(u)*(2+cos(k*pi/6)), sin(u)*(2+cos(k*pi/6)), sin(k*pi/6)), (u,0,2*pi), thickness=5, color='darkred') for k in range(12)]); t+=sum([parametric_plot3d((cos(k*pi/6)*(2+cos(v)), sin(k*pi/6)*(2+cos(v)), sin(v)), (v,0,2*pi), thickness=5, color='cadetblue') for k in range(12)]); show(t)