To a topologist, a coffee cup and a doughnut are the same shape
Spin a circle around an axis lying in its own plane and you trace out a torus, the doughnut shape of swim rings and inner tubes. Topologists care only about holes, so they count the surface of a coffee cup, with its single handle, as a torus too: both have genus one.
Where the axis sits decides the variety. If it misses the circle entirely, you get the familiar ring torus. If it just touches the circle, the result is a horn torus, effectively a doughnut with no hole. If it cuts through the circle twice, the surface crosses itself, forming a spindle torus whose inner shell resembles a lemon and outer shell an apple. Shrink the tube's radius to nothing and a circle remains; shrink the distance to the axis to nothing and you are left with a sphere.
Mathematicians describe a ring torus with two measures: the major radius, from the centre of the whole shape to the middle of the tube, and the minor radius of the tube itself. Their ratio is the aspect ratio; a typical doughnut comes in at about 3 to 2. A pleasing result follows from Pappus's centroid theorem: slice the tube and straighten it, and its area and volume match those of a cylinder, because what the inner side loses the outer side exactly gains.
You can make one without any spinning. Take a flexible rubber rectangle, glue its top edge to its bottom and its left edge to its right, with no half-twists, and you have a torus; add twists and you head toward a Klein bottle. Strictly, the torus is just the hollow surface. Fill it in and you get a solid torus, like an O-ring, a bagel or a non-inflatable lifebuoy.
The word is Latin for something round or swollen. Its geometry also supplied two directions, toroidal around the ring and poloidal around the tube, first used for Earth's magnetic field and now standard in magnetic confinement fusion research.
Source: Torus
