Torus

In geometry, a torus (pl.: tori or toruses) is a special kind of surface made by spinning a circle around an axis in three-dimensional space. Imagine taking a circle and turning it around a line that lies in the same flat plane as the circle. This creates a shape that looks like a ring or a doughnut, called a ring torus. There are different types of tori depending on how the axis touches the circle. When the axis does not touch the circle, it makes a ring shape. If the axis just touches the circle, it makes a special shape called a horn torus. And if the axis cuts through the circle twice, it makes a spindle torus.

Many everyday objects are shaped like tori. Think of swim rings, inner tubes, or ringette rings—they all have this ring-like form. A solid torus is a bit different; it’s made by spinning a flat disk, not just a circle, around an axis. This gives the shape some thickness, like in O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.

In topology, which studies properties of shapes that stay the same even when stretched or bent, a torus is any surface that can be stretched into the shape of a ring torus. For example, the surface of a coffee cup and the surface of a doughnut are both tori because they have the same basic shape structure. One way to imagine building a torus is by taking a flexible strip, like a piece of rubber, and joining its top to its bottom and its left to its right, without twisting it. This creates a surface that loops back on itself in two directions.

Etymology

The word "torus" comes from Latin. It means something round, like a swelling or a bump.

Geometry

A torus is a special shape in geometry made by spinning a circle around a line that runs through the center of the circle but does not touch it. This creates a surface that looks like a ring or a donut.

There are three main types of tori based on the distance between the center of the circle and the line it spins around. When this distance is greater than the radius of the circle, we get a ring torus, which looks like a classic ring. If the distance equals the radius, we get a horn torus, which has no hole in the middle. And if the distance is less than the radius, we get a spindle torus, which looks more like a rounded ball with a pinch in the middle.

Topology

Topologically, a torus is a closed surface made by multiplying two circles: S¹ × S¹.

The surface can also be seen as a piece of the Cartesian plane where opposite edges are glued together, like folding a square and connecting its sides.

Two-sheeted cover

The 2-torus is a special shape that covers the 2-sphere twice, with four special points called ramification points. Each shape that looks the same in a smooth way on the 2-torus can be shown as this double cover of the 2-sphere. The points on the torus that match these ramification points are known as Weierstrass points. The way the torus looks is decided by comparing the positions of these four points.

n-dimensional torus

The torus can also be thought of in higher dimensions, called the n-dimensional torus_. Just like the ordinary torus is made from two circles, the n-dimensional torus is made from n circles put together.

For example, the standard 1-torus is simply a circle. The torus we usually talk about is the 2-torus. The n-torus can also be described by taking an n-dimensional box and gluing the opposite sides together. This creates a shape that loops back on itself in every direction.

These shapes are important in many areas of mathematics.

Flat torus

A flat torus is a special kind of torus, which is a shape made by rotating a circle around a line. It has a flat surface, much like a piece of paper that has been rolled into a cylinder and then connected to form a loop. This flatness means that the surface has no curves or bumps — it stays perfectly smooth everywhere.

Flat tori can be described using math in higher dimensions, such as four-dimensional space, where they appear without needing to stretch or bend the surface. However, showing a flat torus in our normal three-dimensional world requires some clever tricks, like repeatedly folding or corrugating the shape at smaller and smaller sizes. Even with these folds, the surface still keeps its smooth properties, making it a fascinating mix of flatness and complexity.

Genus g surface

Main article: Genus g surface

In the study of shapes, there is a group of objects called "genus" g surfaces. A genus g surface is made by joining g tori together. (The torus itself is a genus one surface.) To join two shapes, you remove a small circle from the inside of each and then stick them together along the edges. This makes two circles become one. You can keep adding more shapes one at a time to make even bigger surfaces. A genus g surface looks like g doughnuts stuck side by side, or a sphere with g handles.

For example, a genus zero surface is a sphere, and a genus one surface is a regular torus. Surfaces with more handles are called n-holed tori. Other names you might hear are double torus and triple torus.

The classification theorem for surfaces says that every simple, connected shape is either like a sphere or made by joining together some tori, disks, and real projective planes.

Toroidal polyhedra

Further information: Toroidal polyhedron

Polyhedra that have the same shape as a torus are called toroidal polyhedra. They follow a special math rule: V − E + F = 0, where V stands for vertices, E for edges, and F for faces.

These special shapes can help us learn about coloring maps on a torus, and one example is the Szilassi polyhedron. Another interesting shape is the Császár polyhedron, which is unique because every line connecting two points is part of the shape itself.

Automorphisms

The shape and ways to move a special kind of round object called a torus are studied in a part of math called geometric topology. These movements can be linked to special patterns of numbers and shapes that stay the same even when turned or shifted.

When looking at how these shapes can stretch and twist but still stay the same, the rules follow patterns that match up with certain number patterns. This makes it possible to understand these shapes better by using simple number rules.

Coloring a torus

A torus can have graphs drawn on it, and these graphs need at most seven colors to make sure no neighboring areas share the same color. This is different from flat surfaces, which only need four colors.

de Bruijn torus

Main article: de Bruijn torus

In combinatorial mathematics, a de Bruijn torus is a special kind of grid made up of symbols, like 0s and 1s. This grid is arranged so that every possible small pattern of numbers appears exactly once. It is called a torus because the edges wrap around, like a circle, when looking for these patterns. The idea comes from something called a De Bruijn sequence, which is a simpler version of this concept.

Cutting a torus

When you cut a solid torus with straight planes, there is a special way to figure out how many pieces you will get. If you use n planes, the number of pieces you can get is given by a special math formula. For example, if you use 0 planes, you get 1 piece (the whole torus). If you use 1 plane, you get 2 pieces, and so on. The numbers go like this: 1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, and more.