Is a toroid a balloon?
IN MATHEMATICS, A DOUGHNUT SHAPE is known as a torus, the three-dimensional generalisation of a ring. A ring lies in a single plane; so topologically speaking there is one closed path around it that lies just outside it (a loop around the ring). Because a torus has one more dimension, you can travel along closed paths around it in two perpendicular directions. If you imagine a doughnut on a plate, one of these is a larger loop around the periphery, parallel to the plate, and the other is a smaller loop through the hole, toward and away from the plate. The generalization of a torus, any closed curve spun in a circle around an axis, is called a toroid. Curiously, there are genuine scientific theories that the universe is toroidal.
Modern cosmology is mathematically modelled through solutions to Einstein's general theory of relativity. Recall that general relativity explains gravity through a mechanism in which matter curves the fabric of space and time. It is expressed in terms of an equation that relates the geometry of a region to its distribution of mass and energy. For example, an enormous star warps space-time much more, and therefore bends the paths of objects in its neighbourhood by a greater amount, than does a tiny satellite.
Soon after general relativity was published, a number of theorists, including Einstein himself, delved for solutions that could describe the universe in general, not just the stars and other objects within it. The researchers discovered a plethora of diverse geometries and behaviours, each a distinct way of characterising the cosmos. Some of these models imagined space as resembling an unbounded plain or endless flat landscapes, only uniform in three directions, not just two. Two parallel straight lines, in such a spatial vista, would just keep going in the same direction indefinitely, like outback railroad tracks. Physicists call these flat cosmologies.
Other solutions possess spaces that curve in a saddle shape, technically known as hyperbolic geometries with negative curvature. This curvature couldn't be seen directly, unless you could somehow step out of three-dimensional space itself, but rather would make itself known through the behaviour of parallel lines and triangles. In a flat geometry (called Euclidean), if you draw a straight line and a point not on it, you can construct just one single line through that point parallel to the first line. For a saddle-shaped geometry, in contrast, there are an infinite number of parallel lines fanning out from that point, like the tracks out of a major city's terminal train station. Moreover, while triangles in flat space have angles that add up to 180 degrees, in saddle-space the angles add up to less than 180 degrees.
Yet another possibility, called positive curvature, resembles the spherical surface of an orange. Like the saddle-shape, its form could be seen only indirectly, through altered laws of geometry.
IN MATHEMATICS, A DOUGHNUT SHAPE is known as a torus, the three-dimensional generalisation of a ring. A ring lies in a single plane; so topologically speaking there is one closed path around it that lies just outside it (a loop around the ring). Because a torus has one more dimension, you can travel along closed paths around it in two perpendicular directions. If you imagine a doughnut on a plate, one of these is a larger loop around the periphery, parallel to the plate, and the other is a smaller loop through the hole, toward and away from the plate. The generalization of a torus, any closed curve spun in a circle around an axis, is called a toroid. Curiously, there are genuine scientific theories that the universe is toroidal.
Modern cosmology is mathematically modelled through solutions to Einstein's general theory of relativity. Recall that general relativity explains gravity through a mechanism in which matter curves the fabric of space and time. It is expressed in terms of an equation that relates the geometry of a region to its distribution of mass and energy. For example, an enormous star warps space-time much more, and therefore bends the paths of objects in its neighbourhood by a greater amount, than does a tiny satellite.
Soon after general relativity was published, a number of theorists, including Einstein himself, delved for solutions that could describe the universe in general, not just the stars and other objects within it. The researchers discovered a plethora of diverse geometries and behaviours, each a distinct way of characterising the cosmos. Some of these models imagined space as resembling an unbounded plain or endless flat landscapes, only uniform in three directions, not just two. Two parallel straight lines, in such a spatial vista, would just keep going in the same direction indefinitely, like outback railroad tracks. Physicists call these flat cosmologies.
Other solutions possess spaces that curve in a saddle shape, technically known as hyperbolic geometries with negative curvature. This curvature couldn't be seen directly, unless you could somehow step out of three-dimensional space itself, but rather would make itself known through the behaviour of parallel lines and triangles. In a flat geometry (called Euclidean), if you draw a straight line and a point not on it, you can construct just one single line through that point parallel to the first line. For a saddle-shaped geometry, in contrast, there are an infinite number of parallel lines fanning out from that point, like the tracks out of a major city's terminal train station. Moreover, while triangles in flat space have angles that add up to 180 degrees, in saddle-space the angles add up to less than 180 degrees.
Yet another possibility, called positive curvature, resembles the spherical surface of an orange. Like the saddle-shape, its form could be seen only indirectly, through altered laws of geometry.
