String Theory: The History of 2-Dimensional Space
One of the most fascinating aspects of string theory is the requirement of extra dimensions to make the theory work. String theory requires nine space dimensions, while M-theory seems to require ten space dimensions. Under some theories, some of these extra dimensions may actually be long enough to interact with our own universe in a way that could be observed.
Contrast this with a 2-dimensional space. Many people think of geometry (the study of objects in space) as a flat, 2-dimensional space that contains two degrees of freedom — up or down and right or left. Throughout most of modern history, this interest has been the study of Euclidean geometry or Cartesian geometry.
Euclidean geometry: Think back to high school geometry
Probably the most famous mathematician of the ancient world was Euclid, who has been called the father of geometry. Euclid’s 13-volume book, Elements, is the earliest known book to have taken all the existing knowledge of geometry at the time of its writing (around 300 BCE). For nearly 2,000 years, virtually all of geometry could be understood just by reading Elements, which is one reason why it was the most successful math book ever.
In Elements, Euclid started off presenting the principles of plane geometry — that is, the geometry of shapes on a flat surface, as in this figure. An important consequence of Euclidean plane geometry is that if you take the measure of all three angles inside of a triangle, they add up to 180 degrees.
Later in the volumes, Euclid extended into 3-dimensional geometry of solid objects, such as cubes, cylinders, and cones. The geometry of Euclid is the geometry typically taught in school to this day.
Cartesian geometry: Merging algebra and Euclidean geometry
Modern analytic geometry was founded by French mathematician and philosopher Rene Descartes, when he placed algebraic figures on a physical grid. This sort of Cartesian grid is shown in this figure. By applying concepts from Euclidean geometry to the equations depicted on the grids, insights into geometry and algebra could be obtained.
Around the same time that Galileo was revolutionizing the heavens, Descartes was revolutionizing mathematics. Until his work, the fields of algebra and geometry were separate. His idea was to display algebraic equations graphically, providing a way to translate between geometry and algebra.
Using the Cartesian grid, you can define a line by an equation; the line is the set of solutions to the equation. In the figure, the line goes from the origin to the point (5, 3). Both the origin (0, 0) and (5, 3) are correct solutions to the equation depicted by the line (along with all the other points on the line).
Because the grid is 2-dimensional, the space that the grid represents contains two degrees of freedom. In algebra, the degrees of freedom are represented by variables, meaning that an equation that can be shown on a 2-dimensional surface has two variable quantities, often x and y.