Why 10 dimensions
This Wikiversity learning project allows participants to explore why some physicists have speculated that our universe might have 10 dimensions.
This project started at Wikipedia. It needs to be adapted to wikipedia.
In our personal human experiences, we seem to exist in a universe with three spatial dimensions. Some theories in physics, including string theory, include the idea that there are additional spatial dimensions. Such theories suggest that there may be a specific number of spatial dimensions such as 10. The question, "Why 10 dimensions?" arises from these theories.
Why 10, 11, or 26 physical dimensions in string theory? 
This is one of the questions discussed by Michio Kaku in his book Hyperspace. That book is an attempt to translate the mathematics of hyperspace theory into ordinary language that can be understood by a wide audience. This article is devoted to the same goal, leaving the details of the mathematics to the hyperspace theory article.
Kaku traces the number of dimensions to Srinivasa Ramanujan's modular functions, but this article will start with some fundamentals and work its way into the mathematics. The goal here is to use ordinary language when possible and be careful to clearly define jargonistic terms that come to us from the mathematics and physics of hyperspace.
Foundation: let's be clear what we are talking about 
All in one place: short definitions
- what are spatial dimensions?
- what is string theory?
- what is hyperspace?
- what is a string?
- what is a modular function?
- Modular functions and string theory
- Ramanujan modular functions
Spatial dimensions 
Macroscopic physical objects such as people are free to move in three independent directions, so we conclude that there are three spatial dimensions. Theoretical physicists have speculated that there might be additional spatial dimensions that escape our notice because they are too small. Some theoretical objects like strings are small enough that they would be able to vibrate in the compact dimensions while moving through the familiar three extended dimensions.
String theory 
String theory is a proposed physical theory. There are several versions or types of string theory. Attempts are being made to discover which version of the theory (if any) is in agreement with observations of the physical universe. All string theories include the idea of a hyperspace of more than three spatial dimensions. The "extra" spatial dimensions are theoretically "compact" or "collapsed" dimensions. This means that they are not as extended in space as the three familiar spatial dimensions. The collapsed dimensions are too small to observe directly. It is not clear how many collapsed dimensions are required for a string theory that is in best agreement with observations of the physical universe, but mathematical constraints currently favor string theories with 10, 11, or 26 dimensions.
What explanatory power comes from including "extra" compact dimensions in a physical theory? Since the time of the first written speculations about the possible existence of an atom, a goal of physics has been to understand the fundamental physical components of the universe. Unfortunately, many subatomic particles (each subject to some combination of the four fundamental forces) have been observed and so attention has turned to theoretical attempts to describe the diversity of subatomic particles in an elegant physical theory. Why are there so many different particles? Why do they have the physical properties that they are observed to have? Have things always been this way or have the properties of subatomic particle changed since the formation of our universe? Do they continue to change? Some theoretical physicists are exploring the idea that the diversity of subatomic particle can be accounted for in terms of symmetry breaking. Maybe under the high energy conditions of the early universe all particles were initially indistinguishable, a condition called supersymmetry. As the universe cooled, some spatial dimensions compactified and particles distributed themselves among the available stable energy states provided by four extended space-time dimensions and six or more compact dimensions. This line of reasoning suggests that it might be possible to explain the diversity of subatomic particles and fundamental forces in terms of a theory of how an original hyperspace "broke" into two "parts"; our extended 4 dimensional space-time and several "invisible" additional compact spatial dimensions. String theory is popular partly because it is a quantum field theory which accommodates gravity in terms of a spin=2 graviton.
Within string theory, a string is an object with one spatial dimension. These hypothetical one-dimensional strings are very small, on the order of the Planck length (about 1.6 × 10-35 meters). String theory explores the behaviours of open strings (with free ends) or closed strings (loops with no free ends). Within string theory, the stable vibrational states of strings are taken to correspond to physical particles like the graviton. When a string moves through spacetime it sweeps out a 2-dimensional surface called the worldsheet. One of the features of string theory that has appeal to physicists is that, when particle interactions are thought of in terms of worldsheets, it is possible to overcome some of the mathematical problems confronted when considering particles as points.
Recap before getting to the mathematics 
String theory grew out of attempts to find a simple and elegant way to account for the diversity of particles and forces observed in our universe. The starting point was to assume that there might be a way to account for that diversity in terms of a single fundamental physical entity (string) that can exist in many "vibrational" states. The various allowed vibrational states of string could theoretically account for all the observed particles and forces. Unfortunately, there are many potential string theories and no simple way of finding the one that accounts for the way things are in our universe.
One way to make progress is to assume that our universe arose through a process involving an initial hyperspace with supersymmetry that, upon cooling, underwent a unique process of symmetry breaking. The symmetry breaking process resulted in conventional 4 dimensional extended space-time AND some combination of additional compact dimensions. What can mathematics tell us about how many additional compact dimensions might exist?
Modular functions 
What is the connection between modular functions and string theory? 
Modular functions are used in the mathematical analysis of Riemann surfaces. Riemann surface theory is relevant to describing the behavior of strings as they move through space-time. When strings move they maintain a kind of symmetry called "conformal invariance"
Conformal invariance (including "scale invariance") is related to the fact that points on the surface of a string's world sheet need not be evaluated in a particular order. As long as all points on the surface are taken into account in any consistent way, the physics should not change. Equations of how strings must behave when moving involve the Ramanujan function.
Ramanujan modular functions 
When a string moves in space-time by splitting and recombining (see worldsheet diagram at right), a large number of mathematical identities must be satisfied. These are the identities of Ramanujan's modular function.
The KSV loop diagrams of interacting strings can be described using modular functions.
The "Ramanujan function" (an elliptic modular function? satisfies the need for "conformal symmetry") has 24 "modes" that correspond to the physical vibrations of a bosonic string.
When the Ramanujan function is generalized, 24 is replaced by 8 (8 + 2 = 10) for fermion strings.