# Dimensions of strings in string theory

In the above image taken from wikipedia, at the string level the strings have been shown as some loops, the article in wikipedia says that in string theory the particles at lower level are broken down into one dimensional strings, but I understand that only a straight line can be one dimensional, how are these loop like strings still said to be one dimensional?

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This would be helpful:math.stackexchange.com/q/465818 – user31782 Dec 24 '13 at 17:35

the article in wikipedia says that in string theory the particles at lower level are broken down into one dimensional strings, but I understand that only a straight line can be one dimensional, how are these loop like strings still said to be one dimensional ?

Maybe this will help:

In mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. For example, a point on the unit circle in the plane can be specified by two Cartesian coordinates, but one can make do with a single coordinate (the polar coordinate angle), so the circle is 1-dimensional even though it exists in the 2-dimensional plane.

Take a string and distort it, the position of the points on the string are into one to one correspondence with the stretched string, thus there is one string coordinate that describes the position of the points on the string, whatever its shape. Given the shape, one can find a mathematical function that will reduce the two or three dimensional description possible for a distorted string's points into one variable, as with the simple example of a circle above.

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But the article on wikipedia does not say anything about taking the coordinates with respect to string only, where it says one-dimensional it has linked to another article which describes cartesian space dimensions but nothing such as what you just described. However if it is as you say then your answer is apt. Can you give some refrence ? – Rijul Gupta Dec 2 '13 at 10:55
I am just extrapolating from the example of how a circle, which in cartesian coordinates is two dimensional can be described by a single independent variable. It is the independent variables that define the dimensions. Maybe a more mathematically inclined contributor will answer your question rigorously. – anna v Dec 2 '13 at 11:45
@rijulgupta The article "one-dimensional" links to is exactly the same anna v has cited. The dimensionality of an object is independent of the space it is embedded in. It is the minimum number of parameters needed in order to uniquely define a point on that geometrical object. One can describe a line, no matter in how many dimensions it is embedded, by a parameter that varies continuously from the beginning to the end point. – Frederic Brünner Dec 2 '13 at 12:38
@anupam Well, I am a physicist, and mathematics is just tool for me. – anna v Dec 11 '13 at 6:29
@NickStauner You are talking of a specific "measurement" for which a circle is a model. In the "measurement" there will be errors in the radius, there will be a thickness in the ring . In the model, which is taken from mathematics "dimension" has a very strict meaning and the number of dimensions are counted from the independent variables in the problem. Constants are constants, not variables. In mathematics all unit circles are the same. Drawing physical circles is another story, but the mathematical models are good within measurement errors. – anna v Mar 18 '14 at 4:25

I'll mention the informal definition of point and line from the work of Euclid :Euclid's Elements.

1. A point is that of which there is no part.

2. And a line is a length without breadth.

3. And a surface is that which has length and breadth only.

consider a straight rectangular rod of length $L$ width $W$ and height $H$. Now to measure the space allocated by this rod you will need to consider its volume given by $L\times W \times H$ . the objects which we come across in day to day life have these three parameters which define the space allocated by them completely. now consider a thin $A4$ size paper of a very fine width as shown. we are not very much interested in its width because its width doesn't allocates much space . we are usually interested in the area allocated by a paper. In mathematics we have such a paper of width $0$ . This $0$ width entity is called a surface in mathematics like gaussian surface. These are called two dimensional . Similarly we have a line in mathematics which have $0$ width and $0$ height but have measurable length. These lines are called one dimensional.
now we come across something called an arc in mathematics which is an extention of a non-straight line but still an arc has no width and no height. The mathematical entity arc can be described by only one coordinate as explained in anna v's answer, does the dimensionality of space in which the entity is described determines the dimensionality of the entity itself NO! the arc is still one dimensional, it is the space in which it is plotted three dimensional or two dimensional, it has still only its length called the arc-length as its unique description. you may signify any number of coordinates to any point allocated on the arc to describe its position in space.
e.g consider a straight line in three different situations
(1) (2) (3) Irrespective of the coordinate system the line will remain one-dimensional.
Similarly your loop like mathematical strings are one dimensional whether you place it in a three or four dimensional space it doesn't matter.

Aliter

An algebraic plane curve given by one polynomial equation $ƒ(x,y) = 0$ is one dimensional.

e.g a circle given by $$ƒ(x,y) = x^2 + y^2 - 1 = 0$$ is one dimensional.
reference : http://en.wikipedia.org/wiki/Plane_curve

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