# Examples of naturally occurring steroids

Digital designers have to work within tighter constraints when it comes to creating a layout; they can’t control the dimensions of the audience’s monitor, after all. A common trick in web design is to use the golden ratio to divide space between the body of the website and the sidebar. Taking the measurements of the space they’re working with, web designers can ensure that the body is times larger than the sidebar by taking the total width of the canvas, dividing it by , and then subtracting that number from the overall width of the canvas.

This abstract isomorphism with a product is not natural, as some isomorphisms of T do not preserve the product: the self-homeomorphism of T (thought of as the quotient space R 2 / Z 2 ) given by ( 1 1 0 1 ) {\displaystyle \left({\begin{smallmatrix}1&1\\0&1\end{smallmatrix}}\right)} (geometrically a Dehn twist about one of the generating curves) acts as this matrix on Z 2 (it’s in the general linear group GL( Z , 2) of invertible integer matrices), which does not preserve the decomposition as a product because it is not diagonal. However, if one is given the torus as a product ( T , t 0 ) = ( S 1 , x 0 ) × ( S 1 , y 0 ) {\displaystyle (T,t_{0})=(S^{1},x_{0})\times (S^{1},y_{0})} – equivalently, given a decomposition of the space – then the splitting of the group follows from the general statement earlier. In categorical terms, the relevant category (preserving the structure of a product space) is "maps of product spaces, namely a pair of maps between the respective components".