# Dimensions: Calculate Ratios of Area and Volume

Calculator for the ratios of sizes with several dimensions: area (2D), volume (3D) and fractals. If you double the size of a one-dimensional thing, like a line, then its length also will double. One meter becomes two. If you double the size of a two-dimensional thing, a field, its area will quadruple, one square meter becomes four. If you double the size of a three-dimensional thing, a space, its area will eightfold, one cubic meter becomes eight.

The formula is: new magnitude = old magnitude * multiplication^{dimension}

n = o * m^{ d} o = n / m^{ d} m = ^{d}√ n/o d = log_{m}( n/o )

Doubling the size of line, area and volume:

The world in which we live in has three dimensions of space. Mathematics doesn't care, this calculation is true for any amount of dimensions, whereas more dimensions can result in very large numbers. Imagination also generally fails beyond three dimensions, since all of our experiences are made in our three-dimensional world. In fact, there are physical theories with more dimensions of space, but where the extra dimensions are wrapped up in a tiny space. An example is eleven-dimensional supergravity. So far, however, these are only models whose relation to reality has neither been proven nor disproved.

There are objects with fractional dimensions, so-called Hausdorff dimensions. These objects are called fractals. An example is the Koch curve, which when tripled quadruples its measures. The Hausdorff dimension there is log(4)/log(3) = 1.261859507142915. Fractals are useful mathematical entities that have their applcations. However, the infinitely fine subdivision that the fractals have is not possible in reality, it only exists in mathematical models.

German: Dimension | Vielfacher Inhalt | Verhältnis | Diagonalen | Flächeninhalt | Rauminhalt | Schneiden | Stapel | Gitter | Anordnung | Rand | Innen-Außen | Lagerung | Ausbreitung | Stufenpyramide