The Hausdorff dimension is a concept in mathematics that generalizes the idea of dimension beyond traditional integers. While a line has a dimension of 1 and a plane has a dimension of 2, the Hausdorff dimension can take non-integer values, allowing for a more nuanced understanding of complex shapes and sets, such as fractals. This dimension is determined by measuring how a set scales as it is covered by smaller and smaller shapes. For example, the Koch snowflake has a Hausdorff dimension of about 1.26, indicating that it is more complex than a line but less than a full plane.