The article explores the concept of Hölder dimension, which measures the size of spaces that are similar in a certain way to a given metric space. The researchers found that for certain types of metric spaces, Hölder dimension is always less than or equal to capacity dimension. This means that for some spaces, Hölder dimension is the same as topological dimension. They also discovered that any compact, doubling metric space can be transformed into Hilbert space in a specific way, preserving certain properties. The researchers provided examples to show that Hölder dimension can be different from topological dimension in some cases, and that it may not always be achievable.