Square: A square consists of nine small squares with a width only one third of its length.

Koch curve (de:Koch-Kurve): Each part of Koch curve is composed of four small curves with the same shape in the ratio of 1:3, so its Hausdorff dimension is an irrational number.

In fact, Hausdorf's calculation is not as simple or even difficult as the above example.

Hausdorf outer measure: Let (x, d) be a metric space and E be a subset of x, and define it.

And e can be covered by set family (aj) K. Then the Hausdorf outer measure of e is defined as:

Hausdorff-Wiehausdorff dimension is defined as the S value corresponding to the jumping point where Hausdorff external metric changes from zero value to non-zero value. Strictly defined as:

In the first part of the paper, Mendelberg discusses how the length between the coastline measured by Angus Frey Richardson and other natural geographical boundaries depends on the measurement scale. Richardson observed that the length L(G) measured at the borders of different countries is a function of the measurement scale g. He collected data from several different examples, and then guessed that L(G) could be estimated by the following form of function:

L(G)=MG 1-D

Meng interprets this result as indicating that the coastline and other geographical boundaries can have statistical self-similarity, and the index d calculates the Hausdorff dimension of the boundary. From this perspective, Richardson's research example has a dimension from the South African coastline 1.02 to the British west coast 1.25.

In the second part of the paper, Mendelberg describes different curves about Koch snowflake, which are standard self-similar graphs. Montfort gave a method to calculate their Hausdorff dimensions, and their dimensions were all between 1 and 2. He also mentioned peano curve, which is full of space and has a dimension of 2, but did not give its structure.

This paper is very important, because it not only shows Mendelberg's early thoughts on fractal, but also is an example of the connection between mathematical objects and natural forms-the theme of many works after Mendelberg.