Let's say we have a line of unit length on the coordinate plane:
As you can see, the line touches one box on the plane (Desmos made each square on the grid $0.5$ in length). But what if we scale it up twice? That makes it touch two boxes:
The dimension of the figure is the log of the boxes touched divided by the log of the scale factor, so in this case $log_{2}(2) = 1$. If we played the same game with a square, we would get 4 boxes touched and have a dimension of $log_{2}(4) = 2$, and if we played it with a cube in 3D space, we woud get 8 cubes touched and have a dimension of $log_{2}(8) = 3$, and so on. But trying this with rough or self-embedded shapes has a different outcome. For example, take the coastline of Britain:
Assuming you had a fine enough grid, you find the dimension of the coastline tends towards about $1.21$, due to the roughness of the shape. Self-similar shapes such as a Sierpinski triangle also have non-integer dimensions. These shapes with fractional dimensions are known as fractals (I reccomend you watch this video for a more in-depth explenation).
But my question is, can we generate fractals themselves from a given dimension? I am aware there are at least countably infinite shapes per dimension, but is it possible to sytstematically generate at least one example of a fractal shape for any dimension?