geometric representation, a family of shapes I propose to

Many important spatial patterns of Nature are either irregular or fragmented to such an extreme degree that ... classical geometry ... is hardly of any help in describing their form. ... I hope to show that it is possible in many cases to remedy this absence of geometric representation by using a family of shapes I propose to call fractals — or fractal sets. [Mandelbrot, "Fractals," 1977]
(Retrieved from https://www.etymonline.com/word/fractal#etymonline_v_11854 )
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1. I'd like to know what "representation" means in that context.

2. I'd like to know if "a family of shapes I propose to" is "a family of shapes which I intend to".

1. I believe he's talking about representing natural forms (shapes seen in the natural world) with fractal sets (mathematical constructions).

2. He's just saying that he's going to call this family of shapes 'fractals'.

by using a family of shapes I propose to call fractals
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I'd also like to know the grammatical construction of the orange part.

He's proposing his newly-invented term "fractals" for the family of shapes he describes in the paper. He hopes it will be adopted by the wider community. (It was. Not many scientists get to add an important new word to the dictionary.)

"By using a family of shapes (which) I propose to call...".

It's like: I got a new dog I propose to name Fido.

I've got a good name for a dog ... :roll: