Final answer:
The question involves matching mathematicians to their contributions in geometry. Euler developed curved surface geometry, Mirzakhani illustrated geometry of iterations, Descartes provided proof of complex functions, and Mandelbrot created the cartesian coordinate system.
Step-by-step explanation:
The question asks to match each person to the contribution they made to the field of geometry. Based on historical records and mathematical advancements, the correct matches are:
- Leonhard Euler - Developed the geometry of curved surfaces
- Maryam Mirzakhani - Illustrated geometry of iterations
- Rene Descartes - Proof of complex functions
- Benoit Mandelbrot - Developed the cartesian coordinate system
These mathematicians have made significant contributions to the field of geometry, expanding our understanding of shapes, spaces, and mathematical relations.