This title improves upon the original by: ?Intrigue: "Beyond Faces" sparks curiosity about what lies beyond a basic feature of simplicial complexes. ?Power: "Power" emphasizes the wider significance of these structures. The piece could delve into face numbers as a foundational concept: ?Counting Faces: Explain how the number of faces (points, edges, triangles, etc.) provides initial insights into a simplicial complex's structure. ?Simplex Homology: Briefly introduce this concept, highlighting how it uses face numbers to reveal hidden properties of the shape. The focus would then shift on the broader applications of simplicial complexes: ?Modeling Complex Systems: Discuss how these structures can represent complex networks, from social connections to the human brain. ?Topological Analysis: Highlight how simplicial complexes allow us to analyze the fundamental properties of shapes independent of size or deformation. ?Applications in Science and Engineering: Explore their use in fields like computer graphics, 3D printing, and even physical simulations. "Beyond Faces" suggests a few content directions: ?Higher Dimensions: Briefly touch upon how simplicial complexes can represent shapes beyond our usual 3D world. ?Visualizations: Emphasize the importance of using visualizations or interactive tools to help readers grasp these concepts. ?Historical Context: Briefly discuss the development of simplicial complexes and their role in the history of mathematics.