I will explore two questions about projections of geometric objects in 4-dimensional spaces:

(1) Let A be a convex body in R^4, and let (p_12, p_13, p_14, p_23, p_24, p_34) be the areas of the six coordinate projections of A to R^2. Which six numbers arise in this way?

(2) Let S be an irreducible surface in (P^1)^4, and let (p_12, p_13, p_14, p_23, p_24, p_34) be the degrees of the six coordinate projections from S to (P^1)^2. Which six numbers arise in this way?

The answers to these questions are governed by the Plucker relations for the Grassmannian Gr(2,4) over the triangular hyperfield T_2. These results suggest a general conjecture on homology classes of irreducible surfaces in smooth projective varieties. Most of the talk will be accessible to general mathematical audience. Joint with Daoji Huang, Mateusz Michalek, Botong Wang, Shouda Wang.