  This article revisits and elaborates the significant role of positive
geometry of momentum twistor Grassmannian for planar N=4 SYM scattering
amplitudes. First we establish the fundamentals of positive Grassmannian
geometry for tree amplitudes. Then we formulate this subject around these four
major facets: 1. Deriving the tree and 1-loop BCFW recursion relations solely
from positivity, after introducing the simple building blocks called positive
components. 2. Applying Grassmannian geometry and Pluecker coordinates to
determine the signs in N$^2$MHV homological identities. Most of them in fact
reflect the secret incarnation of the simple 6-term NMHV identity. 3. Deriving
the stacking positivity relation, which is powerful for parameterizing the
matrix representative in terms of positive variables, of a given Grassmannian
geometric configuration. 4. Introducing a highly refined formalism for tree
BCFW recursion relation, which reveals its two-fold simplex-like structures.
First, the BCFW contour in terms of (reduced) Grassmannian geometry
representatives is dissected into a triangle-shape sum, as only a small
fraction of the sum needs to be explicitly identified. Second, this fraction
can be further dissected, according to different growing modes and parameters.
The growing modes possess the shapes of solid simplices of various dimensions,
with which infinite number of BCFW cells can be entirely captured by the
characteristic objects called fully-spanning cells. We find that for a given k,
beyond n=4k+1 there is no more new fully-spanning cell, which signifies the
essential termination of the recursive growth of BCFW cells. As n increases
beyond the termination point, the BCFW contour simply replicates itself
according to the simplex-like patterns, which enables us to master all BCFW
cells once for all without actually identifying most of them.
