I am trying to understand Rob Kirby's AMS notice https://www.ams.org/notices/200710/tx071001306p.pdf on Boy's surface. From this blogpost https://divisbyzero.com/2020/04/08/make-a-real-projective-plane-boys-surface-out-of-paper/, I understand the octahedron construction of $\mathbb RP^2$.
I have numbered the edges based on the foldable from the blogpost: (I have also colored the foldable's regions with the colors that appear in the above picture)
Kirby then writes the paragraph:
The neighborhood of the top vertex is drawn in Figure 3. This neighborhood can be flattened out to look like the polyhedron in Figure 4; it is still a cone on a figure-8. A neighborhood of the red edge is homeomorphic to the polyhedron in Figure 5, having cones at both ends. Flattening then gives the polyhedron Q in Figure 6. Q is the image of a rectangle, immersed except at the cone points. This can be changed, relative to the boundary, to the immersed image of a rectangle as in Figure 7, where the two cone points have been canceled.
I have drawn in Kirby's red line in the foldable picture above. But I have no idea what he is talking about when he says Figure 3 can be flattened into Figure 4. Did the blue triangle turn into a blue square? What does the gray plane correspond to?
If at all possible, I would appreciate if someone could explain what Kirby is saying; maybe sketching some intermediate diagrams, or taking pictures of a physical model (that one can create by printing, folding and labelling/coloring the foldable)