Took snapshots of spin-vector addition for a closer look.

docs/src/newsreport.md

@@ -157,22 +157,22 @@ gauge3 = float(π)
 
 The geometry of "spin-vector addition" is shown. The spin-vectors exist in a spin-space that is equipped with three operations: scalar multiplication, inner product and addition. The addition of spin-vectors κ and ω results in another spin-vector κ + ω in the spin-space, which has its own flagpole and flag plane. Taking κ and ω as null vectors in the sphere of future null directions, the flagpole of κ is represented by a point (complex number) and the null flag of κ is represented as a point sufficiently close to κ that is used to assign a direction tangent to the sphere at κ.
 
-![addition](./assets/spinspace/addition.png)
+![addition1](./assets/spinspace/addition1.png)
 
 The tails of the flagpoles of κ, ω and κ + ω are in a circle in the sphere of future null directions. The circumcircle of the triangle made by joining the tails of the three spin-vectors makes angles with the flagpoles and null planes. Meaning, the distance between κ and the center of the circle is equal to the distance between ω and the center. Also, the distance of the addition of κ and ω and the circle center is the same as the distance between κ and the center. For the circumcircle, we have three collinear points in the Argand complex plane. However, lines in the Argand plane become circles in sections of the three-dimensional sphere. The angle that the flagpoles of κ and ω make with the circle should be twice the argument of the inner product of the two spin-vectors (modulus 2π with a possible addition of π).
 
-![addition07](./assets/spinspace/addition07.png)
+![addition2](./assets/spinspace/addition2.png)
 
 ```julia
 w = (Complex(κ + ω) - Complex(κ)) / (Complex(ω) - Complex(κ))
 @assert(imag(w) ≤ 0 || isapprox(imag(w), 0.0), "The flagpoles are not collinear: $(Complex(κ)), $(Complex(ω)), $(Complex(κ + ω))")
 ```
 
-![addition08](./assets/spinspace/addition08.png)
+![addition3](./assets/spinspace/addition3.png)
 
 In an interesting way, the argument (phase) of the inner product of κ and ω is equal to half of the sum of the angles that the spin-vectors make with the circle, which is in turn equal to the angle that U and V make with each other minus π (also see the geometric descriptions of the inner product to construct U and V). In the case of spin-vector addition, the angles that the flag planes of κ, ω and κ + ω, each make with the circle are equal. But, be careful with determining the signs of the flag planes and the possible addition of π to the flag plane of κ + ω. For determining flag plane signs, see also Figure 1-21 in page 64 of Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Volume 1: Two-spinor calculus and relativistic fields, (1984).
 
-![addition09](./assets/spinspace/addition09.png)
+![addition4](./assets/spinspace/addition4.png)
 
 For example, the Standard Model is formulated on 4-dimensional Minkowski spacetime, over which all fiber bundles can be trivialized and spinors have a simple explicit description.
 For the Symmetries relevant in field theories, the groups act on fields and leave the Lagrangian or the action (the spacetime integral over the Lagrangian) invariant.