Manuscript VII-E@1 · VII-E.tex · section · b713cf3c

Adapted charts and exact products

One certified four-face slicing exchange › Adapted charts and exact products · lines 2012–2069

VII-E@1#eq:charts, VII-E@1#eq:opposite, VII-E@1#eq:mixed

For each face, use the following Lorentz transformation:

C_{E} C_{N} = 5/4 - 3/4 0 - 3/4 5/4 0 001, = 13/12 0 - 5/12 010 - 5/12 0 13/12, C_{W} C_{S} = 5/4 3/4 0 3/4 5/4 0 001, = 13/12 0 5/12 010 5/12 0 13/12 .

Direct multiplication gives $C_{i}^{T} η C_{i} = η$ , determinant $1$ , and future time orientation. For the slope $v_{i}$ of face $i$ ,

(C_{i} z)^{0} = \frac{T - v _{i} \cdot ( x , y )}{1 - ∣ v _{i} ∣ ^{2}} = s

on that face.

Define $Λ_{i \leftarrow j} = C_{i} C_{j}^{- 1}$ . The opposite pairs are

Λ_{E \leftarrow W} = 17/8 - 15/8 0 - 15/8 17/8 0 001, Λ_{N \leftarrow S} = 97/72 0 - 65/72 010 - 65/72 0 97/72 .

Writing $X_{+} = E, X_{-} = W, Y_{+} = N, Y_{-} = S$ , the mixed pairs are

Λ_{X_{ϵ} \leftarrow Y_{ν}} = 65/48 - ϵ 13/16 ν 5/12 - ϵ 3/4 5/4 0 ν 25/48 - ϵ ν 5/16 13/12 .

Reverse transitions are inverses. Core translations are zero.

Displayed formulas in this passage

VII-E@1#eq:charts · line 2015
$C_{E} C_{N} = 5/4 - 3/4 0 - 3/4 5/4 0 001, = 13/12 0 - 5/12 010 - 5/12 0 13/12, C_{W} C_{S} = 5/4 3/4 0 3/4 5/4 0 001, = 13/12 0 5/12 010 5/12 0 13/12 .$
VII-E@1#eq:opposite · line 2045
$Λ_{E \leftarrow W} = 17/8 - 15/8 0 - 15/8 17/8 0 001, Λ_{N \leftarrow S} = 97/72 0 - 65/72 010 - 65/72 0 97/72 .$
VII-E@1#eq:mixed · line 2058
$Λ_{X_{ϵ} \leftarrow Y_{ν}} = 65/48 - ϵ 13/16 ν 5/12 - ϵ 3/4 5/4 0 ν 25/48 - ϵ ν 5/16 13/12 .$
unlabeled · line 2036
$(C_{i} z)^{0} = \frac{T - v _{i} \cdot ( x , y )}{1 - ∣ v _{i} ∣ ^{2}} = s$