Manuscript VIII-A@3 · VIII-A-v3.tex · section · 2398a747

Primitive core, collar, and particle-star words

The complete fixed-cone construction › Primitive core, collar, and particle-star words · lines 1941–1976

VIII-A@3#A:eq:translation-telescope, VIII-A@3#A:eq:translation-reference

affine translation cancellation. This check is not merely a Lorentz-matrix calculation. Set

A_{i} = (M_{i}, c), M_{i} = O_{i} B_{h} O_{i}^{- 1}, c = - 3 s e_{0}, R_{0} = R (π /2) .

The three transitions in chronological order have pairs

(L_{10}, c - L_{10} c), L_{10} = M_{1} M_{0}^{- 1}, (L_{21}, c - L_{21} c), L_{21} = M_{2} M_{1}^{- 1}, (L_{02}, c - L_{02} c), L_{02} = M_{0} R_{0} M_{2}^{- 1} .

Their Lorentz product is $K = M_{0} R_{0} M_{0}^{- 1}$ , and their translation is

b_{tot} = L_{02} L_{21} (c - L_{10} c) + L_{02} (c - L_{21} c) + (c - L_{02} c) = c - K c .

Thus the chart-based word is $(K, c - K c)$ . Using the affine product law,

A_{0}^{- 1} (K, c - K c) A_{0} = (R_{0}, 0) = J_{0} .

The collar transitions have zero translations and product $(R_{0}, 0)$ directly.

Displayed formulas in this passage

VIII-A@3#A:eq:translation-telescope · line 1960
$b_{tot} = L_{02} L_{21} (c - L_{10} c) + L_{02} (c - L_{21} c) + (c - L_{02} c) = c - K c .$
VIII-A@3#A:eq:translation-reference · line 1970
$A_{0}^{- 1} (K, c - K c) A_{0} = (R_{0}, 0) = J_{0} .$
unlabeled · line 1946
$A_{i} = (M_{i}, c), M_{i} = O_{i} B_{h} O_{i}^{- 1}, c = - 3 s e_{0}, R_{0} = R (π /2) .$
unlabeled · line 1951
$(L_{10}, c - L_{10} c), L_{10} = M_{1} M_{0}^{- 1}, (L_{21}, c - L_{21} c), L_{21} = M_{2} M_{1}^{- 1}, (L_{02}, c - L_{02} c), L_{02} = M_{0} R_{0} M_{2}^{- 1} .$