Manuscript VIII-C@3 · VIII-C-v3.tex · section · 3b070436

Packet-table identities

Exact algebraic and resource checks › Packet-table identities · lines 2309–2346

Let $t = t_{m}$ and $u = arctan t$ . Then

α_{m} = \frac{π}{2} - 2 u .

The double-angle identities give

cos α_{m} = \frac{2 t}{1 + t ^{2}}, sin α_{m} = \frac{1 - t ^{2}}{1 + t ^{2}},

and

cos \frac{α _{m}}{2} = \frac{1 + t}{2 1 + t ^{2}} .

These prove eq:algebraic. For rational $t$ , $e^{i α_{m}}$ has rational real and imaginary parts, as do its finite powers. Furthermore,

e^{i μ_{m}} = i (e^{2 i u})^{3}, e^{2 i u} = \frac{1 - t ^{2} + 2 i t}{1 + t ^{2}},

so the seam rotation has rational entries. The remaining displayed finite data are algebraic.

At $t = 1/64$ ,

1 + t^{2} = \frac{4097}{4096}, cos α_{1} = \frac{128}{4097}, sin α_{1} = \frac{4095}{4097} .

This proves eq:installed,eq:ray. The finer-table index in eq:finer is in $T_{q}$ and has exactly the same rational $t$ .

Displayed formulas in this passage

unlabeled · line 2312
$α_{m} = \frac{π}{2} - 2 u .$
unlabeled · line 2316
$cos α_{m} = \frac{2 t}{1 + t ^{2}}, sin α_{m} = \frac{1 - t ^{2}}{1 + t ^{2}},$
unlabeled · line 2321
$cos \frac{α _{m}}{2} = \frac{1 + t}{2 1 + t ^{2}} .$
unlabeled · line 2329
$e^{i μ_{m}} = i (e^{2 i u})^{3}, e^{2 i u} = \frac{1 - t ^{2} + 2 i t}{1 + t ^{2}},$
unlabeled · line 2338
$1 + t^{2} = \frac{4097}{4096}, cos α_{1} = \frac{128}{4097}, sin α_{1} = \frac{4095}{4097} .$