Solution

Reward: Item or Gold!
You find lines carved into the cold limestone of the dungeon wall.
• u with 0 < u < 1:

 $\begin{array}{l}\left\{u-5,{\mathrm{sin}}^{2}\left(2\pi u\right)\right\}\end{array}$ $\begin{array}{l}\left\{-3,u\right\}\end{array}$ $\begin{array}{l}\left\{\mathrm{sin}\left(5{u}^{20}\right)-1,u\right\}\end{array}$

As your eyes adjust, you discover more and more lines curving all around you—in various steels, glass, tile, paint, and granite. For a moment, you find yourself wishing Transparent Horizon were here . . . but it’s not.

(Note: u,v ∈ ℝ and i ∈ ℤ. Right-click on an expression to obtain raw MathML which may be pasted into, e.g., Mathematica.)

• u,v with 29/2 < u < 29/2 + 2π and 0 < v < 1:

$\left\{-\frac{1}{100}\left(-6u-100{\mathrm{tan}}^{-1}\left(-2u+3\pi +29\right)+9\pi +87\right)\mathrm{cos}\left(❘u-\frac{3\pi }{2}-\frac{29}{2}❘\right),\phantom{\rule{0ex}{0ex}}-\frac{1}{100}\left(-6u-100{\mathrm{tan}}^{-1}\left(-2u+3\pi +29\right)+9\pi +87\right)\mathrm{sin}\left(❘u-\frac{3\pi }{2}-\frac{29}{2}❘\right),\phantom{\rule{0ex}{0ex}}v\right\}$
• u with 0 ≤ u ≤ 2π and ∀ i with -5 ≤ i ≤ -2 or 3 ≤ i ≤ 6:

$\left\{\frac{i-5}{2}+\mathrm{sin}\left(u\right),2\mathrm{cos}\left(u\right)\right\}$
• u,v with 6 ≤ u ≤ 8 and 3 ≤ v ≤ 7:

$\left\{u-6,\frac{1}{8}\left(-{\left(u-7\right)}^{2}-\frac{16}{9}{\left(v-5\right)}^{2}\right),\frac{5\left(v-3\right)}{4}\right\}$
• u with 0 ≤ u ≤ 2π and ∀ i with -3 ≤ i ≤ 8:

$\left\{\frac{3\left(i+4\right)}{2}+\mathrm{sin}\left(u\right),-\frac{{\left(\frac{3\left(i+4\right)}{2}+\mathrm{sin}\left(u\right)\right)}^{6}}{34012224}+\frac{\mathrm{cos}\left(u\right)}{2},\frac{1}{2}\mathrm{cos}\left(\pi \left(i+4\right)-u\right)\right\}$
• u with 0 < u < 1, ∀ i with 1 ≤ iimax + 7:

$\left\{u\left({\mathrm{sin}}^{2}\left(9679i\right)+1\right)\mathrm{sin}\left(15i+\frac{1}{30}\pi {\mathrm{sin}}^{2}\left(7243i\right)\right),u\left({\mathrm{sin}}^{2}\left(9679i\right)+1\right)\mathrm{cos}\left(15i+\frac{1}{30}\pi {\mathrm{sin}}^{2}\left(7243i\right)\right)\right\}$
• u with umin ≤ u ≤ umax:

$\left\{\mathrm{cos}\left(\frac{5}{42}\pi \left(8u-47\right)\right),\phantom{\rule{0ex}{0ex}}\frac{64000000{\left(u-5\right)}^{6}}{5534900853769}-\frac{{\left(u-5\right)}^{4}}{2401}+\frac{1}{252}\left(20u-107\right)-\frac{1}{8}\mathrm{cos}\left(\frac{1}{21}\pi \left(20u-107\right)\right),\phantom{\rule{0ex}{0ex}}-\mathrm{cos}\left(\frac{1}{21}\pi \left(20u-107\right)\right)\right\}$

and ∀ u with 1 ≤ u ≤ 26:

$\left\{\mathrm{sin}\left(\frac{1}{75}\pi \left(14u-39\right)\right),\phantom{\rule{0ex}{0ex}}\frac{4}{15}{\left(\frac{13}{50}-\frac{7u}{75}\right)}^{8}+\frac{53}{29}{\left(\frac{13}{50}-\frac{7u}{75}\right)}^{6}-\frac{111}{83}{\left(\frac{7u}{75}-\frac{13}{50}\right)}^{7}-\frac{227}{81}{\left(\frac{7u}{75}-\frac{13}{50}\right)}^{3}+\frac{23{\left(39-14u\right)}^{2}}{236250}+\frac{2\left(14u-39\right)}{3525}-\frac{37}{50},\phantom{\rule{0ex}{0ex}}-\mathrm{cos}\left(\frac{1}{75}\pi \left(14u-39\right)\right)\right\}$
• u with 16 ≤ u ≤ 20:

$\begin{array}{l}\left\{\frac{u-16}{4}-1,\frac{3\left(u-16\right)}{80}-\frac{3}{20},\frac{u-16}{4}\right\}\\ \left\{\frac{3}{10}-\frac{3\left(u-16\right)}{40},\frac{u-16}{4}-1,\frac{u-16}{4}\right\}\\ \left\{0,\frac{3}{4}-\frac{3\left(u-16\right)}{16},\frac{u-16}{4}\right\}\\ \left\{\frac{u-16}{40}-\frac{1}{20},\frac{16-u}{10}+\frac{2}{5},\frac{16-u}{5}+\frac{11}{10}\right\}\\ \left\{\frac{16-u}{80}+\frac{1}{10},\frac{u-16}{4}-1,\frac{5}{4}-\frac{19\left(u-16\right)}{80}\right\}\\ \left\{\frac{u-16}{2},\frac{4\left(u-16\right)}{5}-1,\frac{5}{4}\right\}\\ \left\{\frac{u-16}{8}+\frac{7}{4},2,\frac{5}{4}-\frac{5\left(u-16\right)}{16}\right\}\\ \left\{\frac{3\left(u-16\right)}{16}+\frac{5}{4},\frac{16-u}{16}+\frac{43}{20},\frac{u-16}{16}\right\}\\ \left\{\frac{9\left(u-16\right)}{80}+\frac{43}{20},\frac{16-u}{8}+\frac{3}{2},\frac{3\left(u-16\right)}{20}+\frac{3}{5}\right\}\\ \left\{\frac{1}{3}\mathrm{sin}\left(\frac{4}{3}\pi \left(2u-37\right)\right)+\frac{43}{20},\frac{\pi \left(\sqrt{2}\mathrm{cos}\left(\frac{2}{3}\pi \left(2u-37\right)\right)+9\right)+2\sqrt{2}\mathrm{sin}\left(\frac{4}{3}\pi \left(2u-37\right)\right){\mathrm{tan}}^{-1}\left(\frac{20}{3}\left(2u-37\right)\right)}{6\pi },-\frac{\mathrm{cos}\left(\frac{2}{3}\pi \left(2u-37\right)\right)}{3\sqrt{2}}+\frac{\sqrt{2}\mathrm{sin}\left(\frac{4}{3}\pi \left(2u-37\right)\right){\mathrm{tan}}^{-1}\left(\frac{20}{3}\left(2u-37\right)\right)}{3\pi }+\frac{3}{5}\right\}\end{array}$
• ∀ u with 0 ≤ u ≤ 2π:

$\left\{-5\mathrm{sin}\left(u\right)+\mathrm{sin}\left(5u\right)+5\mathrm{cos}\left(u\right)-\mathrm{cos}\left(5u\right),5\mathrm{sin}\left(u\right)-\mathrm{sin}\left(5u\right)+5\mathrm{cos}\left(u\right)-\mathrm{cos}\left(5u\right)\right\}$

and ∀ u with 16 ≤ u ≤ umax:

$\left\{\frac{10\left(\mathrm{cos}\left(\frac{9}{10}{\left(u-15\right)}^{2}\right)-\frac{3}{5}\mathrm{sin}\left(\frac{9}{10}{\left(u-15\right)}^{2}\right)\right)}{3{\left(u-15\right)}^{2}}-4,\frac{15}{2}-\frac{10\left(\mathrm{sin}\left(\frac{9}{10}{\left(u-15\right)}^{2}\right)+\frac{3}{5}\mathrm{cos}\left(\frac{9}{10}{\left(u-15\right)}^{2}\right)\right)}{3{\left(u-15\right)}^{2}}\right\}$

and

$\left\{\frac{10\left(\mathrm{sin}\left(\frac{9}{10}{\left(u-15\right)}^{2}\right)+\frac{3}{5}\mathrm{cos}\left(\frac{9}{10}{\left(u-15\right)}^{2}\right)\right)}{3{\left(u-15\right)}^{2}}-\frac{15}{2},4-\frac{10\left(\mathrm{cos}\left(\frac{9}{10}{\left(u-15\right)}^{2}\right)-\frac{3}{5}\mathrm{sin}\left(\frac{9}{10}{\left(u-15\right)}^{2}\right)\right)}{3{\left(u-15\right)}^{2}}\right\}$

You don’t know much about what they represent, but you know what you like. You like particular, identifiable parameter values.

• You like a1 and a2 such that u = a1 (nearer the signature) and u = a2 (farther from the signature) are points along the signed beam where other beams intersect
• You like a3 and a4 such that a3 = umin and a4 = umax are appropriate in 3D
• You like a5 such that a5 = umax yields the correct antenna length
• You like a6 such that i = a6 brings you closest to a baseboard electrical outlet
• You like a7 such that imax = a7 yields the correct number of white lines
• You like a8 and a9 such that (u,v) = (a8, a9) is a reflecting pool when viewed from between a fire-pull-plus-junction-box, a fire alarm, a black sign, and a silver sign
• You like a10 such that the gap you could enter through spans a10 - 1/2 < u < a10 + 1/2
• You like a11 such that, when i = a11, i is treeless but i + 1 and i - 1 are not