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

    { u - 5 , sin 2 ( 2 π u ) } TagBox[RowBox[List["(", "", GridBox[List[List[RowBox[List["u", "-", "5"]], RowBox[List[SuperscriptBox["sin", "2"], "(", RowBox[List["2", " ", "\[Pi]", " ", "u"]], ")"]]], List[RowBox[List["-", "3"]], "u"], List[RowBox[List[RowBox[List["sin", "(", RowBox[List["5", " ", SuperscriptBox["u", "20"]]], ")"]], "-", "1"]], "u"], List[RowBox[List["u", "+", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"], "+", FractionBox["1", "2"]]], RowBox[List[RowBox[List["-", "u"]], "+", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"], "+", FractionBox["1", "2"]]]], List[RowBox[List["u", "+", "2"]], RowBox[List["3", " ", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"]]]], List[RowBox[List["u", "+", "3"]], RowBox[List[FractionBox["3", "2"], "-", RowBox[List["3", " ", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"]]]]]], List[RowBox[List["5", "-", RowBox[List["sin", "(", RowBox[List["\[Pi]", " ", "u"]], ")"]]]], RowBox[List["1", "-", RowBox[List["cos", "(", RowBox[List["\[Pi]", " ", "u"]], ")"]]]]]], Rule[RowSpacings, 1], Rule[ColumnSpacings, 1], Rule[RowAlignments, Baseline], Rule[ColumnAlignments, Center]], "", ")"]], DisplayForm] { - 3 , u } TagBox[RowBox[List["(", "", GridBox[List[List[RowBox[List["u", "-", "5"]], RowBox[List[SuperscriptBox["sin", "2"], "(", RowBox[List["2", " ", "\[Pi]", " ", "u"]], ")"]]], List[RowBox[List["-", "3"]], "u"], List[RowBox[List[RowBox[List["sin", "(", RowBox[List["5", " ", SuperscriptBox["u", "20"]]], ")"]], "-", "1"]], "u"], List[RowBox[List["u", "+", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"], "+", FractionBox["1", "2"]]], RowBox[List[RowBox[List["-", "u"]], "+", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"], "+", FractionBox["1", "2"]]]], List[RowBox[List["u", "+", "2"]], RowBox[List["3", " ", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"]]]], List[RowBox[List["u", "+", "3"]], RowBox[List[FractionBox["3", "2"], "-", RowBox[List["3", " ", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"]]]]]], List[RowBox[List["5", "-", RowBox[List["sin", "(", RowBox[List["\[Pi]", " ", "u"]], ")"]]]], RowBox[List["1", "-", RowBox[List["cos", "(", RowBox[List["\[Pi]", " ", "u"]], ")"]]]]]], Rule[RowSpacings, 1], Rule[ColumnSpacings, 1], Rule[RowAlignments, Baseline], Rule[ColumnAlignments, Center]], "", ")"]], DisplayForm] { sin ( 5 u 20 ) - 1 , u } TagBox[RowBox[List["(", "", GridBox[List[List[RowBox[List["u", "-", "5"]], RowBox[List[SuperscriptBox["sin", "2"], "(", RowBox[List["2", " ", "\[Pi]", " ", "u"]], ")"]]], List[RowBox[List["-", "3"]], "u"], List[RowBox[List[RowBox[List["sin", "(", RowBox[List["5", " ", SuperscriptBox["u", "20"]]], ")"]], "-", "1"]], "u"], List[RowBox[List["u", "+", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"], "+", FractionBox["1", "2"]]], RowBox[List[RowBox[List["-", "u"]], "+", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"], "+", FractionBox["1", "2"]]]], List[RowBox[List["u", "+", "2"]], RowBox[List["3", " ", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"]]]], List[RowBox[List["u", "+", "3"]], RowBox[List[FractionBox["3", "2"], "-", RowBox[List["3", " ", TemplateBox[List[RowBox[List["u", "-", FractionBox["1", "2"]]]], "Abs"]]]]]], List[RowBox[List["5", "-", RowBox[List["sin", "(", RowBox[List["\[Pi]", " ", "u"]], ")"]]]], RowBox[List["1", "-", RowBox[List["cos", "(", RowBox[List["\[Pi]", " ", "u"]], ")"]]]]]], Rule[RowSpacings, 1], Rule[ColumnSpacings, 1], Rule[RowAlignments, Baseline], Rule[ColumnAlignments, Center]], "", ")"]], DisplayForm]

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:

    { - 1 100 ( - 6 u - 100 tan - 1 ( - 2 u + 3 π + 29 ) + 9 π + 87 ) cos ( u - 3 π 2 - 29 2 TemplateBox[List[RowBox[List["u", "-", FractionBox[RowBox[List["3", " ", "\[Pi]"]], "2"], "-", FractionBox["29", "2"]]]], "Abs"] ) , - 1 100 ( - 6 u - 100 tan - 1 ( - 2 u + 3 π + 29 ) + 9 π + 87 ) sin ( u - 3 π 2 - 29 2 TemplateBox[List[RowBox[List["u", "-", FractionBox[RowBox[List["3", " ", "\[Pi]"]], "2"], "-", FractionBox["29", "2"]]]], "Abs"] ) , v }
  • u with 0 ≤ u ≤ 2π and ∀ i with -5 ≤ i ≤ -2 or 3 ≤ i ≤ 6:

    { i - 5 2 + sin ( u ) , 2 cos ( u ) }
  • u,v with 6 ≤ u ≤ 8 and 3 ≤ v ≤ 7:

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

    { 3 ( i + 4 ) 2 + sin ( u ) , - ( 3 ( i + 4 ) 2 + sin ( u ) ) 6 34012224 + cos ( u ) 2 , 1 2 cos ( π ( i + 4 ) - u ) }
  • u with 0 < u < 1, ∀ i with 1 ≤ iimax + 7:

    { u ( sin 2 ( 9679 i ) + 1 ) sin ( 15 i + 1 30 π sin 2 ( 7243 i ) ) , u ( sin 2 ( 9679 i ) + 1 ) cos ( 15 i + 1 30 π sin 2 ( 7243 i ) ) }
  • u with umin ≤ u ≤ umax:

    { cos ( 5 42 π ( 8 u - 47 ) ) , 64000000 ( u - 5 ) 6 5534900853769 - ( u - 5 ) 4 2401 + 1 252 ( 20 u - 107 ) - 1 8 cos ( 1 21 π ( 20 u - 107 ) ) , - cos ( 1 21 π ( 20 u - 107 ) ) }

    and ∀ u with 1 ≤ u ≤ 26:

    { sin ( 1 75 π ( 14 u - 39 ) ) , 4 15 ( 13 50 - 7 u 75 ) 8 + 53 29 ( 13 50 - 7 u 75 ) 6 - 111 83 ( 7 u 75 - 13 50 ) 7 - 227 81 ( 7 u 75 - 13 50 ) 3 + 23 ( 39 - 14 u ) 2 236250 + 2 ( 14 u - 39 ) 3525 - 37 50 , - cos ( 1 75 π ( 14 u - 39 ) ) }
  • u with 16 ≤ u ≤ 20:

    { u - 16 4 - 1 , 3 ( u - 16 ) 80 - 3 20 , u - 16 4 } { 3 10 - 3 ( u - 16 ) 40 , u - 16 4 - 1 , u - 16 4 } { 0 , 3 4 - 3 ( u - 16 ) 16 , u - 16 4 } { u - 16 40 - 1 20 , 16 - u 10 + 2 5 , 16 - u 5 + 11 10 } { 16 - u 80 + 1 10 , u - 16 4 - 1 , 5 4 - 19 ( u - 16 ) 80 } { u - 16 2 , 4 ( u - 16 ) 5 - 1 , 5 4 } { u - 16 8 + 7 4 , 2 , 5 4 - 5 ( u - 16 ) 16 } { 3 ( u - 16 ) 16 + 5 4 , 16 - u 16 + 43 20 , u - 16 16 } { 9 ( u - 16 ) 80 + 43 20 , 16 - u 8 + 3 2 , 3 ( u - 16 ) 20 + 3 5 } { 1 3 sin ( 4 3 π ( 2 u - 37 ) ) + 43 20 , π ( 2 cos ( 2 3 π ( 2 u - 37 ) ) + 9 ) + 2 2 sin ( 4 3 π ( 2 u - 37 ) ) tan - 1 ( 20 3 ( 2 u - 37 ) ) 6 π , - cos ( 2 3 π ( 2 u - 37 ) ) 3 2 + 2 sin ( 4 3 π ( 2 u - 37 ) ) tan - 1 ( 20 3 ( 2 u - 37 ) ) 3 π + 3 5 }
  • ∀ u with 0 ≤ u ≤ 2π:

    { - 5 sin ( u ) + sin ( 5 u ) + 5 cos ( u ) - cos ( 5 u ) , 5 sin ( u ) - sin ( 5 u ) + 5 cos ( u ) - cos ( 5 u ) }

    and ∀ u with 16 ≤ u ≤ umax:

    { 10 ( cos ( 9 10 ( u - 15 ) 2 ) - 3 5 sin ( 9 10 ( u - 15 ) 2 ) ) 3 ( u - 15 ) 2 - 4 , 15 2 - 10 ( sin ( 9 10 ( u - 15 ) 2 ) + 3 5 cos ( 9 10 ( u - 15 ) 2 ) ) 3 ( u - 15 ) 2 }

    and

    { 10 ( sin ( 9 10 ( u - 15 ) 2 ) + 3 5 cos ( 9 10 ( u - 15 ) 2 ) ) 3 ( u - 15 ) 2 - 15 2 , 4 - 10 ( cos ( 9 10 ( u - 15 ) 2 ) - 3 5 sin ( 9 10 ( u - 15 ) 2 ) ) 3 ( u - 15 ) 2 }

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

You’ve changed your mind. You’re glad Transparent Horizon isn’t here.