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<?xml version="1.0" encoding="utf-8" standalone="yes"?>
<Fractal BackgroundVisible="True" BackColor="FFFFFF00" BorderWidth="0" StripBorder="True" PixelSize="0.00548779710473845" Pixels.X="729" Pixels.Y="729" ClassicProcessing="None" OrbitTrapProcessing="ActivateControllers" ClassicMaxDwell="1" CycleDetectionActive="False" SolidGuessing="None">
  <FractalNotes><![CDATA[]]></FractalNotes>
  <FractalEquation Id="adb85431-618d-491f-b8c5-08555b119814" Title="Pixel" MaxPower="1">
    <Instructions><![CDATA[
comment:
  
  This equation can be used when no equation is necessary.
  By setting the MinDwell/MaxDwell to 1 below, we limit
  the iteration to a single step for best performance.
  
global:
  
  FSK.OverrideValue("MinDwell", 1)
  FSK.OverrideValue("MaxDwell", 1)
  
iterate:
  
  z = c
]]></Instructions>
    <PropertyValueOverrides />
  </FractalEquation>
  <TransformationArray>
    <Transformation Id="7fc4bc2c-b961-4431-b074-277d7b29842b" Title="Identity">
      <Instructions><![CDATA[]]></Instructions>
      <PropertyValueOverrides />
    </Transformation>
  </TransformationArray>
  <TransformationArray>
    <Transformation Id="7fc4bc2c-b961-4431-b074-277d7b29842b" Title="Identity">
      <Instructions><![CDATA[]]></Instructions>
      <PropertyValueOverrides />
    </Transformation>
  </TransformationArray>
  <AlternateValueInfo />
  <AlternateValueInfo />
  <OrbitTrapInfo TrapValueNTrapPreProcessing="CenterReflect" TrapValueNTrapStyle="Flat" TrapValueNSmoothing="Linear" TrapValueNTrapPreProcessing1="CenterReflect" TrapValueNTrapStyle1="Flat" TrapValueNSmoothing1="Linear" TrapValueNTrapPreProcessing2="CenterReflect" TrapValueNTrapStyle2="Flat" TrapValueNSmoothing2="Linear" TrapValueNTrapPreProcessing3="CenterReflect" TrapValueNTrapStyle3="Flat" TrapValueNSmoothing3="Linear">
    <OrbitTrapArray TrapIndexMapping="CombinedIndex" TrapDeltaMapping="TrapDelta" BlendValues="False">
      <OrbitTransform />
      <OrbitTransform />
      <TransformationArray>
        <Transformation Id="7fc4bc2c-b961-4431-b074-277d7b29842b" Title="Identity">
          <Instructions><![CDATA[]]></Instructions>
          <PropertyValueOverrides />
        </Transformation>
      </TransformationArray>
      <TransformationArray>
        <Transformation Id="7fc4bc2c-b961-4431-b074-277d7b29842b" Title="Identity">
          <Instructions><![CDATA[]]></Instructions>
          <PropertyValueOverrides />
        </Transformation>
      </TransformationArray>
      <SymmetryTransformation Id="8301dc6c-2273-4fb9-ada8-2ded2833031f" Title="Identity">
        <Instructions><![CDATA[]]></Instructions>
        <PropertyValueOverrides />
      </SymmetryTransformation>
      <OrbitTrapItem>
        <OrbitTrap Id="4e4b2e73-dc52-4107-bb98-f9abb8745748" Title="Unit Circle Group">
          <Instructions><![CDATA[
comment:
  
  trappedPoint.Index is the base circle index: 0 (center), 1-N (in ring)
  trappedPoint.Delta is the level: 0 - Steps-1
  
  See the paper:
    "Evolution of Math into Art via Mobius Transformations"
  by Anne M. Burns, Department of Mathematics,
  Long Island University.
  http://myweb.cwpost.liu.edu/aburns/
  
  Also, see pages 88-89 in the book:
    "Indra's Pearls, The Vision of Felix Klein"
  by David Mumford, Caroline Series, David Wright.
  http://klein.math.okstate.edu/IndrasPearls/
  
global:
  
  Complex ShowCenter[] = LC1,LC2,LC3,LC4,LC5,LC6,LC7,LC8,LC9,LC10,LC11,LC12,LC13,LC14,LC15,LC16
  Complex ShowRing[] = LR1,LR2,LR3,LR4,LR5,LR6,LR7,LR8,LR9,LR10,LR11,LR12,LR13,LR14,LR15,LR16
  AbsV = Sqrt(AbsU^2 - 1)
  u = AbsU * Cis(DegreeToRadian(ArgU))
  v = AbsV * Cis(DegreeToRadian(ArgV))
  Mobius UnitCircleGroup = Mobius(u, v, Conj(v), Conj(u))
  Mobius m[N+1]
  '
  ' Given N, find radius R such that N circles with radius R can be 
  ' placed along the inside of the unit circle, each tangent to the 
  ' unit circle and each of its two adjacent neighbors. The centers 
  ' of all circles are a distance of 1-R from the origin. The centers
  ' of all circles form a regular polygon P with N sides and each edge
  ' has length = 2*(1-R)*Sin(pi/N). Since all N circles are pairwise 
  ' tangent, the edge of P must equal 2*R, so 2*R = 2*(1-R)*Sin(pi/N).
  ' Solving for R yields R = 1/(1+1/Sin(pi/N)).
  '
  r = 1/(1+1/Sin(Math.PI/N))
  
  step = 2*Math.PI/N
  ang = IIf(Shift, step/2, 0)
  '
  ' Generate the Mobius transformation that transforms the
  ' unit circle into the circle in the center of the ring.
  '
  m[0] = Mobius(1-2*r, 0, 0, 1)
  '
  ' Generate the set of N Mobius transformations the  
  ' transform the unit circle into the i'th circle in the 
  ' ring, where i = 1 to N.
  '
  for (i = 1, i <= N, i += 1) {
    rotate = Cis(ang)
    
    m[i] = Mobius.Multiply( \
      Mobius(r*rotate, (1-r)*rotate, 0, 1), \
      UnitCircleGroup \
    )
    ang += step
  }
  '
  ' Assign Total = the total number of circles.
  '
  const Complex Total = 0
  count = N+1
  
  for (i = 0, i < Steps, i += 1) {
    Total += count
    count *= N+1
  }
  const Circle c[Total]
  const Complex index[Total]
  const Complex level[Total]
  Circle UnitCircle = CircleC(0, 1)
  '
  ' Generate the base ring and center circles.
  '
  for (i = 0, i <= N, i += 1) {
    c[i] = Mobius.TransformCircle(m[i], UnitCircle)
    index[i] = i
    level[i] = 0
  }
  count = N+1
  max = 0
  '
  ' Generate the remaining circles.
  '
  if (Steps > 1) {
    for (i = 1, i < Steps, i += 1) {
      min = max
      max = count
    
      for (j = min, j < max, j += 1) {
        for (k = 0, k < N+1, k += 1) {
          c[count] = Mobius.TransformCircle(m[k], c[j])
          '
          ' Skip circles that are too small.
          '
          if (c[count].Radius >= RadiusMin) {
            index[count] = index[j]
            level[count] = i
            count += 1
          }
        }
      }
    }
  }
  '
  ' Reset Total in case you skipped circles that were too small.
  '
  Total = count
  
  CurveTrap.Initialize( \
    Center, DegreeToRadian(Angle), Scale, AlternateAngle, 6, False, LineWidth \
  )
  '
  ' Add the circles to the trap based on user criteria.
  '
  for (i = 0, i < Total, i += 1) {
    lev = level[i]
    idx = index[i]
    '
    ' Note: idx=N for center circles.
    '
    if (idx = 0) {
      if (ShowCenter[lev]) {
        CurveTrap.AddCircle2(c[i], Solid, IIf(Solid, lev, 0), idx, lev)
      }
    } else {
      if (ShowRing[lev]) {
        CurveTrap.AddCircle2(c[i], Solid, IIf(Solid, lev, 0), idx, lev)
      }
    }
  }
  
trap:
  
  trappedPoint = CurveTrap.Apply(z)
  
properties:
  
  divider {
    caption = "General Options"
  }
  option Center {
    type = Complex
    caption = "Center"
    details = "Center of trap"
    default = 0
  }
  option Angle {
    type = Float
    caption = "Angle"
    details = "Angle of rotation"
    default = 0
    range = [-360,360]
  }
  option Scale {
    type = Float
    caption = "Scale"
    details = "Scale factor applied to trap"
    range = (0,)
    default = 2
  }
  option Solid {
    type = Boolean
    caption = "Solid"
    details = "Check to create solid trap"
    default = False
  }
  option AlternateAngle {
    type = Boolean
    caption = "Alternate Angle"
    details = "Use alternate angle calculation"
    default = False
  }
  option LineWidth {
    type = Float
    caption = "Line Width"
    details = "Extent of trap on either side of curve (> 0)"
    range = (0,)
    default = 0.00411522633744855967078189300413
    enabled = ~Solid
  }
  divider {
    caption = "U/V Controls"
  }
  option AbsU {
    type = Float
    caption = "Abs(U)"
    details = "Magnitude of U (1-2)"
    default = 1.1
    range = [1,2]
  }
  option ArgU {
    type = Float
    caption = "Arg(U)"
    details = "Angle of U"
    default = 0
    range = [-360,360]
  }
  option ArgV {
    type = Float
    caption = "Arg(V)"
    details = "Angle of V"
    default = 180
    range = [-360,360]
  }
  divider {
    caption = "Circle Controls"
  }
  option N {
    type = IntegerEnum(3,12)
    caption = "N"
    details = "Number of base circles"
    default = 4
  }
  option Steps {
    type = IntegerEnum(1,16)
    caption = "Steps"
    details = "Number of inversion steps"
    default = 7
  }
  option Shift {
    type = Boolean
    caption = "Shift"
    details = "Check to rotate initial chain by pi/N"
    default = False
  }
  option RadiusMin {
    type = Float
    caption = "Radius Min"
    details = "Minimum acceptable circle radius"
    default = 0
    range = [0,)
  }
  #define ShowLevel(Index)
  
  divider {
    caption = "Level #Index# Options"
  }
  option LR#Index# {
    type = Boolean
    caption = "Show Ring"
    details = "Show ring of circles at level #Index#"
    default = True
    enabled = Steps >= #Index#
  }
  option LC#Index# {
    type = Boolean
    caption = "Show Center"
    details = "Show center circle at level #Index#"
    default = True
    enabled = Steps >= #Index#
  }
  #end
  
  #include ShowLevel("1")
  #include ShowLevel("2")
  #include ShowLevel("3")
  #include ShowLevel("4")
  #include ShowLevel("5")
  #include ShowLevel("6")
  #include ShowLevel("7")
  #include ShowLevel("8")
  #include ShowLevel("9")
  #include ShowLevel("10")
  #include ShowLevel("11")
  #include ShowLevel("12")
  #include ShowLevel("13")
  #include ShowLevel("14")
  #include ShowLevel("15")
  #include ShowLevel("16")]]></Instructions>
          <PropertyValueOverrides>
            <Option Name="Solid" Type="BooleanOption" Value="True" />
            <Option Name="LineWidth" Type="FloatOption" Value="0.00411522633744855" />
            <Option Name="ArgU" Type="FloatOption" Value="180" />
            <Option Name="ArgV" Type="FloatOption" Value="0" />
            <Option Name="Steps" Type="IntegerEnumOption" Value="7" />
            <Option Name="LR1" Type="BooleanOption" Value="False" />
            <Option Name="LR2" Type="BooleanOption" Value="False" />
            <Option Name="LR3" Type="BooleanOption" Value="False" />
            <Option Name="LR4" Type="BooleanOption" Value="False" />
            <Option Name="LR5" Type="BooleanOption" Value="False" />
            <Option Name="LR6" Type="BooleanOption" Value="False" />
            <Option Name="LR7" Type="BooleanOption" Value="False" />
            <Option Name="LR8" Type="BooleanOption" Value="False" />
            <Option Name="LR9" Type="BooleanOption" Value="False" />
            <Option Name="LR10" Type="BooleanOption" Value="False" />
            <Option Name="LR11" Type="BooleanOption" Value="False" />
            <Option Name="LR12" Type="BooleanOption" Value="False" />
            <Option Name="LR13" Type="BooleanOption" Value="False" />
            <Option Name="LR14" Type="BooleanOption" Value="False" />
            <Option Name="LR15" Type="BooleanOption" Value="False" />
            <Option Name="LR16" Type="BooleanOption" Value="False" />
          </PropertyValueOverrides>
        </OrbitTrap>
      </OrbitTrapItem>
    </OrbitTrapArray>
    <OrbitTrapMasterController Id="02699986-4214-4098-b68b-48285874bc4e" Title="⠀">
      <Instructions><![CDATA[
comment:
  
  Return color computed by 1st controller.
  
color:
  
  color = Controller.Color(0)
]]></Instructions>
      <PropertyValueOverrides />
    </OrbitTrapMasterController>
    <OrbitTrapControllerArray>
      <OrbitTrapController Id="29fbd0db-b470-4d0d-ad93-e4d5d937ef86" Title="ⵔ·ⵔ" ApplyDepth="False" Apply3D="False">
        <GradientArray />
        <TextureArray />
        <Instructions><![CDATA[
comment:
  
  This controller colors each trap based on the
  property 'Index Map' and the point's 'Trap Dwell'
  'Trap Index' or 'Trap Delta'. You can define up to
  16 colors and these are mapped to the index map
  values. Offset can be used to shift the set of 
  colors with respect to value 0.
  
  If Cutouts are active, the trap is divided into
  N sectors and a user defined percent of each 
  sector is inverted with respect to the 3D shading
  effect. This gives the appearance that part of
  the trap surface has been cutout or folded over.
  
  The Cutout effect only looks good if the trap 
  is a solid shape. For example, the 'Shape', 
  'Tangent Circles', 'Shapes' and 'Squares'
  traps support a 'Solid' property that is used to 
  produce a solid figure. You should check this
  property to display Cutouts.
  
  Also note that the Cutout effect only works if
  the trap's angle is given relative to the center 
  of the solid shape. This is always the case when
  the trap is composed of a single shape. However, 
  when the trap is composed of several shapes arranged
  in a pattern like the 'Tangent Circles', 'Shapes' and
  'Squares' traps, the angle is given relative to the 
  center of the pattern not the individual shapes. 
  In those cases, you will need to check the 
  'Alternate Angle' property on the properties page 
  for the specific trap so the angle is given relative 
  to the individual shapes.
  
global:
  
  const Complex angleShift = SectorAngle / 360
  const Complex percentSolid = 1 - PercentCutout
  
color:
  
  switch (IndexMap) {
    case IndexMapTypes.TrapDwell: index = Sample.TrapDwellRaw
    case IndexMapTypes.TrapIndex: index = Sample.TrapIndexRaw
    case IndexMapTypes.TrapDelta: index = Sample.TrapDeltaRaw
    case IndexMapTypes.Alternate1Index: index = Sample.Alternate1IndexRaw
  }
  color = Colors[WrapIndex(Offset + index, Count)]
  
  if (ApplyCutouts) {
    r = Sectors * Wrap(Sample.TrapAngle - angleShift)
    r = r - Int(r)
    v = Abs(Sample.TrapValue)
    
    compositeHeight = IIf(r < percentSolid, v, Blend(0.75, 0.25, v))
  } else {
    compositeHeight = Sample.TrapValue
  }
  
properties:
  
  divider {
    caption = "Index Map"
  }
  enum IndexMapTypes {
    TrapDwell, "Trap Dwell"
    TrapIndex, "Trap Index"
    TrapDelta, "Trap Delta"
    Alternate1Index , "Alternate 1 Index"
  }
  option IndexMap {
    type = IndexMapTypes
    caption = "Index Map"
    details = "Color index map"
    default = IndexMapTypes.TrapDwell
  }
  divider {
    caption = "Color Map"
  }
  option Count {
    type = IntegerEnum(1,32)
    caption = "Count"
    details = "Number of colors to use"
    default = 8
  }
  option Colors {
    type = ColorSet[Count]
    caption = "Colors"
    default = FF00AE, FF2020, FF8200, FFFF00, 00FF6C, 00FFFF, 4040FF, AE60FF
  }
  option Offset {
    type = IntegerEnum(0,31)
    caption = "Offset"
    details = "Offset into set of colors"
    default = 0
  }
  divider {
    caption = "Cutouts (Solid traps only)"
  }
  option ApplyCutouts {
    type = Boolean
    caption = "Active"
    details = "Activate cutout effects"
    default = False
  }
  option Sectors {
    type = IntegerEnum(2,16)
    caption = "Sectors"
    details = "Number of sectors"
    default = 2
    enabled = ApplyCutouts
  }
  option SectorAngle {
    type = Float
    caption = "Angle"
    details = "Starting angle of 1st sector"
    default = 0
    range = [-360,360]
    enabled = ApplyCutouts
  }
  option PercentCutout {
    type = Float
    caption = "Percent Cutout"
    details = "Percent of sector that is cutout"
    range = [0,1]
    default = 0.5
    enabled = ApplyCutouts
  }
]]></Instructions>
        <PropertyValueOverrides>
          <Option Name="IndexMap" Type="EnumOption" Value="IndexMapTypes.TrapDelta" />
          <Option Name="Count" Type="IntegerEnumOption" Value="10" />
          <Option Name="Colors" Type="ColorSetOption" Value="A9A9A9,B2B2B2,BABABA,C3C3C3,CBCBCB,D4D4D4,DCDCDC,E5E5E5,EDEDED,F6F6F6" />
        </PropertyValueOverrides>
      </OrbitTrapController>
    </OrbitTrapControllerArray>
    <SampleDataNormalizer>
      <NormalizerDescriptor />
    </SampleDataNormalizer>
    <SampleDataNormalizer>
      <NormalizerDescriptor />
    </SampleDataNormalizer>
    <SampleDataNormalizer>
      <NormalizerDescriptor />
    </SampleDataNormalizer>
  </OrbitTrapInfo>
  <TransformationArray>
    <Transformation Id="7fc4bc2c-b961-4431-b074-277d7b29842b" Title="Identity">
      <Instructions><![CDATA[]]></Instructions>
      <PropertyValueOverrides />
    </Transformation>
  </TransformationArray>
  <TriangleMetricSet>
    <Instructions><![CDATA[metric = p1]]></Instructions>
    <TriangleMetric />
    <TriangleMetric />
    <TriangleMetric />
  </TriangleMetricSet>
</Fractal>