Scripts/newton-basins.ps1
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# Unique Concept: Newton's method visualization for finding polynomial roots in the complex plane. # Colors show which root each point converges to, creating beautiful basin boundaries. $ErrorActionPreference = 'Stop' $esc = [char]27 $reset = "$esc[0m" function Convert-HsvToRgb { param( [double]$Hue, [double]$Saturation, [double]$Value ) $h = ($Hue % 1) * 6 $sector = [math]::Floor($h) $fraction = $h - $sector $p = $Value * (1 - $Saturation) $q = $Value * (1 - $fraction * $Saturation) $t = $Value * (1 - (1 - $fraction) * $Saturation) switch ($sector) { 0 { $r = $Value; $g = $t; $b = $p } 1 { $r = $q; $g = $Value; $b = $p } 2 { $r = $p; $g = $Value; $b = $t } 3 { $r = $p; $g = $q; $b = $Value } 4 { $r = $t; $g = $p; $b = $Value } default { $r = $Value; $g = $p; $b = $q } } return @([int][math]::Round($r * 255), [int][math]::Round($g * 255), [int][math]::Round($b * 255)) } function Clamp { param([double]$Value, [double]$Min, [double]$Max) if ($Value -lt $Min) { return $Min } if ($Value -gt $Max) { return $Max } return $Value } # Newton's method for z^3 - 1 = 0 (simpler, prettier basins) # f(z) = z^3 - 1, f'(z) = 3*z^2 function Get-NewtonIteration { param( [double]$Zr, [double]$Zi, [int]$MaxIter ) $x = $Zr $y = $Zi for ($iter = 0; $iter -lt $MaxIter; $iter++) { # Calculate z^2 $z2r = $x * $x - $y * $y $z2i = 2 * $x * $y # Calculate z^3 = z * z^2 $z3r = $x * $z2r - $y * $z2i $z3i = $x * $z2i + $y * $z2r # f(z) = z^3 - 1 $fr = $z3r - 1.0 $fi = $z3i # f'(z) = 3*z^2 $fpr = 3 * $z2r $fpi = 3 * $z2i # Avoid division by zero $denom = $fpr * $fpr + $fpi * $fpi if ($denom -lt 1e-10) { break } # z = z - f(z)/f'(z) using complex division $divr = ($fr * $fpr + $fi * $fpi) / $denom $divi = ($fi * $fpr - $fr * $fpi) / $denom $xNew = $x - $divr $yNew = $y - $divi # Check convergence $diff = [math]::Sqrt(($xNew - $x) * ($xNew - $x) + ($yNew - $y) * ($yNew - $y)) if ($diff -lt 1e-6) { return @{ X = $xNew Y = $yNew Iter = $iter } } $x = $xNew $y = $yNew } return @{ X = $x Y = $y Iter = $MaxIter } } # Calculate the 3 roots of z^3 = 1 $roots = @() for ($k = 0; $k -lt 3; $k++) { $angle = 2 * [math]::PI * $k / 3.0 $roots += @{ X = [math]::Cos($angle) Y = [math]::Sin($angle) Hue = $k / 3.0 } } $width = 120 $height = 40 $zoom = 2.0 for ($row = 0; $row -lt $height; $row++) { $sb = [System.Text.StringBuilder]::new() for ($col = 0; $col -lt $width; $col++) { $zr = ($col - $width / 2.0) / ($width / 2.0) * $zoom * 1.5 $zi = ($row - $height / 2.0) / ($height / 2.0) * $zoom $result = Get-NewtonIteration -Zr $zr -Zi $zi -MaxIter 50 # Find which root we converged to $closestRoot = 0 $minDist = [double]::MaxValue for ($i = 0; $i -lt $roots.Count; $i++) { $dx = $result.X - $roots[$i].X $dy = $result.Y - $roots[$i].Y $dist = $dx * $dx + $dy * $dy if ($dist -lt $minDist) { $minDist = $dist $closestRoot = $i } } $hue = $roots[$closestRoot].Hue # Brightness based on convergence speed $normalized = 1.0 - ($result.Iter / 50.0) $saturation = Clamp -Value (0.65 + 0.3 * $normalized) -Min 0 -Max 1 $value = Clamp -Value (0.3 + 0.65 * [math]::Pow($normalized, 0.7)) -Min 0.25 -Max 1.0 $rgb = Convert-HsvToRgb -Hue $hue -Saturation $saturation -Value $value # Symbol by convergence speed if ($result.Iter -lt 5) { $symbol = '█' } elseif ($result.Iter -lt 10) { $symbol = '▓' } elseif ($result.Iter -lt 20) { $symbol = '▒' } elseif ($result.Iter -lt 35) { $symbol = '░' } else { $symbol = '·' } $null = $sb.Append("$esc[38;2;$($rgb[0]);$($rgb[1]);$($rgb[2])m$symbol") } Write-Host ($sb.ToString() + $reset) } |