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Tutoring abilities

What happens if you paint one wing of a butterfly on paper, fold it in half, and open it up?

Both sides look exactly the same! The fold down the middle is called a **line of symmetry** or a **mirror line**.

<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="aspect-ratio" content="4:3">
<title>Lines of Symmetry</title>
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: system-ui, -apple-system, sans-serif;
    background: #f8f9fa;
    color: #1e2945;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: space-between;
    min-height: 100%;
    padding: 12px;
    user-select: none;
    -webkit-user-select: none;
  }
  .controls {
    display: flex;
    gap: 8px;
    flex-wrap: wrap;
    justify-content: center;
    width: 100%;
    max-width: 440px;
  }
  .btn-group {
    display: flex;
    background: #e9ecef;
    padding: 3px;
    border-radius: 10px;
    gap: 3px;
  }
  .tab-btn {
    border: none;
    background: transparent;
    padding: 6px 12px;
    font-size: 13px;
    font-weight: 600;
    color: #6b7280;
    border-radius: 8px;
    cursor: pointer;
    transition: all 0.2s ease;
    display: flex;
    align-items: center;
    gap: 4px;
  }
  .tab-btn.active {
    background: #ffffff;
    color: #1e2945;
    box-shadow: 0 2px 4px rgba(0,0,0,0.08);
  }
  .stage {
    position: relative;
    width: 100%;
    max-width: 320px;
    height: 240px;
    display: flex;
    align-items: center;
    justify-content: center;
    margin: 8px 0;
  }
  svg {
    width: 100%;
    height: 100%;
    overflow: visible;
  }
  .status-badge {
    position: absolute;
    bottom: 4px;
    display: flex;
    align-items: center;
    gap: 6px;
    font-size: 13px;
    font-weight: 700;
    padding: 4px 10px;
    border-radius: 20px;
    background: #ffffff;
    box-shadow: 0 2px 8px rgba(0,0,0,0.08);
    transition: all 0.2s ease;
  }
  .status-badge.valid { color: #059669; }
  .status-badge.invalid { color: #dc2626; }
  .fold-btn {
    border: 2px solid #22b7ff;
    background: #ffffff;
    color: #0284c7;
    font-size: 13px;
    font-weight: 700;
    padding: 6px 16px;
    border-radius: 20px;
    cursor: pointer;
    display: flex;
    align-items: center;
    gap: 6px;
    transition: all 0.2s ease;
  }
  .fold-btn:active {
    background: #22b7ff;
    color: #ffffff;
  }
  .shape-btn {
    border: 1px solid #e5e7eb;
    background: #ffffff;
    padding: 6px 10px;
    border-radius: 8px;
    font-size: 13px;
    font-weight: 600;
    color: #4b5563;
    cursor: pointer;
    display: flex;
    align-items: center;
    gap: 4px;
    transition: all 0.2s ease;
  }
  .shape-btn.active {
    border-color: #22b7ff;
    background: #e0f2fe;
    color: #0369a1;
  }
</style>
</head>
<body>

  
  <div class="controls">
    <div class="btn-group">
      <button class="tab-btn active" data-line="vertical">↕ Vertical</button>
      <button class="tab-btn" data-line="horizontal">↔ Horizontal</button>
      <button class="tab-btn" data-line="diagonal">⤢ Diagonal</button>
    </div>
  </div>

  
  <div class="stage">
    <svg id="svgCanvas" viewBox="-120 -120 240 240">
      <defs>
        
        <linearGradient id="baseGrad" x1="0%" y1="0%" x2="100%" y2="100%">
          <stop offset="0%" stop-color="#bae6fd" />
          <stop offset="100%" stop-color="#7dd3fc" />
        </linearGradient>
        <linearGradient id="foldGrad" x1="0%" y1="0%" x2="100%" y2="100%">
          <stop offset="0%" stop-color="#f472b6" stop-opacity="0.8" />
          <stop offset="100%" stop-color="#ec4899" stop-opacity="0.8" />
        </linearGradient>
      </defs>

      
      <g id="baseGroup"></g>

      
      <g id="foldGroup" style="transition: transform 0.6s cubic-bezier(0.4, 0, 0.2, 1); transform-origin: 0px 0px;">
        <path id="foldShape" fill="url(#foldGrad)" stroke="#db2777" stroke-width="2" stroke-dasharray="3 3"/>
      </g>

      
      <g id="mirrorLineGroup">
        <line id="mirrorLine" x1="0" y1="-105" x2="0" y2="105" stroke="#ff5376" stroke-width="2.5" stroke-dasharray="5 4" stroke-linecap="round"/>
      </g>
    </svg>

    <div id="statusBadge" class="status-badge valid">
      <span id="statusIcon">✓</span>
      <span id="statusText">Line of Symmetry</span>
    </div>
  </div>

  
  <div class="controls">
    <button id="foldToggle" class="fold-btn">
      <span id="foldIcon">🗂️</span> <span id="foldLabel">Fold</span>
    </button>
  </div>

  <div class="controls">
    <div class="btn-group">
      <button class="shape-btn active" data-shape="square">🟩 Square</button>
      <button class="shape-btn" data-shape="rectangle">▭ Rectangle</button>
      <button class="shape-btn" data-shape="triangle">🔺 Triangle</button>
      <button class="shape-btn" data-shape="heart">💙 Heart</button>
    </div>
  </div>

<script>
  const SHAPES = {
    square: {
      path: "M -60,-60 L 60,-60 L 60,60 L -60,60 Z",
      symmetry: { vertical: true, horizontal: true, diagonal: true },
      half: {
        vertical: "M 0,-60 L 60,-60 L 60,60 L 0,60 Z",
        horizontal: "M -60,0 L 60,0 L 60,60 L -60,60 Z",
        diagonal: "M -60,-60 L 60,60 L -60,60 Z"
      }
    },
    rectangle: {
      path: "M -85,-50 L 85,-50 L 85,50 L -85,50 Z",
      symmetry: { vertical: true, horizontal: true, diagonal: false },
      half: {
        vertical: "M 0,-50 L 85,-50 L 85,50 L 0,50 Z",
        horizontal: "M -85,0 L 85,0 L 85,50 L -85,50 Z",
        diagonal: "M -85,-50 L 85,50 L -85,50 Z"
      }
    },
    triangle: {
      path: "M 0,-70 L 70,60 L -70,60 Z",
      symmetry: { vertical: true, horizontal: false, diagonal: false },
      half: {
        vertical: "M 0,-70 L 70,60 L 0,60 Z",
        horizontal: "M -35,-5 L 35,-5 L 70,60 L -70,60 Z",
        diagonal: "M -70,60 L 0,-70 L 70,60 Z"
      }
    },
    heart: {
      path: "M 0,65 C -70,15 -75,-55 -35,-65 C -10,-70 0,-40 0,-30 C 0,-40 10,-70 35,-65 C 75,-55 70,15 0,65 Z",
      symmetry: { vertical: true, horizontal: false, diagonal: false },
      half: {
        vertical: "M 0,-30 C 0,-40 10,-70 35,-65 C 75,-55 70,15 0,65 Z",
        horizontal: "M -55,0 C -30,40 0,65 0,65 C 0,65 30,40 55,0 Z",
        diagonal: "M 0,65 C -70,15 -75,-55 -35,-65 Z"
      }
    }
  };

  let currentShape = 'square';
  let currentLine = 'vertical';
  let isFolded = false;

  const baseGroup = document.getElementById('baseGroup');
  const foldGroup = document.getElementById('foldGroup');
  const foldShape = document.getElementById('foldShape');
  const mirrorLine = document.getElementById('mirrorLine');
  const foldToggle = document.getElementById('foldToggle');
  const foldLabel = document.getElementById('foldLabel');
  const statusBadge = document.getElementById('statusBadge');
  const statusIcon = document.getElementById('statusIcon');
  const statusText = document.getElementById('statusText');

  function update() {
    const s = SHAPES[currentShape];
    const isSym = s.symmetry[currentLine];

    // Base shape SVG
    baseGroup.innerHTML = `<path d="${s.path}" fill="url(#baseGrad)" stroke="#0284c7" stroke-width="2" />`;
    
    // Half shape path for fold animation
    foldShape.setAttribute('d', s.half[currentLine]);

    // Mirror line orientation
    if (currentLine === 'vertical') {
      mirrorLine.setAttribute('x1', 0);
      mirrorLine.setAttribute('y1', -105);
      mirrorLine.setAttribute('x2', 0);
      mirrorLine.setAttribute('y2', 105);
    } else if (currentLine === 'horizontal') {
      mirrorLine.setAttribute('x1', -105);
      mirrorLine.setAttribute('y1', 0);
      mirrorLine.setAttribute('x2', 105);
      mirrorLine.setAttribute('y2', 0);
    } else {
      mirrorLine.setAttribute('x1', -85);
      mirrorLine.setAttribute('y1', -85);
      mirrorLine.setAttribute('x2', 85);
      mirrorLine.setAttribute('y2', 85);
    }

    // Fold transform
    if (isFolded) {
      if (currentLine === 'vertical') {
        foldGroup.style.transform = 'scaleX(-1)';
      } else if (currentLine === 'horizontal') {
        foldGroup.style.transform = 'scaleY(-1)';
      } else {
        // Reflection across diagonal line y = x
        foldGroup.style.transform = 'matrix(0, 1, 1, 0, 0, 0)';
      }
      foldLabel.textContent = 'Unfold';
    } else {
      foldGroup.style.transform = 'none';
      foldLabel.textContent = 'Fold';
    }

    // Status message
    if (isSym) {
      statusBadge.className = 'status-badge valid';
      statusIcon.textContent = '✓';
      statusText.textContent = 'Symmetrical';
    } else {
      statusBadge.className = 'status-badge invalid';
      statusIcon.textContent = '✕';
      statusText.textContent = 'Not Symmetrical';
    }
  }

  // Line buttons
  document.querySelectorAll('.tab-btn').forEach(btn => {
    btn.addEventListener('click', () => {
      document.querySelectorAll('.tab-btn').forEach(b => b.classList.remove('active'));
      btn.classList.add('active');
      currentLine = btn.dataset.line;
      isFolded = false;
      update();
    });
  });

  // Shape buttons
  document.querySelectorAll('.shape-btn').forEach(btn => {
    btn.addEventListener('click', () => {
      document.querySelectorAll('.shape-btn').forEach(b => b.classList.remove('active'));
      btn.classList.add('active');
      currentShape = btn.dataset.shape;
      isFolded = false;
      update();
    });
  });

  // Fold toggle
  foldToggle.addEventListener('click', () => {
    isFolded = !isFolded;
    update();
  });

  // Initial draw
  update();
</script>
</body>
</html>

Can we fold shapes in different directions to find new mirror lines?

A **vertical line** goes straight up and down like a standing tree.

When you fold a shape along a vertical line, the left side lands right on top of the right side.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      -webkit-tap-highlight-color: transparent;
      user-select: none;
    }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100vh;
      padding: 12px;
    }
    .nav {
      display: flex;
      gap: 8px;
      margin-bottom: 16px;
      flex-wrap: wrap;
      justify-content: center;
    }
    .tab-btn {
      background: #ffffff;
      border: 2px solid #e2e8f0;
      padding: 8px 14px;
      border-radius: 20px;
      font-size: 14px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 6px;
      transition: all 0.2s ease;
    }
    .tab-btn.active {
      border-color: #22b7ff;
      color: #0284c7;
      background: #e0f2fe;
    }
    .stage-wrapper {
      position: relative;
      width: 220px;
      height: 220px;
      perspective: 800px;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    .shape-card {
      position: relative;
      width: 200px;
      height: 200px;
      background: #ffffff;
      border-radius: 16px;
      box-shadow: 0 4px 16px rgba(0,0,0,0.06);
      transition: transform 0.4s ease;
    }
    .half-container {
      position: absolute;
      inset: 0;
      overflow: hidden;
      border-radius: 16px;
    }
    .foldable-half {
      position: absolute;
      inset: 0;
      border-radius: 16px;
      transform-style: preserve-3d;
      transition: transform 0.6s cubic-bezier(0.4, 0, 0.2, 1);
      z-index: 5;
    }
    .svg-content {
      width: 200px;
      height: 200px;
      display: block;
    }
    .line-of-symmetry {
      position: absolute;
      background: #22b7ff;
      z-index: 10;
      pointer-events: none;
      transition: all 0.4s ease;
    }
    .controls {
      margin-top: 16px;
      display: flex;
      align-items: center;
      gap: 12px;
    }
    .action-btn {
      background: #22b7ff;
      color: #ffffff;
      border: none;
      padding: 10px 20px;
      border-radius: 20px;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
      box-shadow: 0 2px 8px rgba(34, 183, 255, 0.3);
      transition: opacity 0.2s;
    }
    .action-btn:active {
      opacity: 0.85;
    }
  </style>
</head>
<body>

  <div class="nav">
    <button class="tab-btn active" id="btn-vert">
      <span>│</span> Vertical
    </button>
    <button class="tab-btn" id="btn-horiz">
      <span>─</span> Horizontal
    </button>
    <button class="tab-btn" id="btn-diag">
      <span>╱</span> Diagonal
    </button>
  </div>

  <div class="stage-wrapper">
    <div class="shape-card" id="shapeCard">
      
      <div class="half-container" id="baseContainer">
        <svg class="svg-content" viewBox="0 0 200 200">
          <circle cx="100" cy="100" r="82" fill="#f0f9ff" stroke="#bae6fd" stroke-width="4"/>
          
          <circle cx="55" cy="55" r="16" fill="#38bdf8" opacity="0.8"/>
          <circle cx="145" cy="55" r="16" fill="#38bdf8" opacity="0.8"/>
          <circle cx="55" cy="145" r="16" fill="#38bdf8" opacity="0.8"/>
          <circle cx="145" cy="145" r="16" fill="#38bdf8" opacity="0.8"/>
          <polygon points="100,60 135,100 100,140 65,100" fill="#0284c7" opacity="0.25"/>
          <circle cx="100" cy="100" r="14" fill="#0284c7"/>
        </svg>
      </div>

      
      <div class="foldable-half" id="foldingHalf">
        <svg class="svg-content" viewBox="0 0 200 200">
          <circle cx="100" cy="100" r="82" fill="#e0f2fe" stroke="#38bdf8" stroke-width="4"/>
          <circle cx="55" cy="55" r="16" fill="#0284c7"/>
          <circle cx="145" cy="55" r="16" fill="#0284c7"/>
          <circle cx="55" cy="145" r="16" fill="#0284c7"/>
          <circle cx="145" cy="145" r="16" fill="#0284c7"/>
          <polygon points="100,60 135,100 100,140 65,100" fill="#0284c7" opacity="0.4"/>
          <circle cx="100" cy="100" r="14" fill="#0369a1"/>
        </svg>
      </div>

      
      <div class="line-of-symmetry" id="symLine"></div>
    </div>
  </div>

  <div class="controls">
    <button class="action-btn" id="foldBtn">Fold</button>
  </div>

  <script>
    let mode = 'vert'; // 'vert' | 'horiz' | 'diag'
    let isFolded = false;

    const shapeCard = document.getElementById('shapeCard');
    const foldingHalf = document.getElementById('foldingHalf');
    const baseContainer = document.getElementById('baseContainer');
    const symLine = document.getElementById('symLine');
    const foldBtn = document.getElementById('foldBtn');

    const btnVert = document.getElementById('btn-vert');
    const btnHoriz = document.getElementById('btn-horiz');
    const btnDiag = document.getElementById('btn-diag');

    function updateLayout() {
      isFolded = false;
      foldBtn.textContent = 'Fold';

      if (mode === 'vert') {
        shapeCard.style.transform = 'rotate(0deg)';
        baseContainer.style.clipPath = 'polygon(50% 0%, 100% 0%, 100% 100%, 50% 100%)';
        foldingHalf.style.clipPath = 'polygon(0% 0%, 50% 0%, 50% 100%, 0% 100%)';
        foldingHalf.style.transformOrigin = '50% 50%';
        foldingHalf.style.transform = 'rotateY(0deg)';

        symLine.style.top = '-10px';
        symLine.style.bottom = '-10px';
        symLine.style.left = 'calc(50% - 1.5px)';
        symLine.style.width = '3px';
        symLine.style.height = 'calc(100% + 20px)';
        symLine.style.transform = 'none';
      } else if (mode === 'horiz') {
        shapeCard.style.transform = 'rotate(0deg)';
        baseContainer.style.clipPath = 'polygon(0% 50%, 100% 50%, 100% 100%, 0% 100%)';
        foldingHalf.style.clipPath = 'polygon(0% 0%, 100% 0%, 100% 50%, 0% 50%)';
        foldingHalf.style.transformOrigin = '50% 50%';
        foldingHalf.style.transform = 'rotateX(0deg)';

        symLine.style.left = '-10px';
        symLine.style.right = '-10px';
        symLine.style.top = 'calc(50% - 1.5px)';
        symLine.style.height = '3px';
        symLine.style.width = 'calc(100% + 20px)';
        symLine.style.transform = 'none';
      } else if (mode === 'diag') {
        shapeCard.style.transform = 'rotate(-45deg)';
        baseContainer.style.clipPath = 'polygon(50% 0%, 100% 0%, 100% 100%, 50% 100%)';
        foldingHalf.style.clipPath = 'polygon(0% 0%, 50% 0%, 50% 100%, 0% 100%)';
        foldingHalf.style.transformOrigin = '50% 50%';
        foldingHalf.style.transform = 'rotateY(0deg)';

        symLine.style.top = '-20px';
        symLine.style.bottom = '-20px';
        symLine.style.left = 'calc(50% - 1.5px)';
        symLine.style.width = '3px';
        symLine.style.height = 'calc(100% + 40px)';
        symLine.style.transform = 'none';
      }
    }

    function toggleFold() {
      isFolded = !isFolded;
      foldBtn.textContent = isFolded ? 'Unfold' : 'Fold';

      if (mode === 'vert' || mode === 'diag') {
        foldingHalf.style.transform = isFolded ? 'rotateY(180deg)' : 'rotateY(0deg)';
      } else if (mode === 'horiz') {
        foldingHalf.style.transform = isFolded ? 'rotateX(180deg)' : 'rotateX(0deg)';
      }
    }

    function setMode(newMode) {
      mode = newMode;
      [btnVert, btnHoriz, btnDiag].forEach(b => b.classList.remove('active'));
      if (mode === 'vert') btnVert.classList.add('active');
      if (mode === 'horiz') btnHoriz.classList.add('active');
      if (mode === 'diag') btnDiag.classList.add('active');
      updateLayout();
    }

    btnVert.addEventListener('click', () => setMode('vert'));
    btnHoriz.addEventListener('click', () => setMode('horiz'));
    btnDiag.addEventListener('click', () => setMode('diag'));
    foldBtn.addEventListener('click', toggleFold);

    updateLayout();
  </script>
</body>
</html>

What if we fold a shape across from side to side instead?

A **horizontal line** goes straight across from left to right, like the flat ground or the edge of a table.

When you fold along a horizontal line, the top half folds down to match the bottom half.

<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="aspect-ratio" content="4:3">
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: system-ui, -apple-system, sans-serif;
    background: #f8f9fa;
    color: #1e2945;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: space-between;
    min-height: 100%;
    padding: 16px;
    user-select: none;
    -webkit-user-select: none;
  }

  .nav-tabs {
    display: flex;
    gap: 8px;
    width: 100%;
    max-width: 380px;
  }

  .tab-btn {
    flex: 1;
    padding: 8px 4px;
    border: none;
    background: #ffffff;
    color: #6b7280;
    font-size: 14px;
    font-weight: 600;
    border-radius: 8px;
    cursor: pointer;
    box-shadow: 0 1px 3px rgba(0,0,0,0.08);
    transition: background 0.2s, color 0.2s, box-shadow 0.2s;
    text-align: center;
  }

  .tab-btn.active {
    background: #1e2945;
    color: #ffffff;
    box-shadow: 0 3px 8px rgba(30,41,69,0.25);
  }

  /* Stage viewport */
  .stage {
    position: relative;
    width: 220px;
    height: 220px;
    display: flex;
    align-items: center;
    justify-content: center;
    perspective: 800px;
    margin: 12px 0;
  }

  /* Rotating frame to align any symmetry axis horizontally for pure 3D fold */
  .rotator {
    position: absolute;
    width: 180px;
    height: 180px;
    transform-style: preserve-3d;
    transition: transform 0.4s ease;
  }

  /* Base square inside rotator */
  .square-base {
    position: absolute;
    width: 100%;
    height: 100%;
    border-radius: 16px;
    background: #e0f2fe;
    box-shadow: 0 4px 14px rgba(0,0,0,0.06);
    overflow: hidden;
  }

  /* Fixed bottom half */
  .half-fixed {
    position: absolute;
    top: 50%;
    left: 0;
    width: 100%;
    height: 50%;
    background: #bae6fd;
    border-radius: 0 0 16px 16px;
  }

  /* Foldable top half */
  .fold-hinge {
    position: absolute;
    top: 0;
    left: 0;
    width: 100%;
    height: 50%;
    transform-origin: bottom center;
    transform-style: preserve-3d;
    transition: transform 0.05s linear;
  }

  .fold-flap {
    width: 100%;
    height: 100%;
    background: #7dd3fc;
    border-radius: 16px 16px 0 0;
    backface-visibility: visible;
    display: flex;
    align-items: center;
    justify-content: center;
  }

  /* Decorative symmetrical icons */
  .motif {
    position: absolute;
    font-size: 28px;
    opacity: 0.85;
    pointer-events: none;
  }

  .motif.top { top: 22px; left: 76px; }
  .motif.bottom { bottom: 22px; left: 76px; }

  /* Green symmetry line */
  .symmetry-line {
    position: absolute;
    top: 50%;
    left: -12px;
    width: calc(100% + 24px);
    height: 3px;
    border-top: 3px dashed #2bd980;
    transform: translateY(-50%);
    pointer-events: none;
    z-index: 10;
  }

  /* Fold control slider */
  .controls {
    width: 100%;
    max-width: 340px;
    display: flex;
    flex-direction: column;
    gap: 8px;
    align-items: center;
  }

  .slider-row {
    width: 100%;
    display: flex;
    align-items: center;
    gap: 12px;
  }

  .slider-label {
    font-size: 13px;
    font-weight: 600;
    color: #6b7280;
    min-width: 36px;
  }

  input[type="range"] {
    flex: 1;
    -webkit-appearance: none;
    appearance: none;
    height: 6px;
    border-radius: 3px;
    background: #e2e8f0;
    outline: none;
    cursor: pointer;
  }

  input[type="range"]::-webkit-slider-thumb {
    -webkit-appearance: none;
    appearance: none;
    width: 20px;
    height: 20px;
    border-radius: 50%;
    background: #2bd980;
    border: 2px solid #ffffff;
    box-shadow: 0 2px 5px rgba(0,0,0,0.15);
    cursor: pointer;
  }

  .btn-fold {
    background: #ffffff;
    border: 1px solid #e2e8f0;
    color: #1e2945;
    font-size: 13px;
    font-weight: 600;
    padding: 6px 14px;
    border-radius: 20px;
    cursor: pointer;
    box-shadow: 0 1px 3px rgba(0,0,0,0.05);
    display: flex;
    align-items: center;
    gap: 6px;
  }
</style>
</head>
<body>

  
  <div class="nav-tabs">
    <button class="tab-btn active" data-mode="vertical">Vertical</button>
    <button class="tab-btn" data-mode="horizontal">Horizontal</button>
    <button class="tab-btn" data-mode="diagonal">Diagonal</button>
  </div>

  
  <div class="stage">
    <div class="rotator" id="rotator" style="transform: rotate(90deg);">
      <div class="square-base">
        <div class="half-fixed"></div>
        <div class="fold-hinge" id="hinge">
          <div class="fold-flap"></div>
        </div>
      </div>
      <div class="symmetry-line"></div>
      <span class="motif top" id="mTop">✦</span>
      <span class="motif bottom" id="mBottom">✦</span>
    </div>
  </div>

  
  <div class="controls">
    <div class="slider-row">
      <span class="slider-label">Open</span>
      <input type="range" id="foldRange" min="0" max="180" value="0">
      <span class="slider-label" style="text-align: right;">Fold</span>
    </div>
    <button class="btn-fold" id="playBtn">▶ Auto Fold</button>
  </div>

<script>
  const rotator = document.getElementById('rotator');
  const hinge = document.getElementById('hinge');
  const range = document.getElementById('foldRange');
  const playBtn = document.getElementById('playBtn');
  const tabs = document.querySelectorAll('.tab-btn');
  const mTop = document.getElementById('mTop');
  const mBottom = document.getElementById('mBottom');

  const angles = {
    vertical: 90,
    horizontal: 0,
    diagonal: 45
  };

  let currentMode = 'vertical';
  let isPlaying = false;
  let animId = null;
  let foldProgress = 0;
  let foldDirection = 1;

  function updateFold(deg) {
    range.value = deg;
    hinge.style.transform = `rotateX(${deg}deg)`;
  }

  function setMode(mode) {
    currentMode = mode;
    tabs.forEach(t => t.classList.toggle('active', t.dataset.mode === mode));
    
    // Rotate stage so the selected symmetry axis aligns with the hinge
    const rotDeg = angles[mode];
    rotator.style.transform = `rotate(${rotDeg}deg)`;
    
    // Counter-rotate motifs so they stay upright
    mTop.style.transform = `rotate(${-rotDeg}deg)`;
    mBottom.style.transform = `rotate(${-rotDeg}deg)`;
  }

  tabs.forEach(tab => {
    tab.addEventListener('click', () => {
      setMode(tab.dataset.mode);
    });
  });

  range.addEventListener('input', (e) => {
    stopAuto();
    updateFold(e.target.value);
  });

  function autoStep() {
    if (!isPlaying) return;
    foldProgress += 1.8 * foldDirection;
    if (foldProgress >= 180) {
      foldProgress = 180;
      foldDirection = -1;
    } else if (foldProgress <= 0) {
      foldProgress = 0;
      foldDirection = 1;
    }
    updateFold(foldProgress);
    animId = requestAnimationFrame(autoStep);
  }

  function startAuto() {
    isPlaying = true;
    playBtn.textContent = '⏸ Pause';
    foldProgress = parseFloat(range.value);
    animId = requestAnimationFrame(autoStep);
  }

  function stopAuto() {
    isPlaying = false;
    playBtn.textContent = '▶ Auto Fold';
    if (animId) cancelAnimationFrame(animId);
  }

  playBtn.addEventListener('click', () => {
    if (isPlaying) stopAuto();
    else startAuto();
  });

  // Initialize
  setMode('vertical');
</script>
</body>
</html>

Can a mirror line also tilt like a slide?

A **diagonal line** is a slanted line that connects opposite corners of a shape.

A square has diagonal mirror lines: when you fold it corner to corner, it makes two matching triangles.

<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="aspect-ratio" content="4:3">
<title>Lines of Symmetry in a Square</title>
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
    background: #f8f9fa;
    color: #1e2945;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    padding: 12px;
    min-height: 100%;
  }

  .nav-bar {
    display: flex;
    flex-wrap: wrap;
    gap: 6px;
    justify-content: center;
    margin-bottom: 12px;
    width: 100%;
    max-width: 420px;
  }

  .tab-btn {
    appearance: none;
    border: 1px solid #e2e8f0;
    background: #ffffff;
    color: #475569;
    font-size: 13px;
    font-weight: 600;
    padding: 6px 10px;
    border-radius: 8px;
    cursor: pointer;
    transition: background 0.2s, color 0.2s, border-color 0.2s;
  }

  .tab-btn:hover {
    background: #f1f5f9;
  }

  .tab-btn.active {
    background: #9055ff;
    border-color: #9055ff;
    color: #ffffff;
  }

  .stage {
    position: relative;
    width: 240px;
    height: 240px;
    background: #ffffff;
    border-radius: 16px;
    box-shadow: 0 4px 16px rgba(0, 0, 0, 0.06);
    display: flex;
    align-items: center;
    justify-content: center;
  }

  svg {
    width: 100%;
    height: 100%;
    overflow: visible;
  }

  .sym-line {
    stroke: #9055ff;
    stroke-width: 2.5;
    stroke-dasharray: 6 5;
    stroke-linecap: round;
    opacity: 0;
    transition: opacity 0.3s ease;
  }

  .sym-line.visible {
    opacity: 1;
  }

  .sym-fill {
    opacity: 0;
    transition: opacity 0.3s ease;
  }

  .sym-fill.visible {
    opacity: 0.15;
  }

  .caption-area {
    margin-top: 12px;
    height: 24px;
    display: flex;
    align-items: center;
    justify-content: center;
    font-size: 15px;
    font-weight: 600;
    color: #1e2945;
  }

  .auto-controls {
    margin-top: 8px;
    display: flex;
    align-items: center;
    gap: 8px;
  }

  .ctrl-btn {
    border: none;
    background: #ffffff;
    color: #64748b;
    border-radius: 20px;
    width: 28px;
    height: 28px;
    display: flex;
    align-items: center;
    justify-content: center;
    cursor: pointer;
    font-size: 12px;
    box-shadow: 0 2px 6px rgba(0,0,0,0.06);
  }
  .ctrl-btn:hover {
    color: #1e2945;
  }
</style>
</head>
<body>

<div class="nav-bar">
  <button class="tab-btn active" data-mode="vert">Vertical</button>
  <button class="tab-btn" data-mode="horiz">Horizontal</button>
  <button class="tab-btn" data-mode="diag1">Diagonal ⤢</button>
  <button class="tab-btn" data-mode="diag2">Diagonal ⤡</button>
  <button class="tab-btn" data-mode="all">All 4 Lines</button>
</div>

<div class="stage">
  <svg viewBox="0 0 240 240">
    
    <rect x="30" y="30" width="180" height="180" rx="4" fill="#f8fafc" stroke="#cbd5e1" stroke-width="2.5" />

    
    
    <rect id="fill-vert" class="sym-fill visible" x="30" y="30" width="90" height="180" fill="#9055ff" />
    
    <rect id="fill-horiz" class="sym-fill" x="30" y="30" width="180" height="90" fill="#9055ff" />
    
    <polygon id="fill-diag1" class="sym-fill" points="30,30 210,30 30,210" fill="#9055ff" />
    
    <polygon id="fill-diag2" class="sym-fill" points="30,30 210,30 210,210" fill="#9055ff" />

    
    <line id="line-vert" class="sym-line visible" x1="120" y1="12" x2="120" y2="228" />
    <line id="line-horiz" class="sym-line" x1="12" y1="120" x2="228" y2="120" />
    <line id="line-diag2" class="sym-line" x1="18" y1="18" x2="222" y2="222" />
    <line id="line-diag1" class="sym-line" x1="222" y1="18" x2="18" y2="222" />

    
    <circle cx="120" cy="120" r="3.5" fill="#9055ff" />
  </svg>
</div>

<div class="caption-area" id="caption">Vertical Line</div>

<div class="auto-controls">
  <button class="ctrl-btn" id="playBtn" title="Play/Pause">⏸</button>
</div>

<script>
  const modes = [
    { id: 'vert', label: 'Vertical Line', lines: ['line-vert'], fills: ['fill-vert'] },
    { id: 'horiz', label: 'Horizontal Line', lines: ['line-horiz'], fills: ['fill-horiz'] },
    { id: 'diag1', label: 'Diagonal Line', lines: ['line-diag1'], fills: ['fill-diag1'] },
    { id: 'diag2', label: 'Diagonal Line', lines: ['line-diag2'], fills: ['fill-diag2'] },
    { id: 'all', label: '4 Lines of Symmetry', lines: ['line-vert', 'line-horiz', 'line-diag1', 'line-diag2'], fills: [] }
  ];

  let currentIndex = 0;
  let timer = null;
  let isPlaying = true;

  const buttons = document.querySelectorAll('.tab-btn');
  const lines = document.querySelectorAll('.sym-line');
  const fills = document.querySelectorAll('.sym-fill');
  const caption = document.getElementById('caption');
  const playBtn = document.getElementById('playBtn');

  function setMode(index) {
    currentIndex = index;
    const mode = modes[index];

    buttons.forEach((btn, i) => {
      btn.classList.toggle('active', i === index);
    });

    lines.forEach(line => {
      line.classList.toggle('visible', mode.lines.includes(line.id));
    });

    fills.forEach(fill => {
      fill.classList.toggle('visible', mode.fills.includes(fill.id));
    });

    caption.textContent = mode.label;
  }

  function nextMode() {
    setMode((currentIndex + 1) % modes.length);
  }

  function startAuto() {
    if (timer) clearInterval(timer);
    timer = setInterval(nextMode, 3000);
    isPlaying = true;
    playBtn.textContent = '⏸';
  }

  function stopAuto() {
    if (timer) clearInterval(timer);
    timer = null;
    isPlaying = false;
    playBtn.textContent = '▶';
  }

  buttons.forEach((btn, idx) => {
    btn.addEventListener('click', () => {
      stopAuto();
      setMode(idx);
    });
  });

  playBtn.addEventListener('click', () => {
    if (isPlaying) {
      stopAuto();
    } else {
      nextMode();
      startAuto();
    }
  });

  startAuto();
</script>
</body>
</html>

Does every shape fold neatly on a diagonal line?

A rectangle has vertical and horizontal mirror lines, but folding it corner to corner does **not** match up.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Lines of Symmetry</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      user-select: none;
      -webkit-user-select: none;
    }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
    }

    .main-wrap {
      width: 100%;
      max-width: 420px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }

    /* Shape selector tabs */
    .selector-row {
      display: flex;
      gap: 8px;
      width: 100%;
      justify-content: center;
    }
    .tab-btn {
      flex: 1;
      padding: 8px 12px;
      border: 2px solid #e2e8f0;
      background: #ffffff;
      color: #64748b;
      font-weight: 600;
      font-size: 14px;
      border-radius: 10px;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      transition: all 0.2s ease;
    }
    .tab-btn.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
      box-shadow: 0 4px 10px rgba(34, 183, 255, 0.25);
    }

    /* Stage box */
    .stage {
      width: 100%;
      height: 240px;
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e2e8f0;
      position: relative;
      display: flex;
      align-items: center;
      justify-content: center;
      overflow: hidden;
    }

    svg {
      width: 100%;
      height: 100%;
    }

    /* Line selector pills */
    .line-buttons {
      display: flex;
      gap: 8px;
      width: 100%;
    }
    .line-btn {
      flex: 1;
      padding: 10px 6px;
      border: 2px solid #e2e8f0;
      background: #ffffff;
      color: #1e2945;
      font-weight: 600;
      font-size: 13px;
      border-radius: 10px;
      cursor: pointer;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 4px;
      transition: all 0.2s ease;
    }
    .line-btn.active {
      border-color: #1e2945;
      background: #f1f5f9;
    }
    .line-icon {
      font-size: 16px;
      line-height: 1;
    }

    /* Result badge */
    .badge {
      position: absolute;
      top: 10px;
      right: 10px;
      padding: 4px 10px;
      border-radius: 20px;
      font-size: 12px;
      font-weight: 700;
      display: flex;
      align-items: center;
      gap: 4px;
      transition: all 0.25s ease;
    }
    .badge.match {
      background: #e6fcf0;
      color: #059669;
      border: 1px solid #2bd980;
    }
    .badge.no-match {
      background: #fef2f2;
      color: #e11d48;
      border: 1px solid #ff5376;
    }

    /* Animated elements */
    .sym-line {
      stroke: #22b7ff;
      stroke-width: 2.5;
      stroke-dasharray: 6, 4;
      transition: all 0.3s ease;
    }
    .fold-half {
      transition: transform 0.4s ease, opacity 0.4s ease;
      transform-origin: center;
    }
  </style>
</head>
<body>

  <div class="main-wrap">
    
    <div class="selector-row">
      <button class="tab-btn active" id="btn-rect" onclick="setShape('rect')">
        Rectangle
      </button>
      <button class="tab-btn" id="btn-sq" onclick="setShape('square')">
        Square
      </button>
    </div>

    
    <div class="stage">
      <div class="badge match" id="status-badge">✓ Matches</div>
      <svg viewBox="0 0 320 220" id="sym-svg">
        
      </svg>
    </div>

    
    <div class="line-buttons">
      <button class="line-btn active" id="btn-vert" onclick="setLine('vertical')">
        <span class="line-icon">┃</span>
        Vertical
      </button>
      <button class="line-btn" id="btn-horiz" onclick="setLine('horizontal')">
        <span class="line-icon">━</span>
        Horizontal
      </button>
      <button class="line-btn" id="btn-diag" onclick="setLine('diagonal')">
        <span class="line-icon">╱</span>
        Diagonal
      </button>
    </div>
  </div>

  <script>
    let currentShape = 'rect';
    let currentLine = 'vertical';

    function setShape(shape) {
      currentShape = shape;
      document.getElementById('btn-rect').classList.toggle('active', shape === 'rect');
      document.getElementById('btn-sq').classList.toggle('active', shape === 'square');
      render();
    }

    function setLine(line) {
      currentLine = line;
      document.getElementById('btn-vert').classList.toggle('active', line === 'vertical');
      document.getElementById('btn-horiz').classList.toggle('active', line === 'horizontal');
      document.getElementById('btn-diag').classList.toggle('active', line === 'diagonal');
      render();
    }

    function render() {
      const svg = document.getElementById('sym-svg');
      const badge = document.getElementById('status-badge');

      const isRect = currentShape === 'rect';
      const w = isRect ? 180 : 130;
      const h = 130;
      const cx = 160;
      const cy = 110;
      const x1 = cx - w / 2;
      const y1 = cy - h / 2;
      const x2 = cx + w / 2;
      const y2 = cy + h / 2;

      let lineX1, lineY1, lineX2, lineY2;
      let matches = true;
      let foldPolygon = '';
      let foldTransform = '';

      if (currentLine === 'vertical') {
        lineX1 = cx; lineY1 = y1 - 25;
        lineX2 = cx; lineY2 = y2 + 25;
        matches = true;
        // Fold left half over right
        foldPolygon = `${x1},${y1} ${cx},${y1} ${cx},${y2} ${x1},${y2}`;
      } else if (currentLine === 'horizontal') {
        lineX1 = x1 - 25; lineY1 = cy;
        lineX2 = x2 + 25; lineY2 = cy;
        matches = true;
        // Fold top half over bottom
        foldPolygon = `${x1},${y1} ${x2},${y1} ${x2},${cy} ${x1},${cy}`;
      } else {
        // Diagonal: bottom-left to top-right
        lineX1 = x1 - 18; lineY1 = y2 + 18;
        lineX2 = x2 + 18; lineY2 = y1 - 18;
        
        if (isRect) {
          matches = false;
          // Fold bottom-right triangle across diagonal (x1,y2 to x2,y1)
          // Reflected corner of (x2, y2) across diagonal of rectangle
          // For rectangle w=180, h=130, diagonal angle != 45 deg, corner flips outside
          const dx = w;
          const dy = -h;
          const lenSq = dx * dx + dy * dy;
          // Vector from (x1, y2) to (x2, y2) is (w, 0)
          // Reflection of (w, 0) across (w, -h):
          const dot = w * dx; // w * w
          const projX = (dot / lenSq) * dx;
          const projY = (dot / lenSq) * dy;
          const perpX = w - projX;
          const perpY = 0 - projY;
          const refX = x1 + (projX - perpX);
          const refY = y2 + (projY - perpY);
          
          foldPolygon = `${x1},${y2} ${x2},${y1} ${refX.toFixed(1)},${refY.toFixed(1)}`;
        } else {
          matches = true;
          foldPolygon = `${x1},${y2} ${x2},${y1} ${x1},${y1}`;
        }
      }

      if (matches) {
        badge.className = 'badge match';
        badge.innerHTML = '✓ Symmetrical';
      } else {
        badge.className = 'badge no-match';
        badge.innerHTML = '✕ Not Symmetrical';
      }

      let extraContent = '';
      if (!matches) {
        // Show folded flap sticking out
        extraContent = `
          
          <polygon points="${foldPolygon}" fill="#ff5376" fill-opacity="0.25" stroke="#ff5376" stroke-width="2" stroke-dasharray="4,4" />
          <text x="${cx}" y="${y2 + 28}" text-anchor="middle" fill="#ff5376" font-size="12" font-weight="700">Does not match up</text>
        `;
      } else {
        extraContent = `
          
          <text x="${cx}" y="${y2 + 28}" text-anchor="middle" fill="#059669" font-size="12" font-weight="600">Halves match exactly</text>
        `;
      }

      svg.innerHTML = `
        <defs>
          <linearGradient id="shapeGrad" x1="0%" y1="0%" x2="100%" y2="100%">
            <stop offset="0%" stop-color="#f0f7ff" />
            <stop offset="100%" stop-color="#e0f0fe" />
          </linearGradient>
        </defs>

        
        <rect x="${x1}" y="${y1}" width="${w}" height="${h}" rx="4"
              fill="url(#shapeGrad)" stroke="#94a3b8" stroke-width="2" />

        ${extraContent}

        
        <line x1="${lineX1}" y1="${lineY1}" x2="${lineX2}" y2="${lineY2}" class="sym-line" />
        
        
        <circle cx="${lineX1}" cy="${lineY1}" r="3" fill="#22b7ff" />
        <circle cx="${lineX2}" cy="${lineY2}" r="3" fill="#22b7ff" />
      `;
    }

    // Initial render
    render();
  </script>
</body>
</html>

Next time you look at a shape, test it with your fingers: can you fold it up-and-down, side-to-side, or on a slant to make the halves match?

math

Identifying Vertical, Horizontal, and Diagonal Lines of Symmetry. Identify and draw vertical, horizontal, and slanted mirror lines of symmetry on geometric figures.

What is Symmetry? - 2D Shapes and Lines of Symmetry

Identifying Vertical, Horizontal, and Diagonal Lines of Symmetry

Draw and Identify Lines of Symmetry in 2D Figures

Teaching line symmetry effectively across early-to-primary progression requires a multi-sensory concrete-representational-abstract (CRA) approach. Learners should begin with hands-on folding of paper shapes and placing acrylic safety mirrors along potential lines of symmetry before transitioning to drawing dashed mirror lines on dot grids or geometric diagrams. Pay particular attention to differentiating between area equivalence and reflectional symmetry, specifically addressing the rectangle diagonal misconception.

Identifying Vertical, Horizontal, and Diagonal Lines of Symmetry

Up and Down: Vertical Lines

Side to Side: Horizontal Lines