Embedding Explorer: add vector lines from origin + 3D grid axes

- Words shown as vectors (lines from origin to point), not just dots
- Subtle purple gridlines and zero-lines for 3D depth perception
- Aspect mode set to cube for consistent proportions
- Vector lines in both Explore and Arithmetic tabs

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

app.py

@@ -116,15 +116,31 @@ def project_3d(pca, vectors):
     return coords
 
 
+def _axis(title=""):
+    """Subtle 3D axis with gridlines for depth perception."""
+    return dict(
+        showgrid=True,
+        gridcolor="rgba(99,52,141,0.08)",   # very faint purple grid
+        zeroline=True,
+        zerolinecolor="rgba(99,52,141,0.25)",
+        zerolinewidth=2,
+        showticklabels=False,
+        title=title,
+        showspikes=False,
+        backgroundcolor=BG,
+    )
+
+
 def layout_3d(height=560):
     """Shared Plotly 3D layout."""
     return dict(
         scene=dict(
-            xaxis=dict(visible=False),
-            yaxis=dict(visible=False),
-            zaxis=dict(visible=False),
+            xaxis=_axis(),
+            yaxis=_axis(),
+            zaxis=_axis(),
             bgcolor=BG,
             camera=dict(eye=dict(x=1.5, y=1.5, z=1.2)),
+            aspectmode="cube",
         ),
         paper_bgcolor="white",
         margin=dict(l=0, r=0, t=10, b=10),
@@ -139,6 +155,26 @@ def layout_3d(height=560):
     )
 
 
+def add_vectors_from_origin(fig, coords, labels, color=PURPLE, width=3):
+    """Draw vector lines from origin to each point (words ARE vectors)."""
+    for i, label in enumerate(labels):
+        fig.add_trace(go.Scatter3d(
+            x=[0, coords[i, 0]],
+            y=[0, coords[i, 1]],
+            z=[0, coords[i, 2]],
+            mode="lines",
+            line=dict(color=color, width=width),
+            showlegend=False, hoverinfo="none",
+        ))
+    # Origin marker
+    fig.add_trace(go.Scatter3d(
+        x=[0], y=[0], z=[0],
+        mode="markers",
+        marker=dict(size=4, color=GRAY, opacity=0.5),
+        name="Origin", hoverinfo="text", hovertext=["Origin (0,0,0)"],
+    ))
+
+
 def blank(msg):
     """Empty placeholder figure with a centered message."""
     fig = go.Figure()
@@ -197,7 +233,10 @@ def explore(words_text, selected):
     # ── Build figure ──
     fig = go.Figure()
 
-    # Main words
+    # Vector lines from origin to each word
+    add_vectors_from_origin(fig, main_coords, valid)
+
+    # Main words (markers + labels at tips of vectors)
     fig.add_trace(go.Scatter3d(
         x=main_coords[:, 0].tolist(),
         y=main_coords[:, 1].tolist(),
@@ -321,6 +360,28 @@ def arithmetic(expression):
     # ── Build figure ──
     fig = go.Figure()
 
+    # Vector lines from origin for all operands
+    for i, w in enumerate(operands):
+        color = PURPLE if w in pos else PINK
+        fig.add_trace(go.Scatter3d(
+            x=[0, coords[i, 0]], y=[0, coords[i, 1]], z=[0, coords[i, 2]],
+            mode="lines", line=dict(color=color, width=3),
+            showlegend=False, hoverinfo="none",
+        ))
+    # Vector line for top result
+    if result_words:
+        fig.add_trace(go.Scatter3d(
+            x=[0, coords[n_op, 0]], y=[0, coords[n_op, 1]], z=[0, coords[n_op, 2]],
+            mode="lines", line=dict(color=GOLD, width=3),
+            showlegend=False, hoverinfo="none",
+        ))
+    # Origin
+    fig.add_trace(go.Scatter3d(
+        x=[0], y=[0], z=[0], mode="markers",
+        marker=dict(size=4, color=GRAY, opacity=0.5),
+        name="Origin", hoverinfo="text", hovertext=["Origin"],
+    ))
+
     # Positive operands
     pi = [i for i, w in enumerate(operands) if w in pos]
     if pi: