Pushing Cayley's surface

Cayley's cubic surface, 4(x² + y² + z²) + 16xyz = 1, has four nodes (the coral points) at the corners of a regular tetrahedron: near each node the surface looks like two cones meeting tip to tip. At δ = 0 the six edges of the tetrahedron lie on the surface and three more lines lie at infinity, so there are only nine distinct lines; the corner counts each edge four times, and the total is still 27. Now push the right side of the equation from 1 to 1 + δ. Drag δ above zero and the nodes vanish, the surface turns smooth, and four gold lines split off beside each edge: twenty-four, which with the three at infinity make twenty-seven, the number every smooth cubic surface has once complex lines are counted too. Bring δ back to 0 and each group of four closes up into one edge, which then counts four times. Drag δ below zero and the surface is smooth as well, but the only real lines left are the three at infinity, which cannot be drawn: the twenty-four have turned complex, and not one line is left in the picture. The total in the corner stays at 27 throughout. Turn the view to look from another side.