Draw Edge Junction

Draw Edge Junction gives you a set of cities and one goal — connect them all with as little road as possible.

The obvious answer is to connect them directly, finding the shortest spanning tree. This is never the real answer. The real answer requires adding junction points that don't correspond to any city — extra nodes where roads branch. The game shows them as blue draggable points. Drag them. Watch the angle arcs at each junction update in real time. Red arcs mean an angle is too narrow. Blue means too wide. Gold means exactly right. When every arc at every junction turns gold simultaneously — the network is optimal. That is the Steiner tree.

The rule that produces the gold: at every junction in an optimal network, exactly three roads meet, and they always meet at exactly 120° to each other. This is not a heuristic or an approximation. It is a mathematical certainty, proved rigorously. If any junction has angles other than 120°, the total length can be reduced by moving it slightly. The 120° configuration is the unique mechanical equilibrium where no move can shorten the network further.

The square is where the game reveals itself. Four cities at the corners. The natural instinct is one central junction — a plus sign. It cannot be made to lock. The angles refuse to reach 120° simultaneously no matter how carefully the central point is positioned. Move to two junctions instead. Arrange them in an H shape. The total length drops. Drag each junction toward its locked position and watch the angle arcs shift from red and blue toward gold. When both junctions lock at the same moment — the two gold flashes arriving together — the saving is visible in the length counter: about nine percent shorter than the spanning tree. Nobody guesses the H before they discover it.

This is not a coincidence of the square. The H topology — two junctions, each connecting two cities to a shared crossbar — is optimal for any four cities arranged as a rectangle. The crossbar length changes dramatically with the rectangle's proportions. For a square it is moderate. For a very tall rectangle it nearly vanishes, and the two junctions sit almost on top of each other. The game shows this across three levels, making the relationship between geometry and topology tangible.

A soap bubble knows all of this instinctively. Take a piece of glass, drill holes for the cities, insert pins, dip the assembly in soap solution, and lift. The film trapped between two plates of glass forms the Steiner tree every time — all angles exactly 120°, total length minimal. Surface tension minimises area, which in two dimensions is equivalent to minimising length. The soap solves in one millisecond a problem that, for large inputs, has no known efficient algorithm. The general Steiner tree problem is NP-hard. The soap film does not care.

A faint arrow on each junction shows the gradient — the direction that would reduce total length if followed. Following the arrows converges toward the locked configuration. The arrows make the physics tangible: the system has a force pulling it toward equilibrium, exactly as surface tension pulls a soap film.

No formulas shown. No algorithm explained. Just the junctions, the arcs, and the gold that arrives when the geometry is right.