Three utilities: from the plane to a mug
Connect three houses to gas, water, and electricity. The plane refuses—but a mug, a donut, and a square with joined edges reveal a way through.
Three houses each need gas, water, and electricity. Can we draw all nine pipes on the plane without crossings?
Pipes may bend and meet at a shared endpoint. They cannot overlap, cross, or pass through another house or utility. No tunnels or bridges are allowed.
Explore Try it on the plane below and its Info panel. The eight-connection example leaves one pipe missing: does rerouting seem to make room for it?
Try it on the plane
0 / 9 pipesChoose a house, click points along your path, then choose a utility. We’ll draw a smooth curve through the points. You can also trace a path by dragging between endpoints.
Info
Select a house, choose points for a path, then select a utility. You can also drag between endpoints. Select an existing pair without points to remove its pipe. Undo restores the last change.
Can you draw all nine pipes without a crossing? Pipes may meet only at their own endpoints.
A failed drawing does not settle the question. Perhaps a different arrangement would work. We need an obstruction that applies to every arrangement.
Count regions instead of trying more routes
Replace the houses and utilities by vertices, and the pipes by edges. The graph is \(K_{3,3}\): two groups of three vertices, with every vertex in one group connected to every vertex in the other. It has \(V=6\) vertices and \(E=9\) edges.
A crossing-free drawing of a connected graph in the plane satisfies Euler’s formula
\[V-E+F=2,\]where \(F\) includes the unbounded outside region. Our drawing would therefore have five regions.
Every region needs at least four edge-sides around its boundary. There are no loops or parallel edges. Every cycle alternates between houses and utilities, so it has even length and cannot be a triangle. Nor is any edge a bridge: each belongs to a four-edge cycle. A two-sided boundary from retracing a bridge is therefore impossible too.
Each edge supplies two sides to region boundaries. Nine edges supply eighteen sides, while five regions require at least twenty:
\[2E\geq4F\qquad\Longrightarrow\qquad18\geq20.\]No planar drawing exists. Moving the endpoints or making the pipes curvier cannot change this count. The difficulty in the plane experiment comes from the surface itself.
What changes on a mug?
Put the same houses and utilities on the whole boundary surface of a thick mug with one handle: outside, inside, rim, underside, and handle. Its bowl has a bottom; the opening is an indentation, while the handle supplies one through-hole.
That surface can be continuously reshaped into a donut, or torus, without cutting or joining it. The surface transformation follows the hole through the change.
Follow the hole
A mug
Drag to look into the bowl. Its bottom, inner wall, rim, outside and handle form one surface. The handle supplies the one through-hole.
Info
Drag or use arrow keys to rotate. Turn on the surface grid to follow the same points as the bowl lifts and the mug rounds into a donut. This stage has no cuts. The purple cut opens the ring into a tube; the green slit opens the tube into a square. Move the slider or choose a stage. Reset returns to the mug.
The first stage only deforms the surface. The later stages cut it open so we can work with a flat picture: open the ring into a tube, slit the tube lengthwise, and flatten it into a rectangle, then a square.
The cut edges retain their original pairings. Recovering the torus means identifying
Top matches bottom at the same horizontal position; left matches right at the same vertical position. No edge is reversed. Leaving through one side and returning through its partner is a continuous journey on the torus.
Use the joined edges for the missing pipes
Seven pipes stay inside the square. The remaining two use the edge pairings (1).
In the figure, House 1 → Water continues from the top to the matching point on the bottom. House 3 → Gas continues from the right to the matching point on the left. Each pair of seam dots represents one point of the torus, rather than a gap in a pipe.
All nine routes now join the required endpoints without crossings. Their seam continuations use distinct points, so joining the edges introduces no new intersection. Reshaping the torus back into the mug carries the pipes with it.
The handle has changed which routes are available. The planar counting obstruction still holds on the plane; the mug admits a drawing because its surface is different.
John M. Lee’s Introduction to Topological Manifolds, opening chapter, illustrates the coffee-cup–torus deformation and introduces the distinction between shape and topology.