Shortest paths on the surface

An ant sits at one corner of a $3 \times 4 \times 5$ box. It wants to reach the diagonally-opposite corner. It can only walk along the outside surface, not through the box. What is the shortest path?

This puzzle is delightful because the "obvious" answer is wrong, and the right answer uses nets , exactly the unfolding we just learned.

Concept

Why we need to unfold. The shortest path along the surface might cross several faces. On the 3D box, this is a sequence of broken straight lines, hard to optimise directly. But once you unfold (some of) the faces into a flat net, the path that was broken on the surface becomes a straight line in the net. And the shortest path between two points in a plane is just the straight line.

Procedure.

1. Identify the start and end points on the 3D box.
2. Choose a path that crosses some faces. Unfold those faces flat (the surrounding faces stay attached at the shared edge).
3. Mark the start and end in the net. Connect them with a straight line.
4. Its length is the candidate shortest path. Different unfoldings give different lengths , try a few.
5. The minimum over all possible unfoldings is the true shortest surface distance.

Example. A box of size $3 \times 4 \times 5$. Ant at corner $A = (0,0,0)$, target at corner $B = (3, 4, 5)$.

There are three ways to unfold for the path:

1. Unfold via $3 + 4 = 7$ and $5$: straight line of length $\sqrt{7^2 + 5^2} = \sqrt{74} \approx 8.60$.
2. Via $3 + 5 = 8$ and $4$: length $\sqrt{8^2 + 4^2} = \sqrt{80} \approx 8.94$.
3. Via $4 + 5 = 9$ and $3$: length $\sqrt{9^2 + 3^2} = \sqrt{90} \approx 9.49$.

Shortest: $\sqrt{74} \approx 8.60$. The route uses the $3$ and $4$ sides plus the $5$ side. Through the inside (which the ant can't do), the diagonal would be $\sqrt{50} \approx 7.07$.

Generalising. For a cuboid $a \times b \times c$ with the ant going corner to corner, the three candidate path lengths are:

• $\sqrt{(a+b)^2 + c^2}$
• $\sqrt{(b+c)^2 + a^2}$
• $\sqrt{(a+c)^2 + b^2}$

The smallest is the shortest. Often the middle-sized face combinations win.

Other surfaces. Shortest paths on cylinders, cones, and pyramids work the same way: unfold, draw a straight line, measure.

Worked examples

Example 1. Ant on a $1 \times 1 \times 1$ cube, corner to corner via surface. Find shortest path.

• Three candidates: $\sqrt{(1+1)^2 + 1^2} = \sqrt{5}$, three identical.
• Shortest: $\sqrt{5} \approx 2.236$.

Example 2. $2 \times 3 \times 4$ box, corner to corner.

• Candidates: $\sqrt{(2+3)^2 + 4^2} = \sqrt{41} \approx 6.40$, $\sqrt{(3+4)^2 + 2^2} = \sqrt{53} \approx 7.28$, $\sqrt{(2+4)^2 + 3^2} = \sqrt{45} \approx 6.71$.
• Shortest: $\sqrt{41}$.

Example 3. A spider in a $20 \times 20 \times 20$ room, sitting at one corner of the floor. A fly is at the diagonally opposite corner of the ceiling. Shortest surface path?

• For an unfolding through two walls plus floor or ceiling: $\sqrt{40^2 + 20^2} = \sqrt{2000} \approx 44.72$.
• A second unfolding: $\sqrt{30^2 + 30^2} = \sqrt{1800} \approx 42.43$. (Counterintuitive: a "$Z$-shaped" path is shorter.)

Example 4. On a cylinder of radius $1$ and height $3$, an ant at the bottom edge wants the diametrically-opposite top edge. Unfold the cylinder into a $2\pi \times 3$ rectangle. Half-circumference $= \pi$. Path: $\sqrt{\pi^2 + 9} \approx \sqrt{18.87} \approx 4.34$.

Try it yourself

1. Cube of side $3$. Find shortest corner-to-corner surface distance.
2. Cuboid $2 \times 5 \times 6$. Find shortest surface distance.
3. Cube of side $1$, ant going from a vertex to centre of the opposite face?
4. Cylinder $r = 2$, $h = 5$. Shortest from bottom rim to diametrically opposite top rim.
5. A long thin box $1 \times 1 \times 10$. Shortest path corner-to-corner?
6. Square pyramid base $4$, slant edge $5$. Shortest path along the surface from one base vertex to the opposite base vertex.
7. Cube of side $5$, ant on centre of one face going to centre of opposite face along surface.
8. Compare with the "through-the-inside" straight-line distance for the $3 \times 4 \times 5$ box ($\sqrt{50}$).

Activity / Insight

Real-life unfolding. Take a small box and a piece of string. Mark two opposite corners. Try wrapping the string around the box to find the shortest path. Now unfold the box and draw the same string path as a straight line. The straight line should match the wrapped string exactly , and any other route should be longer.