Shortest Path Intuition Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
An ant on the surface of a unit cube wants to travel from vertex A=(0,0,0)A = (0,0,0) to the opposite vertex B=(1,1,1)B = (1,1,1) along the surface. What is the shortest surface path, and what is its length?

Solution

  1. 1
    Step 1: The ant must traverse two faces. Unfold two adjacent faces into a flat rectangle. One such unfolding maps AA at (0,0)(0,0) and BB at (2,1)(2,1) (or (1,2)(1,2) depending on which faces are unfolded).
  2. 2
    Step 2: Shortest surface path in the unfolded net =(2−0)2+(1−0)2=5= \sqrt{(2-0)^2 + (1-0)^2} = \sqrt{5} for the (2,1)(2,1) unfolding.
  3. 3
    Step 3: Checking other unfoldings: (1,2)(1,2) also gives 5\sqrt{5}. Both give the same minimum length 5\sqrt{5}.

Answer

Shortest surface path =5≈2.24= \sqrt{5} \approx 2.24 units.
Surface shortest paths (geodesics on polyhedra) are found by unfolding the surface into a flat net and drawing a straight line. The ant's path crosses exactly two faces; unfolding those faces flat makes the geodesic a straight line of length 5\sqrt{5}.

About Shortest Path Intuition

The minimum-length route connecting two points, whose form depends on the geometry of the underlying space.

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Example 3 easy

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