Rigid vs Flexible Shapes Math Example 5

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

Example 5

hard
A hexagonal frame with all sides equal and all angles at 120°120° is rigid only if all angles are fixed. If the frame's joints are flexible (angles can change), how many degrees of freedom does it have, and how could you brace it minimally?

Solution

  1. 1
    Step 1: A hexagon has n=6n = 6 sides. Degrees of freedom with fixed sides and free angles =n−3=3= n - 3 = 3.
  2. 2
    Step 2: To triangulate a hexagon (make it rigid), add internal diagonals to create triangles. The minimum number of diagonals needed to fully triangulate a convex nn-gon is n−3=3n - 3 = 3 diagonals.
  3. 3
    Step 3: For a hexagon: add 33 non-crossing diagonals, dividing it into 44 triangles. This removes all 33 DOF.

Answer

The hexagonal frame has 33 degrees of freedom. It can be made rigid by adding 33 internal diagonal braces.
A convex nn-gon with free joints has n−3n-3 degrees of freedom. Triangulation with n−3n-3 non-crossing diagonals creates n−2n-2 triangles and eliminates all flexibility. This is the minimum bracing needed.

About Rigid vs Flexible Shapes

A rigid shape cannot be deformed without breaking — its sides and angles are locked. A triangle is always rigid because its three side lengths uniquely determine its angles. A rectangle, by contrast, is flexible: it can collapse into a parallelogram because four side lengths do not fix the angles.

Learn more about Rigid vs Flexible Shapes →

More Rigid vs Flexible Shapes Examples

Example 1 medium

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Example 2 hard

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Example 3 medium

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Example 4 easy

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