Draw a “clean” scribble – one in which all regions are easily discernible
| D || Dots |
(where lines segments cross or the end of line segments)
| L || Lines |
(line segment between 2 dots)
| R || Regions |
(area surrounded by closed line segments)
For the scribble shown D = 10, L = 14 and R = 5
Investigate the formula connecting D, L and R.
Leonhard Euler discovered a formula relating the number of edges, vertices, and faces of a convex polyhedron, and hence of a planar graph.
- Find a formula for a straight line with varying numbers of dots placed on it.
- Find a formula for regular polygons.
- Find a formula for a bicycle wheel with varying numbers of spokes.
- Consider other groups of scribbles.
- Formulate a formula for all scribbles.
- Test your formula.
- Start with a single dot – is your formula true for this scribble?
- If you have a scribble which satisfies your formula, find the four simplest ways that the scribble can be made slightly larger.
- By carefully considering each of these changes, show that each of the larger scribbles also satisfies your formula.
- How does this argument help you generalise your formula?
- This formula is called the scribble theorem.
- There are some limitations on the use of the scribble theorem (because of assumptions that you have made in the proof) – can you find them?
The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical objects), and is related to the genus of the object.
The study and generalization of this formula, specifically by Cauchy and L'Huillier, is at the origin of topology.
This is one of the many Enrichment tasks on this site.
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