Graph theory

Planarity asks whether crossings are necessary, not whether they are visible.

This exploration starts with train-track networks that students can physically untangle, then moves towards the deeper question: are crossings just a problem with the drawing, or are they forced by the structure?

The aim is not to remove the thinking by giving the theorem first. Students should experience the frustration, search for patterns, propose laws and then see how graph theory connects maps, transport, topology and the world around them.

Start with a real map

The maths begins when the scenery disappears.

Imagine this is a small rail network. Stations can connect by tracks, and tracks are only allowed to meet at stations. A crossing in the middle of a track would need a bridge, tunnel or redesign.

Station

Later this becomes a vertex.

Track

Later this becomes an edge.

Question

Can this network be drawn flat without tracks crossing?

trains can run

Watch the trains move.

Ignore the scenery.

Keep only connections.

Joy in the process

The point is not just to finish. It is to notice, test and return.

These tools are invitations to explore. A good mistake, a surprising pattern or a question you cannot yet answer is part of the work, not a failure of it.

The challenge is deliberate: the site should support thinking, not remove the need for it.

Before changing a setting, pause and predict what you think will happen.

Change one thing at a time. What stayed the same, and what changed?

Try to create a surprising case, a broken case, or a beautiful pattern.

Ask what this connects to outside the page: maps, movement, nature, systems or decisions.

Reset, then try again with a new question in mind.

Guided exploration

Move slowly: predict, test, explain, then look for the wider connection.

Before moving anything, predict whether the crossings are removable or unavoidable.

Try the planar examples first. What did you change when the drawing finally untangled?

Now try K3,3 and K5. What does repeated failure make you suspect?

Use Euler's formula and the edge limits to explain what the dragging experiment suggested.

Where else do you see networks that are really about connections rather than exact shape?

Future extensions

This can become the first full exploration template for the site.

Add staged planarity puzzles with increasing difficulty and saved best crossing counts.

Add a teacher display mode that hides theorem checks until students have made conjectures.

Add Kuratowski-style subdivisions of K5 and K3,3 as an expanded exploration.

Add a proof journey for Euler's formula using cutting, stretching and counting faces.

Add more map-colouring puzzles and links to the four-colour theorem.