It all started with this drawing.

And with the help of AI and computers, the very enemies I am trying to defeat here, see the creation of maps without F2, F3, or F4.

Watch a planar map grow, live, on a sphere, visualizing the combinatorics behind the Four Color Theorem This is an interactive visualization of a growing cubic planar map, rendered on a fixed-size sphere (“Waterworld colonization”). Starting from the smallest possible seed, a theta graph with 2 faces, the simulation adds one new region at a time, like a coastline expanding across an island, until the map wraps all the way around the globe. Each new region is carved out by choosing an arc of the current coastline and building a new border beyond it, following simple combinatorial rules:

• A region’s face-degree (its number of sides) always follows n = k + 3, where k is how many existing coastal edges it swallows.
• A sealing rule guarantees no region with fewer than 5 sides ever gets fully enclosed (“buried”) by its neighbors, a key structural invariant used in Four Color Theorem-style reduction arguments.
• The running quantity Σ(6−n)·Fₙ − C = 6 is tracked live in the “Ship’s log” panel and stays mathematically invariant at every single step, no matter how the coastline grows, a live, visual proof-of-invariant.

The camera is fully interactive: drag to orbit the globe, scroll or pinch to zoom, and use the Start/Stop and speed controls to watch the island grow at your own pace. Growth halts automatically once the island has expanded to fill the sphere. This is part of an ongoing exploration of graph-theoretic approaches to the Four Color Theorem, built around Tait’s edge-coloring equivalence between 3-edge-colorings of cubic planar graphs and 4-colorings of maps.