A skyrmion is a tiny, particle-like whirl of magnetization in which the spins twist smoothly from one direction at the centre to the opposite direction at the edge, wrapping all orientations exactly once, giving it a protected topological charge. It is compared to a hedgehog because in a Neel-type skyrmion the spins point radially outward or inward from the core, exactly like a hedgehog's spines pointing in all directions.
The Mandelbrot set is the collection of complex numbers for which a specific iteration formula stays bounded; its boundary is a fractal with infinite detail. Zooming in reveals recurring motifs including tiny baby Mandelbrot shapes, a form of self-similarity. Romanesco broccoli shows the same property: each floret is a scaled, spiral-arranged miniature of the whole head, built from smaller cones that look like the larger cone.
The Fibonacci sequence is 1, 1, 2, 3, 5, 8... where each term is the sum of the two before it; ratios of successive terms approach the golden ratio ~1.618. A nautilus shell grows by adding chambers, keeping the same shape while scaling up, an approximate logarithmic spiral. The steady multiplicative growth resembles the golden spiral, making the nautilus a favourite visual echo of Fibonacci-like scaling in nature.
In topology, a torus (donut shape) and a coffee cup with one handle are considered identical, because each has exactly one hole. You can continuously deform one into the other without cutting or gluing. A topologist cannot tell them apart! This is the most famous example of topological equivalence.
A hexagonal tiling covers the plane with congruent hexagons, three meet at each vertex, filling space without gaps or overlaps. Bees build their combs in the same pattern: for cells of equal size, hexagons minimise the total wall length (the honeycomb optimality theorem), conserving wax while maximising storage. Evolution and mathematics arrived at the same answer.
A Voronoi tessellation divides space into regions around seed points: every location in one region is closer to its own seed than to any other. This produces irregular polygonal patterns that arise wherever growth or territory is controlled by proximity. The patches on a giraffe's skin are a near-perfect Voronoi diagram, each dark patch is the territory of one pigment centre.
Topology Matching Game
Topology studies properties that survive stretching and bending but not tearing or gluing. One famous example: a donut and a coffee cup are topologically identical because each has exactly one hole. But mathematics is full of other deep structural patterns too, like self-similarity in fractals, optimal tilings in nature, and nearest-neighbour diagrams in animal skin. This game asks you to match objects that share one of these six hidden mathematical connections.
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What to look forEach pair shares a deep mathematical property, not just a visual resemblance. Think about holes, growth patterns, symmetry, or tiling rules.
After you winClick any face-up card to read a full explanation. All six connections are also revealed below the board.
How to think about it
The six pairs are: a donut and a coffee cup (same number of holes), a fractal and a vegetable (self-similarity), a number sequence and a shell (logarithmic growth), a physics pattern and an animal (radial symmetry), a mathematical tiling and a beehive (optimal packing), and a geometric diagram and an animal skin (nearest-neighbour regions).
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Flip two cards to find a matching pair.
All 6 connections explained
How to play
Click any face-down card to flip it. Flip a second card: if the two share a topological property they stay face-up as a matched pair. If not, they flip back. Find all 6 pairs to win.
After winning, click any revealed card to read its full explanation.
What is topology?
Topology studies properties that do not change when you stretch, bend or twist an object, but would change if you tore or glued it. A circle and a square are topologically identical. A donut and a coffee cup are too.
A topologist cannot tell their coffee cup from their donut.
Where topology appears in everyday life
DNA and enzymes - topoisomerases cut and rejoin DNA strands to change their topological type, allowing cell replication. Without them, cell division fails.
The recycling symbol - deliberately designed as a Mobius strip in 1970; one side, one edge, infinite loop: topology as graphic design.
GPS and maps - the sphere has a different topological type from the plane, which is why no flat map of Earth can avoid distortion.
Internet routing - network topology (ring, mesh, star) determines resilience and efficiency; graph theory is a branch of topology.
The key idea - topological invariants
Two objects are homeomorphic (topologically equivalent) if there exists a continuous, invertible map between them — that is, one can be deformed into the other without tearing or gluing. The donut and coffee cup pair in this game is a genuine example of this. The other five pairs are not topological equivalences in the strict sense: they share a structural or geometric pattern — self-similarity, logarithmic growth, optimal partitioning, radial symmetry, or nearest-neighbour regions. These patterns are studied within the broader toolkit of geometry and mathematical structures, of which topology is one important branch. A useful topological invariant is the Euler characteristic: for a surface, chi = V - E + F. A sphere has chi = 2, a torus has chi = 0 — which is why they are genuinely different despite both being smooth closed surfaces.
The pairs in this game use a broader notion: shared structural patterns such as self-similarity (fractals), optimal partitioning (tilings), proximity diagrams (Voronoi), and topological equivalence (holes). All are studied within topology and geometry.
✓
Topology Game
You found all 6 pairs!
Scroll down after closing to read the full explanations for all pairs.
