Teaching objectives
The geometry you learned in school isn't the only one possible: it depends on the shape of the space you live in. How would an inhabitant discover this if they could never leave their world to see it "from outside"? This upper-secondary lab answers with a recurring character — the angle-counting ant, who can only measure angles and distances — and travels through three complete geometries (flat, spherical, hyperbolic) across 24 rooms, using real 2D and 3D renders (Three.js) instead of formulas alone.
What you'll learn
- Students start with the flat triangle: however its vertices are dragged, the sum of its angles is always 180° — the signature of Euclid's world, proved by drawing a parallel through the opposite vertex.
- The question that drives the whole lab is posed: can we get more than 180°? — answered by rotating and enlarging a triangle on a real sphere, where the sum grows with the triangle's size.
- Students explore the Poincaré disk, the flat model of the hyperbolic world where the boundary represents infinity and a triangle's angle sum always falls below 180°.
- The disk is connected to a real, three-dimensional surface: the hyperboloid, showing that both are two different maps of the same curved world.
- Euclid's fifth postulate (parallels) is studied, along with its failure in both directions: on the sphere there are no parallels at all (every pair of geodesics meets), while in the hyperbolic disk infinitely many parallels pass through the same point (Lobachevsky, Bolyai, 1830-1832).
- Students discover the link between {p,q} tilings and curvature: whether the polygons around a vertex leave a gap, overlap, or fit exactly determines whether the resulting world is spherical, hyperbolic, or flat — all decided by the inequality 1/p + 1/q versus 1/2.
- Girard's theorem is introduced: a spherical triangle's area is computed purely from its angle excess, with no ruler needed — along with its hyperbolic counterpart, angle defect.
- Classic puzzles are tackled: the shortest path between two cities (why transatlantic flights "curve" on the map), the 10 km puzzle (why the North Pole isn't the only answer) and why maps lie (Mercator inflates Greenland).
- The lab closes by asking what shape the real universe has, using the angular size of cosmic microwave background spots as the ant's cosmic protractor.
- A photo-tiling room lets students see, through a custom shader, how an uploaded image repeats infinitely toward the edge of the hyperbolic disk — as in Escher's engravings.
Key mathematical ideas
- A triangle's angle sum depends on the curvature of space: exactly 180° in the plane (K=0), greater than 180° on the sphere (K>0), less than 180° in the hyperbolic disk (K<0).
- Euclid's fifth postulate (existence of a unique parallel) is independent of the other four: negating it in two different ways produces the other two consistent geometries.
- Girard's theorem (spherical area = angle excess × R²) and its hyperbolic counterpart (area = angle defect) let you measure surface without a ruler, using only a protractor.
- Regular {p,q} tilings exist exactly when 1/p + 1/q is greater than, equal to, or less than 1/2, corresponding to spherical, flat, and hyperbolic worlds respectively.
- An inhabitant living inside a curved world can deduce its geometry by measuring only local angles and distances, with no need to "step outside" into a higher dimension — the core idea of intrinsic curvature (Gauss, Riemann).
- Any projection of a sphere onto a plane necessarily distorts something (angles, areas, or distances): there is no perfect map of a curved world.
Room-by-room contents
Room 1 · The flat triangle
Students freely drag a triangle's vertices on a 2D canvas while the 🐜 angle-counting ant measures its three interior angles. However much the triangle's shape changes, the sum always stays at 180° — the signature of Euclid's flat world.
Student tasks
- Drag the vertices, trying several triangle shapes.
- Verify that the sum of the three angles is always 180°.
Room 2 · The ant walks the triangle
A second ant walks along the triangle's three sides and, at each vertex, adds the interior angle to a visible counter. Once it completes the loop, the counter always reads 180°, reinforcing from a different angle — the walk, not a static measurement — the same property from room 1.
Student tasks
- Move the vertices and watch how the counter changes during the walk.
- Verify the accumulated counter always reaches 180° when the loop closes.
Room 3 · Euclid's proof
The classic proof of why a triangle's angle sum is 180° is presented: a parallel to side AB is drawn through vertex C, and the alternate interior angles with secants AC and BC turn out equal to angles A and B, together forming a straight angle.
Student tasks
- Follow step by step the construction of the parallel through C.
- Identify the alternate interior angles and explain why their sum with the angle at C forms 180°.
Room 4 · Can we get more than 180°?
Conceptual bridge: Euclid's proof depends on being able to draw a parallel — the fifth postulate. The question that drives the rest of the lab is posed: what would happen on a surface where no parallels exist?
Student tasks
- Read why Euclid's proof needs the fifth postulate.
- Formulate a hypothesis about what would happen with no parallels.
Room 5 · The fat triangle
On a real 3D globe (Three.js), "straight" means great circles. Students rotate the sphere and enlarge the triangle with a slider: its angle sum exceeds 180° and grows with the triangle's size — something the ant can notice without ever leaving its spherical world.
Student tasks
- Rotate the globe and view the spherical triangle from different angles.
- Enlarge the triangle with the slider and verify the angle sum grows above 180°.
Room 6 · The shortest path
The flight from New York to Madrid doesn't trace a straight line on the flat map — it arcs over Greenland. Students learn that on the sphere the shortest route between two points is always a great circle, never a parallel of latitude or a straight line on the flat map.
Student tasks
- Compare the flight's real route with a straight line drawn on the flat map.
- Explain why the great circle is the shortest path on the sphere.
Room 7 · No parallels on the sphere
The ant walks the sphere in a straight line (a great circle) while a telephoto camera starts with a very narrow field of view (10°), where the sphere looks flat, then widens up to 45°, revealing the curvature. Students then mark two points and trace their own great circle.
Student tasks
- Watch how the perception of curvature changes as the field of view widens from 10° to 45°.
- Mark two points of your own and trace the great circle joining them.
Room 8 · Parallels on the sphere
Two planes leave the equator perpendicular to it, 1,000 km apart. Although they start out in "parallel" directions, their routes (meridians) converge at the North Pole: on the sphere there are no parallel lines — every pair of geodesics ends up crossing.
Student tasks
- Follow the simulation of the two planes leaving the equator.
- Explain why two routes that start out parallel end up crossing on the sphere.
Room 9 · The thin triangle
The Poincaré disk is introduced: the boundary represents infinity and "lines" are arcs that cross it perpendicularly. Students drag the vertices of a hyperbolic triangle and verify that here the angle sum never reaches 180° — the world of negative curvature.
Student tasks
- Drag the triangle's vertices inside the Poincaré disk.
- Verify the angle sum always stays below 180°.
Room 10 · The hyperboloid
The Poincaré disk is just a flat photo of a real curved surface: the hyperboloid. Students rotate this negatively-curved 3D surface and enlarge the triangle, seeing simultaneously the triangle on the surface and its shadow projected onto the disk — two maps of the same world.
Student tasks
- Rotate the hyperboloid in 3D and observe the triangle on the surface.
- Compare the triangle on the hyperboloid with its projection on the Poincaré disk.
Room 11 · Infinitely many parallels: Lobachevsky
In the hyperbolic disk the fifth postulate also fails, but the opposite way from the sphere: through a point outside a line, infinitely many geodesics pass that never cross it. Students drag a point P and watch the dynamic parallels — a discovery made independently by Lobachevsky (1830) and Bolyai (1832).
Student tasks
- Drag point P and watch how the geodesics parallel to the given line change.
- Count how many distinct parallels can be drawn through P.
Room 12 · The hyperboloid: parallels in 3D
Hyperbolic geometry has a three-dimensional model: the upper sheet of the hyperboloid. Every point of the Poincaré disk corresponds to a point on that surface, and the geodesics from the previous room now become real curves on the hyperboloid. From M, infinitely many parallels to D pass through.
Student tasks
- Rotate the 3D hyperboloid and locate point M and line D.
- Verify that, just as in the disk, infinitely many parallels to D pass through M.
Room 13 · Hyperbolic distances
Five circles all have exactly the same size on screen. In the hyperbolic metric of the Poincaré disk, space "compresses" toward the boundary, so what looks equally large on screen can be hyperbolically much bigger the closer it sits to the edge.
Student tasks
- Compare the five circles' apparent size on screen.
- Explain why, in the real hyperbolic metric, they do not all represent the same size.
Room 14 · Which tilings fit?
Students place q polygons of p sides around a single point. If they leave a gap, the resulting world closes up into a sphere; if they fit exactly, it's flat; if they overlap, they only fit by curving inward (hyperbolic). The same inequality 1/p + 1/q versus 1/2 decides which world you're in.
Student tasks
- Try different combinations of p and q and observe whether they leave a gap, fit exactly, or overlap.
- Relate the result to the value of 1/p + 1/q compared with 1/2.
Room 15 · Curve the space
The vertex figure from the previous room is picked back up and the space is curved in 3D: when there's leftover gap, the polygons fold and close up forming the corner of a polyhedron (sphere); when they overlap, they don't fit in the plane and need the hyperbolic world to accommodate them.
Student tasks
- Curve the space with a vertex figure that left a gap and watch it close into a polyhedron.
- Repeat with a figure that overlapped and verify it needs the hyperbolic world.
Room 16 · A tessellation without end
Students see what happens when tiling the hyperbolic plane with regular polygons: the polygons appear to shrink toward the edge of the disk, but in real hyperbolic geometry they are all identical — it is the Poincaré disk that compresses infinity inside a circle.
Student tasks
- Observe the full hyperbolic tiling inside the disk.
- Explain why polygons that look smaller near the edge are actually the same size as those in the center.
Room 17 · Tile your own photo
A WebGL shader draws a {p, q} hyperbolic tiling pixel by pixel. Students upload their own photo and choose a mode: "Full photo" shows it whole (curved) in every cell, "Kaleidoscope" fragments it into mirrors. The image repeats infinitely toward the edge, as in Escher's engravings.
Student tasks
- Upload a photo and try both modes (full photo and kaleidoscope).
- Change the p and q values and observe how the resulting tiling changes.
Room 18 · Area without a ruler
Girard's theorem is presented: a spherical triangle's area equals its angle excess (the sum minus 180°) multiplied by R². So the ant can compute the area of a region of its world using only a protractor, with no ruler needed.
Student tasks
- Measure the three angles of a given spherical triangle.
- Compute its area applying Girard's theorem (excess × R²).
Room 19 · The 10 km puzzle
A classic orientation puzzle: you walk 10 km north, 10 km east and 10 km south, and end up exactly where you started. Where are you? The best-known answer is the North Pole, but students discover there are infinitely many more solutions, near the South Pole.
Student tasks
- Verify why the North Pole satisfies the conditions of the puzzle.
- Explore the additional family of solutions near the South Pole.
Room 20 · Hyperbolic area
In the hyperbolic disk, a triangle's area equals its angle defect: π minus the sum (A+B+C). The larger the triangle, the smaller its angles become, up to the ideal triangle, with zero angles and maximum area π·R².
Student tasks
- Compare hyperbolic triangles of different sizes and their angles.
- Compute a triangle's area from its angle defect and verify the limiting case of the ideal triangle.
Room 21 · Maps lie
It's impossible to flatten a sphere onto a plane without distorting something. The Mercator projection preserves angles but hugely inflates regions near the poles: on the map, Greenland looks the size of Africa, even though Africa is actually 14 times larger.
Student tasks
- Compare Greenland's and Africa's size on the Mercator projection.
- Explain why no flat projection of the sphere can avoid all distortion.
Room 22 · The ant measures curvature
An inhabitant living inside a curved world can't see it from outside, but can measure local angles. The key: in a curved world, a large triangle's angle sum drifts away from 180° more the bigger the triangle gets. Students move a control and watch that curvature signal grow.
Student tasks
- Move the triangle-size control and watch the angle sum drift from 180°.
- Relate the size of that deviation to the curvature of the world.
Room 23 · What shape is the universe?
The cosmic microwave background (CMB) is the oldest light in the observable universe. The angular size of its hot and cold spots depends on the curvature of space: greater than 1° if curvature is positive, exactly 1° if flat, less than 1° if negative — the ant's cosmic protractor.
Student tasks
- Read how the CMB spots' angular size reveals the universe's curvature.
- Relate the three cases (K>0, K=0, K<0) to the three worlds studied in the lab.
Room 24 · The three worlds
Final summary: flat, spherical and hyperbolic are three consistent geometries, each with its own answer to the questions posed by Euclid. Students review a comparison table and check that they already know each of its key properties.
Student tasks
- Review the comparison table of the three worlds.
- For each property in the table, identify which lab room it was discovered in.
Rooms to project
The most striking ones to show and discuss in class.