Teaching objectives
What would a world with four spatial dimensions look like? Starting from a question that seems impossible, this Bachillerato lab builds the answer step by step, using the same trick a 2D being would use to imagine 3D space: dimensional analogy and projecting shadows.
What you'll learn
The intuition that the fourth spatial dimension is as coherent and geometric as the third, even though it escapes our direct perception.
- Block A — Shadows (rooms 1-5): a rotating 3D cube casts 2D shadows; a rotating tesseract casts 3D shadows. Students learn to “read” the 4D structure from the shadow.
- Block B — 4D polytopes (rooms 6-11): the regular 4D polytopes — 5-cell, tesseract, 16-cell, 24-cell, 120-cell, 600-cell — and their geometric properties. Room 13 brings the full gallery together: each polytope can be selected and its 3D projection explored in real time, rotating it with the mouse.
- Block C — Construction and sections (rooms 12-13): the tesseract is built by analogy (point → segment → square → cube → tesseract) and cut with hyperplanes to reveal 3D sections, including the cuboctahedron.
- Room 14 — 4D superpowers: what could a 4D being do in our world? See inside a closed room, undo knots that are impossible in 3D.
- Room 15 — Final review: a visual quiz on the vertices, edges, faces and cells of the polytopes.
- Room 16 — Calabi-Yau manifolds: the most striking room of the lab. String theory needs 10 dimensions; the 6 extra ones are compactified in a Calabi-Yau manifold. The exact mathematical visualisation is shown, using complex trigonometry: cos(x+iy) and sin(x+iy) with n² rainbow-coloured sheets. Two sliders — the degree n and the 4D projection angle — transform the figure in real time.
Key mathematical ideas
- Dimensional analogy: the progression 0D→1D→2D→3D→4D guides the construction of the tesseract. The vertices double (2ⁿ), and the edges follow the rule 2n.
- 4D→3D perspective projection: just like 3D→2D. The distance to the “eye” along the fourth coordinate determines the relative scale of the inner and outer cubes of the tesseract.
- 4D Euler characteristic: V−E+F−C = 0 for all convex regular 4D polytopes (the analogue of V−E+F = 2 in 3D).
- Regular polytopes in 4D: there are exactly 6 (in 3D there are 5). The 24-cell is the only one with no analogue in any other dimension.
- Calabi-Yau: z₁ = e^(2πik₁/n)·cos(x+iy)^(2/n), z₂ = e^(2πik₂/n)·sin(x+iy)^(2/n). 4D→3D projection: X=Re(z₁), Y=Re(z₂), Z=Im(z₁)·cos(a)+Im(z₂)·sin(a). The quintic n=5 (25 sheets) is the most relevant for string theory.
Room-by-room contents
Room 1 · Shadows lose one dimension
A 3D cube spins and casts its 2D shadow on the “floor”, with the projection rays drawn; it can be rotated by dragging. The shadow changes with the angle even though the cube does not. In the same way, we (3D beings) can only see 3D shadows of 4D objects. Multiple-choice question: which polygons can be the shadow of a cube. Only quadrilaterals and hexagons: the central symmetry of the cube forces an even number of sides. After a wrong answer, buttons appear to show the square, the rectangle and the hexagon.
Student tasks
- Rotate the cube by dragging and watch how the shadow changes.
- Mark which polygons can be the shadow of a cube (quadrilateral and hexagon).
Room 2 · Tetrahedron shadows
A regular tetrahedron (4 vertices, 6 edges) spins above the floor with its 2D shadow; it can be rotated by dragging. Multiple-choice question: which shapes it can cast as a shadow. A triangle (seen face-on) and a square (looking from the midpoint of one edge towards the opposite edge): a square from a solid made only of triangles. A shadow with 4 vertices cannot have more than 4 sides.
Student tasks
- Rotate the tetrahedron until the square shadow appears.
- Mark the possible shadows (triangle and square).
Room 3 · Octahedron shadows
A regular octahedron (6 vertices, 12 edges), the dual of the cube, with its 2D shadow. Multiple-choice question: which shapes it can cast. A square (looking from a vertex), a rhombus (intermediate angles) and a regular hexagon (looking from the centre of a face); never a triangle.
Student tasks
- Find the direction from which the shadow is a regular hexagon.
- Mark the possible shadows (square, rhombus and hexagon).
Room 4 · Cube cross-sections
A plane cuts a 3D cube. Two sliders: tilt (from a straight cut to a diagonal one) and position of the plane; a label names the resulting section. Multiple-choice question: which shapes can be a cross-section of a cube. Square and rectangle (straight cuts), triangle and hexagon (diagonal cut) and pentagon (oblique cut); an octagon never appears. This prepares the cross-sections of the tesseract.
Student tasks
- Move the sliders until you get a triangle, a pentagon and a hexagon.
- Mark the possible sections (all except the octagon).
Room 5 · Diagonal sections: cube and tesseract
Two animated panels side by side. On the left, a diagonal plane x+y+z = t sweeps through the cube (point → triangle → regular hexagon); on the right, the 4D analogue: a diagonal hyperplane x+y+z+w = t sweeps through the tesseract (point → tetrahedron → truncated tetrahedron → regular octahedron). A tilt slider goes from the diagonal cut to the straight one, and tapping a panel pauses it. Multiple-choice question: which shapes can be a 3D section of the tesseract (cube, parallelepiped, tetrahedron, truncated tetrahedron, cuboctahedron and octahedron; never an icosahedron or a dodecahedron).
Student tasks
- Pause both panels at equivalent moments and compare the section of the cube with that of the tesseract.
- Mark the possible 3D sections of the tesseract.
Room 6 · Euler’s characteristic
The tetrahedron, the cube and the octahedron, one after another. For each one the student enters the vertices (V), edges (E) and faces (F); if needed, buttons highlight the elements so they can be counted. A table collects the results and the V − E + F column always gives 2: the Euler characteristic of convex polyhedra.
Student tasks
- Count V, E and F for the tetrahedron (4, 6, 4), the cube (8, 12, 6) and the octahedron (6, 12, 8).
- Check that V − E + F = 2 for all three.
Room 7 · How to count edges and vertices
A guided example with the octahedron, step by step: 8 triangles × 3 sides = 24 sides; each edge is shared by 2 triangles, so 24 ÷ 2 = 12 edges; there are 24 triangle vertices and 4 meet at each vertex, so 24 ÷ 4 = 6 vertices. Then, “Now you: the icosahedron” (20 triangles).
Student tasks
- Calculate the edges of the icosahedron (20 × 3 ÷ 2 = 30).
- Calculate its vertices, knowing that 5 triangles meet at each one (20 × 3 ÷ 5 = 12).
Room 8 · Duality: cube and octahedron
Animation: the centre of each face of the cube is marked and those centres form an octahedron; the centres of the 8 faces of the octahedron in turn form a smaller cube. Then the same process with the tetrahedron. It can be watched again. Question: what is the dual of the tetrahedron. It is another tetrahedron: it is the only self-dual regular polyhedron.
Student tasks
- Follow the animation cube → octahedron → cube.
- Answer what the dual of the tetrahedron is (another tetrahedron).
Room 9 · The 5-cell: the simplest 4D polytope
3D shadow (perspective projection) of the 5-cell, or pentachoron: 5 vertices, 10 edges, 10 triangular faces and 5 tetrahedral cells. It is the analogue of the tetrahedron in 3D and of the triangle in 2D. It is rotated by dragging and with three 4D rotation sliders (XW, YW and ZW planes). Numerical question: how many tetrahedra it contains. A button highlights them one by one, including the outer cell that wraps around the others.
Student tasks
- Rotate the 5-cell in 3D (dragging) and in 4D (sliders).
- Answer how many tetrahedra it contains (5, one for each vertex left outside).
Room 10 · The tesseract
The 4D hypercube: 16 vertices, 32 edges, 24 square faces and 8 cubic cells. Same rotation controls as the previous room. Numerical question: how many cubes it contains. A button highlights the cubes one by one (cells x−, x+, y−, y+, z−, z+, w−, w+). An answer of 6 gets the reminder that this is the number of faces of the cube.
Student tasks
- Rotate the tesseract and look at the inner and outer cube of the shadow.
- Answer how many cubes it contains (8: 2 for each of the 4 dimensions).
Room 11 · The 16-cell
The 16-cell, or hexadecachoron, dual of the tesseract: 8 vertices, 24 edges, 32 triangular faces and 16 tetrahedral cells. Its vertices are (±1, 0, 0, 0) and their permutations: one at each pole of each 4D axis. Its shadow is a star of octahedra. Two questions: what shape its 16 cells have (tetrahedra, just as the 3D octahedron has 8 triangular faces) and how many vertices it has (8).
Student tasks
- Choose the shape of the cells of the 16-cell (tetrahedron).
- Answer how many vertices it has (8) and compare with the 3D octahedron (6 vertices, one at each pole of each axis).
Room 12 · The 24-cell
The 24-cell, or icositetrachoron: 24 vertices (all permutations of (±1, ±1, 0, 0)), 96 edges, 96 triangular faces and 24 octahedral cells. It is its own dual and the only regular polytope that belongs to the fourth dimension alone: it has no analogue in any other. When rotated in XW or YW, its shadow goes from a star to a cuboid and back. Choice question: what shape its 24 cells have (octahedron).
Student tasks
- Rotate in XW or YW and watch how the shadow changes.
- Choose the shape of the cells (octahedron).
Room 13 · Building by analogy
From 0D to 4D: a point is swept to form a segment, the segment a square, the square a cube and the cube a tesseract. A canvas shows the five steps with their vertices, edges, faces and cells (V, E, F, C). Each step doubles the vertices. Two multiple-choice questions: how many vertices a 5D hypercube would have (32 = 2⁵) and how many tesseracts it contains (10, following the series 2 vertices, 4 sides, 6 faces, 8 cubes: rule 2n).
Student tasks
- Follow the dimensional construction up to the tesseract.
- Answer the two questions about the 5D hypercube (32 vertices and 10 tesseracts).
Room 14 · 4D polytope gallery ★
Five buttons select the polytope (5-cell, tesseract, 16-cell, 24-cell and 600-cell) and its rotating 3D projection is shown in a large viewer; it can be dragged to rotate. A statistics panel (V, E, F, C) updates with each polytope. In 4D there are exactly 6 regular polytopes; the 120-cell is not in the gallery. Free-answer question: what V − E + F − C equals for any regular 4D polytope (always 0).
Student tasks
- Explore the five polytopes and compare their projections.
- Calculate V − E + F − C for two or three polytopes and check the pattern.
- Enter the answer (0).
Room 15 · The 4D being’s superpowers
A Flatland-style illustration: a 2D being shut inside a “closed box” that we see from above. By analogy, a 4D being would see the inside of your closed room, could touch your heart without opening the skin or remove a ring without taking it off. A question with non-obvious distractors (folding space to shorten distances, perceiving time as one more spatial direction, transmitting force without contact) against the correct answer: undoing the knot in a closed loop of rope, which is impossible in 3D.
Student tasks
- Understand the 2D → 3D analogy as preparation for 3D → 4D.
- Identify which of the four “powers” really requires an extra spatial dimension.
Room 16 · What shadow do you see?
Final review: three single-answer questions, each followed by an explanation, about the 16-cell (8 vertices, one at each pole of each axis), the cuboctahedral section of the tesseract rotated 45° in the xw plane and cut at w = 0, and the 24-cell as the only regular polytope with no analogue in any other dimension.
Student tasks
- Answer the three questions using what has been learnt in the lab.
Room 17 · Hidden dimensions: Calabi-Yau manifolds ★
String theory requires 10 dimensions; the 6 we do not perceive are compactified, at every point of space, into a Calabi-Yau manifold (3 complex dimensions = 6 real ones, at the Planck scale). It is visualised with the exact parametrisation using complex trigonometry: z₁ = e^(2πik₁/n)·cos(x+iy)^(2/n), z₂ = e^(2πik₂/n)·sin(x+iy)^(2/n), projected to 3D with an angle a. A degree slider n (2-6, n² coloured sheets) with a note for each degree (from the toric variety to the sextic) and a 4D angle slider (from 0 to π). The shape rotates on its own. The quintic (n = 5, 25 sheets) is the most studied one in string theory.
Student tasks
- Look at the quintic n = 5 (25 sheets).
- Move the degree slider n and count how many sheets appear.
- Move the 4D angle slider and see how the manifold deforms.
Rooms to project
The most striking ones to show and discuss in class.