Flatland: the maths
Exact geometry: slices of spheres, cones and cubes by a lower-dimensional space, projections, and ray casting for A Square's view. The story is Abbott's 1884 analogy.
Equations
- r = √(R² − z²)
- The circle A Square sees when a sphere of radius R is at height z above his plane (nothing when |z| ≥ R).
- corners = 2ⁿ, edges = n · 2ⁿ⁻¹
- An n-dimensional cube: each step doubles the shape and joins every corner to its copy.
- tilt < 45° → ellipsoid; = 45° → paraboloid; > 45° → hyperboloid
- Slicing a 4D cone at a tilt: the 3D cousins of the ellipse, parabola and hyperbola.
- tesseract ∩ {x + y + z + w = d}
- A tesseract passing corner-first through our space: its slice starts as a tetrahedron and is an octahedron halfway.
Constants
| dimensions | 2 (Flatland) to 5 |
Worked examples ✓ checked on every change
| Sphere R = 3 at height 2.7: the slice's radius | 1.308 | √(R² − z²) |
| Tesseract (n = 4): corners and edges | 16 corners, 32 edges | 2⁴ and 4 · 2³ |
| 4D cone sliced at exactly 45° | paraboloid | the conic at the cone's own slope |
| Tesseract corner-first, just past the corner: slice's corners | 4 (a tetrahedron) | 4 edges meet at a corner |
| …halfway through | 6 (an octahedron) | the 6 corners with two +1s and two −1s |
What's simplified
- Orthographic projection for the 3D views; the Flatland world is drawn from above for us, and as a single line for A Square.
Where it breaks
It doesn't — the geometry is exact. The analogy is the part to hold lightly: tagged ANALOGY.
Sources: E. A. Abbott, Flatland (1884); H. S. M. Coxeter, Regular Polytopes (1973).