Flatland to High Dimensions: Why Spheres Become Skin and Distance Loses Meaning

There is a 140-year-old novella that has resurfaced recently: Flatland by Edwin A. Abbott, published in 1884. Its world has only length and width, inhabited by triangles, squares, pentagons; the narrator is a Square.

What the Square Sees: A Sphere Passing Through Flatland

A Sphere visits Flatland and tries to explain it comes from a world with three directions. The Square cannot comprehend "up" because it does not exist in his plane. The Sphere then passes through the plane from below. The Square sees a point grow into a small circle, expand to a maximum, then shrink back to a point and vanish.

This can be expressed with a formula: sphere radius R, distance of sphere center from plane z, the cross-sectional circle radius is sqrt(R^2 - z^2). As z goes from -R to R, the radius grows from 0 to R and back to 0. The Square sees a sequence of cross-sections of the sphere in his world.

The Square refuses to believe: "That's just a circle that changes size."

We Are the Square: Extending to Four Dimensions

Now raise the analogy by one dimension. If a four-dimensional "hypersphere" passed through our three-dimensional space, we would see exactly the same phenomenon: a point becoming a small sphere, growing, then shrinking and disappearing.

We cannot "see" four dimensions, just as the Square cannot see three. But mathematics gives us two tools: look at cross-sections and count .

Invisible but Countable: The Hypercube

Start counting with a cube:

A point is 0-dimensional.

Drag the point along one direction to get a line segment: 2 vertices, 1 edge.

Drag the segment perpendicularly to get a square: 4 vertices, 4 edges.

Drag the square along a third direction to get a cube: 8 vertices, 12 edges.

The pattern: each drag doubles the vertices; old edges are duplicated, and each old vertex spawns a new edge. Extending this:

A four-dimensional hypercube (tesseract) has 16 vertices and 32 edges. We have never seen one, yet we can count its components precisely.

This is how mathematics works: where imagination cannot reach, regularity can take you.

First Oddity: The Sphere Is Almost All Skin

In high dimensions, some behaviors contradict intuition completely.

Experiment: peel a 1% thick shell from a sphere (thickness = 0.01 × radius). What fraction of the volume does this shell occupy?

Volume scales with radius ^n. After peeling, the inner radius is 0.99 of original, so remaining volume is 0.99^n. The shell fraction is 1 - 0.99^n.

In 3D, the shell is only 3%.

In 100D, it occupies 63%.

In 1000D, it occupies 99.996%.

In a 1000-dimensional world, peeling 1% of the radius leaves almost nothing. This is the same principle behind the "60 cm egg tart is all crust" example — the effect becomes extreme as dimension increases.

Second Oddity: The Sphere Peaks at Five Dimensions

Consider a unit-radius sphere. 1D "sphere" is a line segment of length 2. 2D sphere is a circle of area π. 3D sphere volume ≈ 4.19.

One might expect volume to grow with dimension. Actually:

Volume peaks at 5 dimensions (~5.26), then declines steadily; at 20 dimensions it is only 0.026, tending to zero.

Reason: as dimension increases, the sphere fills less and less of its enclosing cube; most of the cube's volume is pushed into the corners.

Why This Matters: The Curse of Dimensionality

You might ask: we don't live in high dimensions, so why care?

It matters enormously. A user on an e-commerce platform is represented by hundreds of numbers: age, city, browsing history, purchases — a point in hundreds of dimensions. Recommendation systems, face recognition, large models all search for "nearest points" in high-dimensional space.

High-dimensional space has a major problem. The author simulated 2000 uniformly random points in a unit cube (seed 7) and measured the ratio of farthest to nearest distance from a random query point:

In 2D, the farthest point is 80+ times farther than the nearest. In 1000D, it is only 1.1 times farther. In high dimensions, all points are roughly equidistant; "nearest neighbor" loses meaning.

This is the curse of dimensionality in machine learning. A key engineering task is to compress hundreds of dimensions down to the few truly important directions so that distance becomes meaningful again.

The Square's Fate

In the book, the Sphere lifts the Square out of the plane, letting him see three dimensions directly. The Square returns and preaches the "third direction", only to be imprisoned as a madman.

Reading 140 years later, the Square's predicament is ours: there are always dimensions we cannot see. Mathematics cannot make us "see" them, but it lets us examine their cross-sections piece by piece, count their components one by one, and compute their properties bit by bit.

Can you imagine four dimensions? Type 1 if yes, 2 if no.

References [1] Edwin A. Abbott. Flatland: A Romance of Many Dimensions . 1884. [2] The "near and far" data in the text come from the author's simulation (2000 uniformly random points in a unit cube, random seed 7).