If you’ve flipped through a textbook on high-dimensional probability or geometry, you’ve seen the obligatory drawing of an $n$-dimensional cube. It is usually rendered as a strange spiky object: a sea urchin, a thorny diamond, a star with way too many points. Why does the high-dimensional cube get drawn like that?

Some of it is concentration of measure (the phenomenon where, in high dimensions, almost all of a set’s mass sits in a thin shell rather than being spread out), but there is a much simpler counting argument behind a particular family of these drawings. It is just the binomial coefficient in disguise.

Counting corners

The unit cube in $n$ dimensions is the set $[0,1]^n$. Its corners are the points where every coordinate is either $0$ or $1$. There are exactly $2^n$ of them.

• $n = 1$: $2$ corners, namely $(0)$ and $(1)$.
• $n = 2$: $4$ corners, namely $(0,0), (0,1), (1,0), (1,1)$. The unit square.
• $n = 3$: $8$ corners. The familiar 3D cube.
• $n = 10$: $1{,}024$ corners.
• $n = 30$: just over a billion.

Now group the corners by Hamming weight, the number of coordinates equal to $1$. Define

How many such corners are there? You’re picking which $\ell$ of the $n$ slots get the $1$, so

That is the point: the level distribution of cube corners is exactly the binomial coefficient.

What does $\binom{n}{\ell}$ look like?

A bell. Peaked at $\ell = n/2$, symmetric around it, with $C_0 = C_n = 1$ at the extremes and $C_{\lfloor n/2 \rfloor} \approx 2^n / \sqrt{\pi n / 2}$ in the middle.

For large $n$, the de Moivre–Laplace theorem (a classical, pre-CLT special case of the central limit theorem) gives

In other words, the count of corners at level $\ell$, viewed as a function of $\ell$, converges to a Gaussian with mean $n/2$ and variance $n/4$, exactly the mean and variance of $n$ independent fair coin flips.

Drawing the cube

Now suppose we draw the cube on a 2D page. One natural convention: stack the levels vertically and draw each level as a horizontal band whose width is proportional to $C_\ell$.

For small $n$ this is a chunky diamond. For large $n$ it is a vertical bell curve, narrow at the top and bottom (only one corner each, at $\ell = 0$ and $\ell = n$) and bulging in the middle. The ratio between the middle width and the tip width is $\binom{n}{\lfloor n/2 \rfloor}$, which is roughly $2^n / \sqrt{n}$, exponentially huge.

Slide $n$ and watch the chunky diamond at $n = 4$ sharpen into a needle-tipped spindle:

That’s the spike. The drawing has sharp points at the top and bottom because there is exactly one corner at each end, and a dramatic bulge in the middle because the binomial is dramatic about its peak. As $n$ grows, the tips become invisibly thin while the middle balloons out.

The small-$n$ end of the slider gives something more familiar. At $n = 2$ the levels have widths $1, 2, 1$, so the silhouette is a diamond. At $n = 3$ the widths are $1, 3, 3, 1$, giving a hexagonal outline that looks like the way we usually sketch a 3D cube on paper. The spiky cross-section is built the same way; it only starts to look spiky once the binomial peak pulls far away from the tips.

A few observations follow immediately:

• Almost all corners have roughly $n/2$ ones. Since the standard deviation is $\sqrt{n}/2$, a deviation of $t\sqrt{n}$ is $2t$ standard deviations, so the fraction of corners with $|\ell - n/2| > t \sqrt{n}$ falls off Gaussian-fast in $t$.
• The tips are vanishingly rare. Out of $2^n$ corners, only two (the all-zero and the all-one) sit at the extreme levels. They are the spike points that give the drawing its name.
• Two random corners are typically at Hamming distance $\approx n/2$: they disagree in each coordinate independently with probability $1/2$, so the number of disagreements is again binomial.

References

Versions of the spiky-cube drawing appear all over high-dimensional probability and data-science texts. One good place to look:

• Roman Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge University Press, 2018. The first few chapters set up exactly this geometric picture before deploying it for concentration inequalities.

The drawings vary author to author, but the underlying combinatorics is always the same: corners count as a binomial, and a binomial peaks sharply.