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, 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 punchline: 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 $n/4$.

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.

Try it yourself. 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.

A few observations follow immediately:

• Almost all corners have roughly $n/2$ ones. 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 $\approx n/2$, which drops out of the same binomial picture for free.

References

Versions of the spiky-cube drawing appear all over high-dimensional probability and data-science texts. A few good places 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.
• Avrim Blum, John Hopcroft, Ravi Kannan, Foundations of Data Science. Cambridge University Press, 2020. Chapter 2 is a friendly tour of the geometry of high dimensions; available freely on the authors’ webpages.
• Stéphane Boucheron, Gábor Lugosi, Pascal Massart, Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, 2013.

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