Big Maths Ideas · Beta

Kepler and sphere packing

Compare square stacks with close-packed layers of equal spheres.

Change the layer sequence to compare arrangements of identical spheres.

δ=π18≈0.740480\delta=\frac{\pi}{\sqrt{18}}\approx0.740480
Change the layer sequence to compare arrangements of identical spheres.Close-packed layers: ABCABC…

The spheres are identical. Colours distinguish layers; this perspective diagram is schematic.

Ideal bulk density
74.05%
Visible layer spacing (radius r)
√(8/3)r
The idea behind the experiment

Kepler’s conjecture is now a theorem: no packing of equal spheres in three-dimensional space has a higher density than close packing. Both FCC (ABCABC) and HCP (ABAB) attain π/√18. The number shown is bulk density, not the exact best fill of an arbitrary small container.

Thomas Hales on the Kepler proof
Predict, test and explain

Step 1 of 3

Predict

Do ABAB and ABCABC close-packed layers have the same ideal bulk density?

InvestigateWhy can both ABAB and ABCABC layers have the same bulk density?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Do ABAB and ABCABC close-packed layers have the same ideal bulk density?

δHCP=δFCC\delta_{\rm HCP}=\delta_{\rm FCC}

Hint

Equal spheres can form square or triangular layers.

Reveal answer

Both place each new layer in triangular holes and reach π/√18.

π18\frac{\pi}{\sqrt{18}}

Build an explanation

Hint 1

Equal spheres can form square or triangular layers.

Hint 2

A close-packed layer fits into holes between the previous layer’s spheres.

Hint 3

Bulk density and exact filling of a finite container are different questions.

Worked example

This example uses fixed values, separate from the diagram controls.

Compare the ideal bulk densities.

  1. Simple cubic density is π/6, about 0.524.
  2. Both close-packed sequences have density π/√18, about 0.740.
  3. The close-packed arrangement fills a larger fraction of space.

Answer: About 52.36% versus 74.05%.

Connect this idea

Watch out: Different layer sequences can share the same maximum bulk density.