Artificial Wasteland  ·  the life seam  ·  a showing, checked

What the Bees Don’t Know

Bees build hexagons, and the reason is real: of all the ways to divide a surface into equal cells, the hexagon walls off the most area with the least wall. But the two famous sequels — that bees also nail the perfect three-dimensional cell, and that they once out-computed a mathematician — are both myths. Here is the geometry that is true, the geometry that isn’t, and every number recomputed in front of you.

Instrument 1 — the wall-economy contest

Every cell below has the same area. The only thing that changes is how much wall it costs to fence them off. Tap a shape; the bar chart updates with the total wall length per unit of floor. Less wall = less wax. Watch the hexagon win — and watch the circle cheat.

Triangles Squares Hexagons Circles

Hexagons: wall per unit area = 1.861 — the most economical tiling there is.

Pappus of Alexandria noticed this around 340 CE. In his Collection he wrote that bees, “in their wisdom,” chose the cell “that has the most angles… perceiving that it would hold more honey for the same expenditure of material.” He compared the only three regular shapes that tile the plane edge-to-edge — the equilateral triangle, the square, the regular hexagon — and the hexagon won. Fifteen centuries later Darwin, in the Origin of Species, called the result perfect:

“…the comb of the hive-bee, as far as we can see, is absolutely perfect in economising wax.” Charles Darwin, On the Origin of Species (1859), ch. VII, “Instinct”

And the contest above shows why, in numbers: a hexagonal grid spends about 7% less wall than squares and 18% less than triangles for the same floor. But notice the fourth option. The circle beats every one of them — a disk has the least perimeter of any shape enclosing a given area (the isoperimetric champion). Bees don’t build circles, and they’re right not to: circles can’t tile. Pack them as tightly as geometry allows and 9.3% of the floor is still wasted gap. The hexagon is the best shape that also leaves no gap — it is the circle’s honest compromise with tiling.

The part that stayed open for 1,659 yearsPappus checked three shapes. The theorem is about all of them.

Here is the catch that the popular telling skips. Pappus only compared the three regular tilings. But why should the walls be straight? Why six sides and not a wavy boundary, or a mix of pentagons and heptagons, or some cunning curved partition no one has drawn? To prove the hexagon is best, you have to beat every way of carving a plane into equal areas — including ones with curved walls. That problem, the Honeycomb Conjecture, stayed open from Pappus until 1999, when Thomas C. Hales proved it (published 2001): any partition of the plane into regions of equal area has total wall length at least that of the regular hexagonal grid. Curved walls don’t help; irregular cells don’t help. The hexagon is genuinely, provably optimal — and the proof is younger than most of the people reading this.

---

Into the third dimensionThe cell has a back wall. That’s where the story turns.

A honeycomb is two layers of cells back to back, and a real cell is not a flat-bottomed tube. Its base is closed by three rhombi tilted into a shallow point, so the bases of one layer interlock with the bases of the other — the same way the faces of a rhombic dodecahedron pack. That three-rhombus base is where bees are supposed to show their second feat of genius: of all the ways to angle those rhombi, one angle uses the least wax. Tilt the base yourself and find it.

Instrument 2 — find the wax-cheapest base

The curve is the cell’s wax cost (surface area at fixed volume) as you change the obtuse angle of the base rhombus. Slide to hunt for the bottom of the valley. The dashed mark is where Maraldi measured the real comb in 1712; the cool mark is the angle a mathematician first computed for it.

Rhombus obtuse angle Snap to the optimum

Angle 118.0° — wax cost 1.430. The minimum is 1.4142, reached at 109.47°.

The bottom of that valley sits at 109.47° — exactly arccos(−1/3), the angle of a regular tetrahedron, the angle of a rhombic dodecahedron’s faces. Three independent routes land on it: minimise the cost curve numerically; solve cos θ = 1/√3 by calculus; or note that the optimal rhombus has diagonals in the ratio √2 : 1. All give 109.4712°.

The eighteenth century’s favourite anecdote

In 1712 the astronomer Giacomo Maraldi, at the Paris Observatory, gave the comb’s rhombus angles as 109°28′ and 70°32′. Réaumur suspected those angles minimised wax and put the problem to the mathematician Johann Samuel König, who in 1739 computed an optimum of about 109°26′ — tantalisingly two arc-minutes off Maraldi’s comb. In 1743 Colin Maclaurin re-derived it cleanly and got 109°28′, matching Maraldi and the true optimum arccos(−1/3).

From that two-arc-minute gap grew one of science’s most repeated bedtime stories: that the bees had the angle right and the mathematician had it wrong — nature out-computing a Fellow of the Royal Society. It is a wonderful story. It is also false, and the way it’s false is the whole point of this page.

Three myths, weighed honestlyWhat the bees don’t know

And the deepest oneDo bees even “know” the hexagon at all?

Here the honest answer is: we’re still arguing. Two camps, both with real evidence:

The wax flows. Bees first build roughly circular cells. Warmed near its melting point, the wax behaves like a slow liquid, and surface tension pulls the walls at each three-way junction into 120° meeting angles — hexagons forming the way soap bubbles do in a foam, no geometry required of the bee (Pirk et al., 2004; Karihaloo et al., 2013).

The bees build. Against this: thermographic recordings of comb under construction put the wax at 33.6–37.6°C — below the ~40°C needed for it to flow freely — while the bees were filmed actively shaping each wall with antennae, mandibles and legs in a regular sequence (Bauer & Bienefeld, 2013). Later work shows bees deliberately building non-hexagonal five- and seven-sided cells to stitch together regions of different size — the mark of a builder solving a problem, not a fluid settling into a default (Smith, Napp & Petersen, 2021).

The current weight of evidence leans toward active construction, with a real strand of self-organising physics surviving in the middle (Gallo & Chittka, 2021, call it openly unresolved). So the title is doubled: the bees may not know the hexagon in any sense at all; the order may be partly handed to them by physics, partly worked by hand — and the one sharp optimum that was theirs to claim, the perfect cell base, they never reached.

The check — what was recomputed, and what is only cited

Recomputed here, in front of you and in research/what-the-bees-dont-know/verify.mjs (16/16):

• Wall per unit area for the three regular tilings: hexagon √(2√3) ≈ 1.861 < square 2 < triangle √(3√3) ≈ 2.280. The bars and the contest read straight off these.
• The circle’s perimeter-per-area is lower than any tiler, but densest disk packing covers only π/√12 ≈ 90.7% of the plane — the 9.3% gap is real arithmetic.
• The 3-D base optimum arccos(−1/3) = 109.4712°, found three independent ways (numeric minimum of the cost curve = √2; the calculus condition cosθ=1/√3; the √2:1 diagonal ratio), and matching Maraldi’s measured 109°28′ to the arc-minute. Instrument 2 plots that exact curve.

Cited, not reproduced (and flagged as such): Hales’ 2-D theorem over all partitions (a 20-page proof, not a browser computation); Fejes Tóth’s 0.035% 3-D improvement; the wax-temperature measurements and the construction-vs-flow evidence. Sources below. Two arc-minute figures (König’s 109°26′, the cause of his error) are historical and not independently re-derived here; the “bee beat the mathematician” debunking follows D’Arcy Thompson and is presented as the scholarly consensus, not a fresh finding.

ApparatusSources & honest edges

Geometry checked in research/what-the-bees-dont-know/verify.mjs. The two “myths” are not strawmen — they are the standard popular claims, each with a real, citable correction. Where the science is unsettled (why hexagons form) the page says so rather than picking a winner.

1. T. C. Hales, “The Honeycomb Conjecture,” Discrete & Computational Geometry 25 (2001) 1–22. arXiv:math/9906042 (1999 preprint). Plain-language: “Cannonballs and Honeycombs,” Notices AMS 47(4) (2000) 440–449.
2. Pappus of Alexandria, Collection (Synagoge), Book V, c. 340 CE — the bees’ “geometrical forethought” and the three-tiling comparison.
3. C. Darwin, On the Origin of Species, 1st ed. (1859), ch. VII “Instinct”: “…absolutely perfect in economising wax.” Darwin Online. (The often-quoted “labour and wax” conflates this with an adjacent sentence; the verbatim clause ends at “wax.”)
4. G. Maraldi, “Observations sur les abeilles,” Mém. Acad. Roy. Sci. (1712) — the 109°28′/70°32′ angles. C. Maclaurin, “Of the bases of the cells…,” Phil. Trans. R. Soc. 42 (1743) 561–568. On J. S. König & Réaumur: MacTutor.
5. D’A. W. Thompson, On Growth and Form (1917/1942), honeycomb chapter — debunks the “bee was right, mathematician wrong” legend (traceable to Lord Brougham, 1858). Project Gutenberg.
6. L. Fejes Tóth, “What the bees know and what they do not know,” Bull. Amer. Math. Soc. 70 (1964) 468–481 — the better two-hexagon-two-rhombus base. AMS (PDF). The 0.035% figure (≈1 part in 2,850) follows the standard secondary accounts; the finite-wall-thickness flip is Weaire & Phelan (1994).
7. C. W. W. Pirk, H. R. Hepburn, S. E. Radloff & J. Tautz, “Honeybee combs: construction through a liquid equilibrium process?,” Naturwissenschaften 91 (2004) 350–353. B. L. Karihaloo, K. Zhang & J. Wang, “Honeybee combs: how the circular cells transform into rounded hexagons,” J. R. Soc. Interface 10 (2013) 20130299 — the surface-tension camp.
8. D. Bauer & K. Bienefeld, “Hexagonal comb cells of honeybees are not produced via a liquid equilibrium process,” Naturwissenschaften 100 (2013) 45–49 (wax measured 33.6–37.6°C). M. L. Smith, N. Napp & K. H. Petersen, “Imperfect comb construction reveals the architectural abilities of honeybees,” PNAS 118(31) (2021) e2103605118. V. Gallo & L. Chittka, PNAS 118(33) (2021) — the question framed as open.