The dance angle is the food bearing minus the sun's azimuth. Move the clock and watch the angle drift.
Sun azimuth (from N, clockwise)-Sun altitude-Declination-Day length (geometric)-
Sun azimuth and altitude through the day
Dance angle for the selected patch
Dance angle α = bearing − azimuth (wrapped to ±180°). A dancer must keep updating α as the sun moves; the colony's compass is the sun, transposed onto gravity.
Waggle-run angle α-Waggle-run duration T-Dance circuits n-Trip profitability-Decoded-position RMS error-
Noise and calibration
Decoded angle vs bearing
Distance code: run duration vs distance
Red dots on the map are 120 recruits' decoded destinations for the selected patch.
Foragers in the field-Dancing now-Recruits sent / lost trips-Sugar delivered, dance-Sugar delivered, no dance-Net energy ratio (dance / none)-
Cumulative net energy
Where the foragers are
Efficiency ratio vs patchiness
Press “Sweep patchiness”: 6 levels x 6 landscapes, 6 h each, with and without dance.
A forager dances on the comb. Read the waggle-run angle and duration, add the angle to the sun's azimuth, and click the food on the map.
Round-Sun azimuth now-Waggle-run angle?Waggle-run duration?Last error-Score0Best (this browser)-
Press Start. Scoring: 100 points at zero error, 0 at 1.5 km or more.
Why hexagons? Compare regular shapes of equal area: the cell wall length per sqrt(area) is smallest for the hexagon among the shapes that tile the plane.
Cell area-Perimeter-Wax wall area / comb area-
Worker cell width about 5.2–5.4 mm and wall thickness about 0.07–0.1 mm are approximate textbook values.
The science: the dance code
A forager that found a rich patch runs a figure-eight on the vertical comb. The straight waggle run carries the message (von Frisch, 1967):
alpha = wrap180( bearing - sun_azimuth )
Straight up on the comb means “towards the sun”; alpha degrees clockwise from vertical means a food bearing alpha degrees clockwise from the sun. The code is transposed from the sky to gravity, so the dance angle drifts as the sun moves.
T_run = d / k (k about 0.75–1.0 km per s)
Distance is coded in the duration of the run. The slope k differs between honeybee races and habitats; the default 0.9 km/s here is approximate. The number of circuits grows with the profitability of the patch.
Recruits follow the dancer, decode angle and duration with noise (angles of about 10–15° SD, distance of about 10 %) and fly the vector; radar tracking of recruits showed they do fly it (Riley et al., 2005).
The science: the sun's azimuth
sin(alt) = sin(phi) sin(dec) + cos(phi) cos(dec) cos(H)
az = atan2( sin H, cos H sin(phi) - tan(dec) cos(phi) ) + 180°
phi latitude, dec solar declination (Cooper approximation, dec = 23.44° sin(360°(284+n)/365)), H = 15°(t - 12 h) the hour angle in solar time. At the equator at an equinox the sun's average azimuth rate is 15°/h; at higher latitudes it is slow near noon in winter and fast near noon in summer. Simplification: solar time, no refraction, no equation of time. Bees also use polarised sky light and landmarks when the sun is hidden.
The science: colony foraging model
Agent-based model (see the Colony tab): idle foragers either scout (random heading, a corridor of 120 m + patch radius 200 m), or follow a dance and fly a noisy copy of the vector, or return to a remembered patch (site fidelity). Profitability:
P = ( E_load - P_fly t_fly ) / ( t_fly + t_collect + t_unload )
with E_load = 40 mg x sugar fraction x 16.5 J/mg, flight speed 7 m/s, flight power about 0.05 W, collection time falling with nectar concentration and rising with crowding (1 + n/15). A forager dances only if P exceeds a threshold, with circuits growing with P. Positive feedback concentrates foragers on good patches; crowding makes returns diminishing, so the distribution tends towards the ideal free distribution (Fretwell & Lucas, 1970; Seeley, 1995). Colonies hold about 20,000–50,000 bees in summer; here a few hundred agents stand for the foragers (all numbers approximate).
Honest limits: the map is flat, no weather, no nectar depletion over the day, no odour or landmark learning, constant flight speed, and the dance threshold and circuit counts are illustrative.
The science: hexagons
P = sqrt(4 n tan(pi/n)) sqrt(A) : n=3 4.559, n=4 4, n=6 3.722
Among regular polygons that tile the plane the hexagon needs least wall for a given area; Hales proved in 2001 that no tiling by equal-area cells does better (the honeycomb conjecture). A circle (3.545) is better but does not tile. Real bees get hexagons partly from wax physics, not planning.
Facts (approximate) and references
Worker flight speed about 6.5–7.5 m/s; nectar load up to about 40 mg; waggle wag frequency about 13–15 Hz (the animation is slowed so you can see it); forager range typically a few km, up to 10 km.
von Frisch, K. (1967) The Dance Language and Orientation of Bees. Seeley, T. D. (1995) The Wisdom of the Hive. Riley, J. R. et al. (2005) Nature 435, 205. Dyer, F. C. (2002) Annu. Rev. Entomol. 47, 917. Fretwell & Lucas (1970) Acta Biotheor. 19, 16. Hales, T. C. (2001) Discrete Comput. Geom. 25, 1. Models built in Blender 5.2; Three.js r160 + WebGL.