Fig 1.
Actual structures made of beeswax on a wood ceiling (a-d) and schematic honeycomb images (e-j) taken during the first stage of the hive construction process, where panels e-g represent two-dimensional (2D) patterns on a plane (e.g., ceiling), and panels h-j represent 3D patterns. We consider the tetrapod structure (a), the size of which is of a similar order to that of a honeybee, to be the basic building block. On the plane, several tetrapods connect horizontally (b) in one direction (x-direction), which then elongates (c) in the perpendicular direction (y-direction). The structure also grows (d) in the vertical direction (z-direction). The schematic images e-g correspond to a-c, respectively. Thus, the pattern grows simultaneously in each direction.
Fig 2.
(a) An excavator on the silhouette figure of a honeybee. The excavator consists of a body segment (blue), head segment (yellow), and the antennae (green). Hereafter, the dual antennae are treated and referred to as a single organ. The black circle indicates the connection point between segments. (b) Excavator movement. The head segment can rotate around the connection point in the range of −π/2 to π/2. The excavator can move forward and backward along the axis of the body segment. (c) Excavation zone (EZ). The yellow region indicates the EZ in which the attached wax is removed, and the green band whose width is ds shows the area detected by the excavator’s antennae. The black circle indicates the rotation center.
Fig 3.
Motion of excavation zones (EZs).
Schematic overview describing the EZ motion (yellow), where white regions show clusters of wax. (a) EZs can move freely in the system far from the wax. All move at the same speed. (b) When one EZ collides headfirst into another EZ, the latter (EZ 2 in this example) is instantly transported to the edge of the system. (c) When two EZs meet head on, one of the EZs (EZ 4 in this example) is instantly transported to the edge of the system. (d) When an EZ (EZ 5) detects a cluster of wax within ds, it approaches the wax by rotating its body. (e) When an EZ (EZ 6) detects wax thinner than dw, it avoids digging wax at that location by rotating its body. (f) When an EZ (EZ 7) is surrounded by thin wax, it stops in place (local equilibrium state).
Table 1.
Parameters for numerical simulations, where l denotes the length of each side of the system.
Fig 4.
Wax growth without the excavators.
Simulation results of wax growth without the excavators for different anisotropic parameters px = 0.5, 0.7, and 0.9. The system surface (i.e., off-state) is indicated by brown, and the lattice cells that contain attached wax (i.e., on-state) are shown in white. The simulation parameters are summarized in the Methods section.
Fig 5.
Wax growth obtained in the attachment-excavation model simulation. The frame of these results is the center of nine squares partitioned within the system. The brown area shows the surface of the system and the white region shows the attached wax. The EZs are not displayed in the figures. The simulation parameters are summarized in the Methods section. A movie showing this time evolution (with and without EZs) is uploaded as S2 Video.
Fig 6.
Wax growth (anisotropic cases).
Wax growth obtained via the attachment-excavation model simulation for anisotropic parameters px = 0.5, 0.6, 0.7, 0.8, and 0.9, where px = 0.5 corresponds to isotropic growth. Other simulation parameters are summarized in the Methods section. The brown area denotes the system surface and the white region represents the attached wax. The EZs are not included in these figures. A movie showing this time evolution is uploaded as S3 Video.
Fig 7.
Linear sequence of constructed tripods in anisotropic cases.
Schematic illustration of the tripod pattern (a-d) constructing process and their 1D connection. EZs are shown in the upper four panels but omitted from the bottom ones. The brown area denotes the surface of the system, the white region denotes the attached wax, and the yellow object denotes the EZ.
Fig 8.
Time dependence of the wax amounts for different probabilities px = 0.5, 0.6, 0.7, 0.8, and 0.9. The amounts are measured as the number of on-state lattice cells. These results correspond to Fig 6.