angles
angles of the rhombus
Withe the formulas of the goniometric numbers in a rectangular triangle we can calculate the angles of the rhombusses.
The angles = 70.53° en = 109.47° corresponding the extremum matches the values experimentally measured by Maraldi in 1712.
angle of the rhombusses with the horizontal plane
With the formulas in a rectangular triangle we can calculate the angle with the horizontal plane.
For x = we find:
= 35.26°
theory and practice
The perfect six angled honeycombs of domestic bees aren't built from scratch by the bees themselves. A wax foundation is provided by the beekeepers.
Wild honeybees, however, don’t have the luxury of nestboxes fitted with wax foundations. Their honeycomb cells, though still very regular, aren’t always hexagonal—sometimes pentagons and heptagons creep in. After choosing an irregularly-shaped tree cavity, many worker bees will start independently building the hive’s hexagons at once, meeting in the middle along seams. It is along these seams where the five and seven-sided irregularities and other defects tend to appear.
The American science professor H. Randall Hepburn executed experimental research in South-Africa. He tried with variously sized hexagonal foundations. The largest caused the bees to to build gorgeous rosettes—they filled each large hexagonal foundation with five or six cells (again, typically five or seven-sided) surrounding a central cell. The next size down caused the bees to build in an irregular pattern, while the size smaller than that led to the bees building hexagonal cells on each vertex of the foundational hexagons, and leaving a hexagonal void or ‘false cell’ in the centre.
In these experiments the bees proved to be flexible problem solvers. Hepburns results are available in a book.
more information
Mathematicians did supplemantary theoretical research on alternative shapes and minimalisation. You can read some of it on honeycomb-geometry. The magazine Nature to published an interesting article by the Italian scientist Francesco Nazzi. As you can expect from Nature you can find a lot of detailes information and references on the geometry of bees.