Beauty and the Bees Mathematics in the Apiary Beauty and the Bees Mathematics in the Apiary Rachel Payne July 12, 2018 Once upon a time... “Nothing takes place in the world whose meaning is not that of some maximum or minimum.” – Leonhard Euler • Bee space • Top-bar hive construction • Smaller angle bees attached comb • Larger angle longer top bars and less comb • Bees stopped attaching comb at 120° Honey Bee Biology Drone Family Tree Drone Family Tree ퟏퟑ 8 5 3 2 1 1 Fibonacci sequence Fibonacci Numbers Nest Construction • Wax is expensive. Bees need to maximize area and minimize perimeter. • Known as early as 500 BC that only 3 regular polygons tile the plane Honeycomb Conjecture A regular hexagonal grid or honeycomb is the best way to divide a surface into regions of equal area with the least total perimeter. • 200 BC – Zenodorus – Of all regular polygons of equal perimeter, that is the greatest in area which has the most angles. • 36 BC – Marcus Terentius Varro – First record of Honeycomb Conjecture • 290-350 – Pappus of Alexandria – “Bees…by virtue of a certain geometrical forethought…know that the hexagon is greater than the square and the triangle, and will hold more honey for the same expenditure of material.” • 1943 – L. Fejes Tóth proved regular hexagon gives smallest perimeter for pattern made of polygons with straight edges. • 1999 – Thomas C. Hales proved complete conjecture. What about volume? Maximum volume, minimum surface area • 1712 – Maraldi measured cells. Angles of rhombus were 109° 28’ and 70° 32’. • Reaumur speculated bees’ choice of M might optimize space in the cell and posed a question. • Koenig found angles of rhombuses should be 109° 26’ and 70° 34’. • 1743 – Maclaurin was bothered by the discrepancy. Found Maraldi was correct. Would it be more economical for a bee cell to end in another shape? Yes! But only very slightly. Honey Bee Communication Waggle Dance Honey Bee Algorithm A biologically inspired approach to internet server optimization • 1988 – Seeley talked about honey bee foraging behavior on NPR • 1991 – Tovey joined Seeley in NY to test model • 2002 – Nakrani was working on a way to optimize server allocation Honey Bee Algorithm • Performs up to 20% more efficiently than others and distributes web transactions more smoothly and quickly for users • Has increased revenues for the companies in a global market worth more than $50 billion. • 2016 – Golden Goose Award to Bartholdi, Nakrani, Seeley, Tovey, and Vande Vate “The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.” – Aristotle ...and she lived happily ever after. The End References Some Historical Aspects of Honeycomb Mathematics - Stuart A. Baldwin http://coloradobeekeepers.org/csba/images/2013/Honeycomb-mathematics.pdf The Mathematics of the Honeycomb - Dr. Crypton http://www.sunday5april2015.info/geodesic/mathematics_of_honeycomb.pdf A Pollination Moment http://www.pollinatorparadise.com/market/zenith.htm Golden Goose Award Goes to Scientists Who Use Bees for Web Hosting https://www.aaas.org/news/golden-goose-award-goes-scientists-who-use-bees-web-hosting On Honey Bees and Dynamic Server Allocation in Internet Hosting Centers https://www.researchgate.net/publication/247757127_On_Honey_Bees_and_Dynamic_Server_Allocation_in_Internet_Hosting_Centers https://www.goldengooseaward.org/awardees/honey-bee-algorithm https://www.youtube.com/watch?v=elTfueXcYaU Top Bar Beekeeping – Les Crowder and Heather Harrell The Mathematical Universe – William Dunham.