mathematics in the apiary beauty and the bees · •290-350 –pappus of alexandria –“bees…by...
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Beauty and the BeesMathematics in the Apiary
Rachel Payne
July 12, 2018
Beauty and the BeesMathematics in the Apiary
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
𝟏𝟑
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
• 200 BC – Zenodorus – Of all regular polygons of equal perimeter, thatis the greatest in area which has the most angles.
• 36 BC – Marcus Terentius Varro – First record of HoneycombConjecture
A regular hexagonal grid or honeycomb is the best way to divide a surface into regions of equal area with the least total perimeter.
• 290-350 – Pappus of Alexandria – “Bees…by virtue of a certaingeometrical forethought…know that the hexagon is greater than thesquare and the triangle, and will hold more honey for the sameexpenditure of material.”
• 1943 – L. Fejes Tóth proved regular hexagon gives smallest perimeterfor 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
• 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
A biologically inspired approach to internet server optimization
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 exhibitorder, symmetry, and limitation; and these are thegreatest forms of the beautiful.”
– Aristotle
...and she lived happily ever after.The End
ReferencesSome 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