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