Math-bees: Min and Max | etwinning-2015-2016

Mission: Find out the hidDen Min-Max!

WHERe is it ? Part 1

THE BEE'S activity

1. by TS. Schimatari

Solved! A bee-buzzing, honey-licking 2,000-year-old mystery that begins here, with this beehive. Look at the honeycomb in the photo and ask yourself: (I know you've been wondering this all your life, but have been too shy to ask out loud ... ) Why is every cell in this honeycomb a hexagon?

Bees, after all, could build honeycombs from rectangles or squares or triangles ...

But for some reason, bees choose hexagons. Always hexagons.

And not just your basic six-sided hexagon.

They like "perfect" hexagons,

meaning all six sides are of equal length.

They go for the jewelers' version — precise, just so.

Why?

Well, this is a very old question. More than 2,000 years ago, in 36 B.C., a Roman soldier/scholar/writer, Marcus Terentius Varro, proposed an answer, which ever since has been called "The Honeybee Conjecture." Varro thought there might be a deep reason for this bee behavior. Maybe a honeycomb built of hexagons can hold more honey. Maybe hexagons require less building wax. Maybe there's a hidden logic here.

I like this idea — that below the flux, the chaos of everyday life there might be elegant reasons for what we see. "The Honeybee Conjecture" is an example of mathematics unlocking a mystery of nature, so here, with help from physicist/writer Alan Lightman, (who recently wrote about this in Orion Magazine) is Varro's hunch.

The Essential Honeycomb

Honeycombs, we all know, store honey. Honey is obviously valuable to bees. It feeds their young. It sustains the hive. It makes the wax that holds the honeycomb together. It takes thousands and thousands of bee hours, tens of thousands of flights across the meadow, to gather nectar from flower after flower after flower, so it's reasonable to suppose that back at the hive, bees want a tight, secure storage structure that is as simple to build as possible.

So how to build it? Well, suppose you start your honeycomb with a cell like this ... a totally random shape, no equal sides, just a squiggle ...

If you start this way, what will your next cell look like? Well, you don't want big gaps

between cells. You want the structure tight. So the next cell will have to be customized

to cling to the first, like this ...

... this method of constructing a honeycomb would require that the worker bees

work sequentially, one at a time, first making once cell, then fitting the next cell to

that, and so on.

But that's not the bee way. Look at any YouTube version of bees building a

honeycomb, and you won't see a lot of bees lounging about, waiting for their turn

to build a cell.

Instead, everybody's working. They do this collectively, simultaneously and constantly.

So a "squiggle cell plan" creates idle bees. It wastes time. For bees to assemble a honeycomb the way bees actually do it, it's simpler for each cell to be exactly the same. If the sides are all equal — "perfectly" hexagonal — every cell fits tight with every other cell. Everybody can pitch in.

That way, a honeycomb is basically an easy jigsaw puzzle. All the parts fit.

OK, that explains why honeycomb cells are same-sized. But back to our first question: Why the preference for hexagons? Is there something special about a six-sided shape?

Some shapes you know right away aren't good. A honeycomb built from spheres would have little spaces between each unit ...

.. creating gaps that would need extra wax for patching. So you can see why a honeycomb built from spheres wouldn't be ideal. Pentagons, octagons also produce gaps. What's better?

"It is a mathematical truth that there are only three geometrical figures with equal sides that can fit together on a flat surface without leaving gaps: equilateral triangles, squares and hexagons."

So which to choose? The triangle? The square? Or the hexagon? Which one is best? Here's where our Roman, Marcus Terentius Varro made his great contribution. His "conjecture" — and that's what it was, a mathematical guess — proposed that a structure built from hexagons is probably a wee bit more compact than a structure built from squares or triangles. A hexagonal honeycomb, he thought, would have "the smallest total perimeter." He couldn't prove it mathematically, but that's what he thought.

Compactness matters. The more compact your structure, the less wax you need to construct the honeycomb. Wax is expensive. A bee must consume about eight ounces of honey to produce a single ounce of wax. So if you are watching your wax bill, you want the most compact building plan you can find.

And guess what?

Two thousand thirty-five years after Marcus Terentius Varro proposed

his conjecture, a mathematician at the University of Michigan, Thomas Hales,

solved the riddle. It turns out, Varro was right. A hexagonal structure is

indeed more compact. In 1999, Hales produced a mathematical proof that

said so.

As the ancient Greeks suspected, as Varro claimed, as bee lovers have always thought, as Charles Darwin himself once wrote, the honeycomb is a masterpiece of engineering. It is "absolutely perfect in economizing labor and wax."

The bees, presumably, shrugged. As Alan Lightman says, "They knew it was true all along."

Alan Lightman's essay "The Symmetrical Universe," originally published in Orion Magazine, will be included in his new book The Accidental Universe: The World You Thought You Knew to be published early next year. I also recommend Ivars Peterson's essay in Science News, The Honeycomb Conjecture.

http://www.npr.org/sections/krulwich/2013/05/13/183704091/what-is-it-about-bees-and-hexagons

"..The honeycomb is absolutely perfect in economizing labor and wax..."

Charles Darwin

The honeycomb conjecture states that

a regular hexagonal grid or

honeycomb

is the best way to divide a surface

into regions of equal area

with the least total perimeter.

The conjecture was proven in 1999

by mathematician Thomas C. Hales.

Statement

Let Γ be a locally finite graph in R2, consisting of smooth curves, and such that R2 \Γ has infinitely many bounded connected components, all of unit area. Let C be the union of these bounded components.[1]

Then

Equality is attained for the regular hexagonal tile.

Honeycomb up close

Using a honeycomb grown at a research facility in Beijing, the researchers were able to carefully ward off the bees and photograph the bare honeycomb seconds after formation, providing the first clear evidence that cells naturally start as circles. They then observed honeybees heating the wax after the initial cell formation — a phenomenon identified in previous studies, but never analyzed in close detail — and found this to be the key step in hexagon-formation.

By heating the cells, the bees cause the wax to become molten and flow like lava. Once the wax starts flowing, the cell walls naturally fall flat and take on the shape of a hexagon, like adjoining bubbles in a bath. This is physically the simplest and most stable way for cylinders to merge.

The team still does not know exactly how the bees go about heating each cell, and explored the mechanics of two plausible scenarios: One in which the bees focus their heat only at points where neighboring cells touch (a total of six points per cell), and another in which the bees heat the entire cell all at once.

"My own feeling is that nature tries to minimize energy spent, and from that point of view, I would think that scenario one is most probable," was written in LiveScience. "But, on the other hand, from the perspective of the bees, they might just want to warm the whole thing and get away with it. That remains to be shown."

See more at: http://www.livescience.com/38242-why-honeybee-honeycombs-are-perfect.html#sthash.72xGxW2f.dpuf

Bees and Maths

by Valluri Fossano

1.

2.

Have you ever wondered why bees to build honeycomb cells use the hexagon shaped ?

The hexagon is the form that allows you to hold more honey because it has a larger area than the square and the equilateral triangle of equal perimeter, although all these three forms could be linked without leaving gaps and without overlapping.

Besides bees do not use the form to the circle because it would remain empty spaces, while the hex does not leave holes, because the form in each vertex three angles of 120°.

3.

Steiner tree problem.

by Vallauri-Schimatari

Today minimizing is crucial in economics : make faster , at the lowest possible cost . If a railway company has to connect different locations , the goal will be to try to decrease the length of the track to shorten travel times and at the same time reduce construction costs . Minimize the length of pipes, cables , how?

If you have to connect the two cities , there is no problem : everyone knows that the line segment will be the best course . But what is the best network ( shortest ) connecting three or four cities ? Unfortunately , as just increases the number of points to be connected , also increases the amount of possible different paths to consider! Once again we come to the aid of soap films .

We soak in soapy water two transparent sheets ( Plexiglas ) flanked by three pins : what form will emerge ? ( The students conjectures : circle ? )

The soap , to reduce the surface tension , it assumes a configuration that minimizes the total distance . In other words , it gives us the searched pattern.

The solution of the problem to three points was determined for the first time from the mathematical and physical Italian Evangelista Torricelli in 1650 , the inventor of the thermometer .

What will happen with 4 screws:

and with 5 screws and so on ?

The soap film course assumes minimal configuration: n points to be connected they always form n - 2 points Fermat -Torricelli or point of Steiner ( Jacob Steiner , 1800 Swiss mathematician who had worked on optimization problems) .

The connections are always among three branches and will always determine angles of 120 ° .

We can observe these minimal surfaces in space.

If we use a frame in the shape of tetrahedron we will obtain the same structure of methane molecule