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Like the hive, sustainble building/architecture takes into account natural, sustainable resources, energy efficiency, reduction of waste, solar proximity, and the well being of it's inhabitants.
EcoFriendly Building
Green Building, also known as green construction or sustainable building, is the practice of creating structures and using processes that are environmentally responsible and resourceefficient throughout a building's lifecycle: from siting to design, construction, operation, maintenance, renovation, and deconstruction. This practice expands and complements the classical building design concerns of economy, utility, durability, and comfort.[1]
Although new technologies are constantly being developed to complement current practices in creating greener structures, the common objective is that green buildings are designed to reduce the overall impact of the built environment on human health and the natural environment by:

Efficiently using energy, water, and other resources

Protecting occupant health and improving employee productivity

Reducing waste, pollution and environmental degradation[1]
A similar concept is natural building, which is usually on a smaller scale and tends to focus on the use of natural materials that are available locally.[2] Other related topics include sustainable design and green architecture.
Reducing Environmental Impact
Green building practices aim to reduce the environmental impact of buildings. Buildings account for a large amount of land use, energy and water consumption, and air and atmosphere alteration. Considering the statistics, reducing the amount of natural resources buildings consume and the amount of pollution given off is seen as crucial for future sustainability, according to EPA.The environmental impact of buildings is often underestimated, while the perceived costs of green buildings are overestimated. A recent survey by the World Business Council for Sustainable Development finds that green costs are overestimated by 300 percent, as key players in real estate and construction estimate the additional cost at 17 percent above conventional construction, more than triple the true average cost difference of about 5 percent.
http://environmentecology.com/environmentandarchitecture/80greenbuilding.html
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Building Most efficient Shape in Nature
After nearly two thousand years, mathematicians have now finally proved that honeybees are among the world's most efficient builders.
The 4th century geometer Pappus was one of several ancient Greek mathematicians who suspected that the elegant shape of the honeycomb was a result not of an innate beesense of geometric beauty but of nature's efficiency. The repeating pattern of sixsided figures you see in a crosssection of a honeycomb, Pappus guessed, used the least amount of wax to build the walls.
His guess, in an essay on "the sagacity of bees", became known as the Honeycomb Conjecture. It resisted all attempts to prove it until a few weeks ago, when mathematician Thomas Hales of the University of Michigan announced that he had cracked the puzzle.
Not until the advent of closeup film techniques did scientists know for certain how bees build their honey stores. It is a remarkable feat of high precision engineering. Young worker bees excrete slivers of warm wax, each about the size of a pinhead. Other workers take the freshly produced slivers and carefully position them to form vertical, sixsided, cylindrical chambers (or cells). Each wax partition is less than 0.1mm thick, accurate to a tolerance of 0.002mm. Each of the six walls is exactly the same width, and the walls meet at an angle of precisely 120 degrees, producing one of the "perfect figures" of geometry, a regular hexagon.
Why don't bees make each cell triangular, or square, or some other shape? Why have straight sides in the first place? After all, the warm wax could just as well be formed into curved walls.
Although a honeycomb is a threedimensional object, because the individual cells are all cylindrical, the total area of the wax walls depends solely on the shape of the crosssection of the cells. Thus, the mathematical problem is one of twodimensional geometry  the kind we learn in school. What it boils down to is finding the twodimensional shape that can be repeated endlessly to cover a large flat area, for which the total length of all the cell perimeters is the least (so that the total area of the honeycomb walls is as small as possible)
Some facts are easy to establish. For instance, only three kinds of regular polygons (straightedged figures whose edges are all the same length and whose angles are all the same) can be fitted together sidebyside to cover a plane: equilateral triangles, squares, and regular hexagons. Others will leave gaps. Of the three spacefilling figures, squares give a smaller total perimeter than triangles and hexagons do even better than squares.
It is also easy to show that, if you restrict yourself to hexagons, the regular ones (ie the ones with equal sides and angles all 120 degrees) give a smaller perimeter than non regular ones.
But if you allow combinations of polygons of all kinds, or edges that are not straight lines, then things become a lot more complicated. Relatively little was known until 1943, when Hungarian mathematician L Fejes Toth used an ingenious argument to prove that the regular hexagon pattern does give the smallest total perimeter for all patterns made up of any combination of straightedged polygons.
What happens if the edges can be curved? Toth thought the regular hexagon pattern would still be more efficient than anything else, but he could not prove it.
Thought of in terms of a single honeycomb cell, if a wall bulges out, you can store more honey in that cell for the same wall area than if the wall were straight. But when all the cells are packed together, a wall that bulges out for one cell bulges in for the adjacent cell. Could there be an entire honeycomb of cells with bulging walls where the net increase in efficiency of the outward bulges outweighs the net decrease caused by the inward bulges? If there were such a pattern, the honeycomb conjecture would be false.
Intuitively, outward bulges would exactly balance out inward bulges, which is why Toth thought the hexagonal pattern would be the best. But as mathematicians observed, things are not quite as simple as they might seem.
Nevertheless, that is exactly what Hales has proved: the bulges do cancel out. He has posted his 19 page proof at: www.math.lsa.umich.edu/~hales and experts who have seen it say it seems correct.
Last year, Hales proved that the most efficient way to pack equalsized spheres together in a large crate is to arrange them in a regular pattern of staggered layers, the way greengrocers the world over stack oranges, so that the oranges in each higher layer sit in the hollows made by the four oranges beneath them.
That problem, which was first raised by the astronomer Johannes Kepler in 1611, resisted many attempts at a solution until Hales took 250 pages and a three gigabyte computer package (program plus data) to prove what supermarket employees know instinctively.
But the instincts of small creatures go way beyond human capabilities. Take the Tunisian desert ant. This tiny creature can wander across the desert sands for up to 50 metres until it stumbles across the remains of a dead insect, whereupon it bites off a piece and takes it directly back to its nest  a hole no more than one millimetre in diameter.
How does it find its way back? By the process the Apollo astronauts used to plot their course to the Moon: dead reckoning (strictly "ded" reckoning from "deductive reckoning). The idea of dead reckoning is to calculate your position relative to your starting point from a knowledge of your speed and your direction of travel.
Some ants lay down a scent trail to keep track of their path but not the Tunisian desert ant. If it is moved after it finds the food, it will head off in exactly the direction it should have to find its nest if it had not been moved, and will stop and start a bewildered search for its nest when it has covered the precise distance that should have brought it home.
Many birds that migrate over featureless seas navigate by the stars. Humans need optical equipment, star charts, and trigonometric calculations. The birds seem to learn the pattern of the stars  which has to be learned, since it changes faster than evolutionary adaptation could adjust for  by gazing up at the sky as fledglings in the nest.
Those who struggle with mathematics sometimes marvel at the way birds or insects can perform "mathematical" feats. But these creatures do not use mathematics as we do. Natural selection has produced a rich diversity of creatures that can eachperform a particular range of activities to survive. Human beings have extended their own range of instinctive behaviours so that they can mimic some of the activities of their fellow creatures. Using mathematics, science, and technology, we too can navigate around the globe and build elaborate places to live.
The fact that it has taken 2,000 years of effort and 19 pages of advanced mathematics to show that the familiar honeycomb is the most efficient pattern for the storage of honey is as much a marvelous testament both to the wonders of nature and to our mathematical ingenuity.
Building the Hive
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