Almost all buildings are shaped like boxes but, apart from simplicity of visualization, when you get right down to it the only thing a rectangular box has going for it is that it's a good place to store smaller rectangular boxes.
Take a look at DaVinci's Vitruvian Man, drawn to illustrate human physiology mathematics. We can see that a circle roughly accomodates the location of the eyes and reach of arms and legs, ie: everything is within immediate visual and physical reach. But circles don't stack properly, and if people lived or worked in tubes they'd bump into each other a lot while walking around.
Hexagons on the other hand stack quite well, can be stretched in height or width while maintaining stackability, and the floors are flat. Stairways, rather than occupying a large portion of floorspace, take up no floorspace whatsoever being directly mounted onto the canted wall. All the shelves on a full-height cupboard are both viewable and reachable. There's more stretching room for the same area.
Immediate applications include ships, cottages, holes dug into the side of extraterrestrial mountains, that sort of thing.-- FlyingToaster, Aug 04 2012 http://en.wikipedia.../wiki/Vitruvian_man [FlyingToaster, Aug 04 2012] stacked hexagons http://www.rdwarf.c...ular/screenshot.gif [FlyingToaster, Aug 04 2012] That was quick . . .-- FlyingToaster, Aug 04 2012 Don't look at me.-- Phrontistery, Aug 04 2012 I have some sympathy with this thought and i like the idea of hexagonal architecture. It reminds me, however, of a problem which arose with geodesic domes. Rectangular boxes are good for other rectangular shapes, but the thing is, as a result of our rectangular obsession that's what we've got, in various proportions and sizes. Furniture is going to fit along the walls of such rooms but not the corners unless altered or specially designed. That means the space would still be inefficiently used, less so in fact than rectangular space. The alternative is to stop using rectangular items and replace or modify all of them. The same applies to the exterior of the rooms. As the outside is rectangular, on the whole this will waste space in cities.
Also, hexagons are not inherently strong, a feature they share with rectangles but not with triangles. Besides that, there's the question of the ratio of perimeter to area. Squares and equilateral triangles are both worse than hexagons in this respect, but of course dodecahedra are better than any other Platonic polyhedron. They have other disadvantages.
Concerning Vitruvian man, since we have no tails we are pentagons rather than hexagons. This makes people in general hard to tesselate.
I'm not completely hostile to this. I think abandoning rectilinearity is a very stimulating measure. I just doubt that hexagons are the answer, mainly because it's too big a break with the past and it's not three-dimensional. I think we trap ourselves in harmful approaches by thinking in terms of flat surfaces as well as in terms of right angles, and we need to abandon both where that's practical.-- nineteenthly, Aug 04 2012 //pentagons// //triangles// //duodecahedra// None of those fit together and retain viable floors. Except a 12-gon with a skinny little floor, but that won't fill at all.
//Vitruvian man// may show that we have 5 lateral protrusions, but while the legs and head aren't splayed in normal use, the arms are, that being what arms are for.
//break with the past// The pilot applications are ships and cottages, where built-ins are already de rigeur.
Here's what it means: assuming the same square-footage cross-section as a square...
Shelving: Shelf depth is greatest at middle-height and least at top and bottom, which means no getting down on hands and knees or standing on a footstool to view or reach items, and most of the stuff is actually on the convenient middle shelf.
Countertops, sinktops, working surfaces: are deeper. Again, anything placed above or below on the walls are easier to reach
Stairs between levels: don't take up any floorspace whatsoever; there's no "stairwell"; while steeper than a standard staircase, they're *much* less so than a "ship's staircase". They proceed directly up the lower wall and exit into the adjacent upper hexagon through a hole in its lower-wall, all while maintaining the same amount of headroom that's in the rooms.
Walking: less floorspace but more elbow room; roughly the same. There's an advantage in the case of two people passing each other, both carrying parcels: nobody has to turn sidewards.-- FlyingToaster, Aug 04 2012 Aren't you proposing hexagonal prisms though? As far as they're concerned, triangles and hexagons are equally viable. There is, however, a problem with large surface area with an equilateral triangle, in that, for instance, it will heat up and cool down more quickly than a hexagon. Pentagons are just not practical if arranged across a plane. We can agree on that. With a dodecahedron, what's to stop the insertion of floors at several levels, and with a semiregular icosahedron even more so? Although i think that's a battle i've fought and lost here.
The legs do tend to move around a bit in certain circumstances. We don't just jump everywhere bipedally.
Concerning a break with the past, we can agree that the chance to begin again in a golden land of opportunity and adventure in the off world colonies would enable innovation in architecture. There are some places on this planet where this would also be possible, such as the bottom of the ocean or Antarctica.
I admit that my previous impression was of horizontal rather than vertical hexagons, so to some extent we were talking at cross-purposes. Even so, i wish to hold out for the possibilities of various uniform convex polyhedra with twelve or more sides, in particular the semiregular icosahedron, regular icosahedron and the regular dodecahedron, just because they're beautiful. The only regular polyhedron with hexagonal faces is infinite. Dodecahedra have the advantage of small surface area for their size, making lighting and temperature control more efficient and minimising the materials needed to enclose a space.-- nineteenthly, Aug 04 2012 O_O wait, what ?
Beehive, honeycombs, standing on its side; blow it up to people-size, say 8 ft from land to land within each "cell". Extend the comb backwards aways, forming long hexagonal corridors, build staircases between cells where needed.
<link> - a horizontal view.
Rectangles and Hexagons are the only 2D figures that work in a prism for human beings: the other ones that can be stacked without space in between (triangles, pentagons[edit: not pentagons ,they don't tile]) you can't count on having proper floors for all the cells. So it's down to 4gons and 6gons.
Hexagons fit the human shape better than rectangles. You can do a stairwell-less flight of stairs between floors. For odd outside pressures, hex cross-sections are much stronger than rectangles.
Sure, you could do octagonal cross-section with interstitial diamonds, but that would require lots of explanation and it would mess up the stairs between adjacent cells.-- FlyingToaster, Aug 04 2012 I have now gleaned what you mean. I also think that the third dimension could be better exploited. Regular pentagons can't tile flat surface at all although they can be sort of combined hexagons and squares at the cost of a sloping floor if they're vertical. However, there are also polyhedra which pack together without leaving any room between them, and i think one of those is the regular dodecahedron. I think it's partly a question of scale, and i've written about this elsewhere. I'll post a link.-- nineteenthly, Aug 07 2012 // This makes people in general hard to tesselate. //
[marked-for-tagline]-- 8th of 7, Aug 13 2012 random, halfbakery