Math Mutation 73:  The Future That Never Arrived

    If you've gone to Disney's Epcot center, or seen other visions of
the future created in the 1950's and 1960's, you've probably seen a
geodesic dome.  These are the big domes made up of triangles, that
were thought by many half a century ago to represent the future of
architecture.  They still do look cool, but never quite caught on, and
most of us still live in rectilinear houses.  What are these geodesic
domes, though, and what makes them so interesting?
    In general a dome is an efficient design, as
judged by the relationship of interior volume to surface area, since
in three dimensions, spherical figures enclose the most volume with
the least surface.  A geodesic dome is built using a network of linear
elements forming portions of great circles around a sphere, where a
"great circle" is a circle like the equator that has a diameter equal
to the sphere itself.  The intersecting network of great circles forms
a large number of triangular elements.  The most important
property of these geodesic domes is that these great circles
distribute stress across the structure, making them naturally very
strong.  However, due to the large network of intersecting elements,
small flaws can cause stress to be transferred in very odd ways:
Wikipedia references an incident where a snow plow bumped into one of
these buildings in Princeton, New Jersey, and the damage appeared on
the opposite side.  But properly built domes have been seen to stand
up extremely well against earthquakes, hurricanes, and other risks. 
    Since this self-reinforcing structure is inherently strong, if you
do it right, it is apparently possible to build one out of very
lightweight materials; in the show notes, you will see a reference to
a scitoys.com site that shows you how to build one out of straws and
gumballs.  
    But you've most likely heard about geodesic domes more
recently due to 'buckyballs', the 60-atom carbon molecules discovered
in the last few decades that have geodesic-dome-like shapes.  Their
extreme structural stability leads to many potentially interesting
applications, such as using them to contain and transport simpler
molecules, including densely packed hydrogen.  Buckyballs were named
after architect Buckminster Fuller, thought to be the inventor of
geodesic dome architecture.  He certainly patented and popularized
them, though there is some dispute about whether certain German
designs from the 1920's technically beat him to it.  
    Fuller led a successful but very unusual life.  While he gained
international fame from his exotic but extremely strong dome-shaped
buildings, he attributed more universal significance to his
geometrical observations.  These included principles of close-packing 
spheres, tetrahedrons, and octahedrons, which he called "tensegrity
structures", in addition to understanding geodesic domes.  He
considered himself not just an architect, but a philosopher, and
believed that understanding the interdependent nature of these
geometric structures would lead naturally to environmentalism,
universal government, and world peace.  He called this philosophy
Synergetics, explained as "The integration of geometry and philosophy
in a single conceptual system providing a common language and
accounting for both the physical and metaphysical."  I tried to read
his long, ponderous book on the topic, which described this philosophy
in detail, but didn't quite make it through.  As Wikipedia states,
Fuller often created "long run-on sentences and used unusual compound
words (omniwell-informed, intertransformative,
omni-interaccommodative, omniself-regenerative) as well as terms he
himself coined."  He also considered his own thoughts and actions so
significant that he made an effort to document every 15 minutes of his
entire life between 1915 and 1983, titling the resulting work the
"Dymaxion Chronofiles".  These files included items ranging from
personal journal entries to dry cleaning bills.  But the fact that he
went off the deep end philosophically should not detract from his
architectural achievements. 
    So anyway, if these geodesic domes are so great, why aren't we all
living in them in the 21st century?  To start with, domes have some
obvious downsides:  the sloping shapes mean that you will have a lot
of wasted areas without enough headroom to be practical living space.
And most building materials come in rectangular shapes, so it's hard
to build a dome without a lot of waste.  And laws and regulations
regarding things like fire escapes, windows, and electrical wiring are
ill-suited to these structures, adding extra expense.  Similar issues
come due to the geometic conflict with commonly constructed furniture:
where can you put a sofa in a dome without wasting the space behind
it?  None of these problems were enough to stop Fuller himself from
living in one of his domes, of course, but the rest of us who aren't
quite as rich as him have to consider these practical issues.
    And this has been your math mutation for today.




References:

<li><a href="http://en.wikipedia.org/wiki/Geodesic_dome">Geodesic
Domes at Wikipedia</a>
<li><a href="http://www.thirteen.org/bucky/dome.html">Another geodesic
dome site</a>
<li><a
href-"http://www.bfi.org/our_programs/who_is_buckminster_fuller/design_science/design_science_today/buckyballs_as_hydrogen_containers">An
application of buckyballs</a>
<li><a
href="http://sci-toys.com/scitoys/scitoys/mathematics/dome/dome.html">Build
your own geodesic dome</a>
<li><a
href="http://en.wikipedia.org/wiki/Buckminster_Fuller">Buckminster
Fuller at Wikipedia</a>
<li><a href="http://www.grunch.net/synergetics/map/dymax.html">Fuller
projections</a>
<li><a href="http://www.grunch.net/synergetics/">Synergetics on the
web</a>