struck: Part 2 - Fundamentals of Elastic Interval Geometry

Ken G. Brown kbrown at
Wed Oct 14 02:55:33 UTC 1998

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>Date: Tue, 13 Oct 1998 22:27:55 +0200
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>From: Gerald de Jong <gerald at>
>Subject: struck: Fundamentals of Elastic Interval Geometry
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>at first glance it might seem sensible to consider elastic intervals to be
>unidirectional in their exertion of impulses, since that seems simpler than
>bidirectionality, but an essential element of the intervals is that they
>know nothing about their world other than that it consists of two
>connections.  if an interval currently has a span less than the span it
>desires it will try to achieve the desired span.  unidirectionality would
>presume that one of the connections is considered to be an anchor, but that
>implies that the interval knows too much about its connections.  an
>interval may not discriminate.  to treat its connections differently, an
>interval would have to contain knowledge about the difference, but this is
>extra information and the goal is to arrive at a minimal definition.
>geometry is very often associated with the notion of angle, but it is
>actually one of the most important defining characteristics of EIG that
>angle is a completely derivative attribute.  for some purposes it might be
>interestng to measure angles in an EIG network, but as far as the intervals
>are concerned there is no such thing.  to actually account for angle in EIG
>would be a fascinating process in itself, because due to the inherent
>elasticity we would be forced to abandon static precise angle in favor of a
>very different definition that is also elastic.  when considered in this
>light, it is abundantly clear that 'elastic angle' is a clearly only
>derived from the configuration of the intervals and their elasticity.  the
>actual value of 'elastic angle' should probably be based on the cosine law,
>which calculates angle based purely on the lengths of the edges of a
>triangle, but this immediately introduces spans raised to the power of two,
>further complicating the determination of an angle's elasticity.
>the fascinating forms that can easily be built within the simplicity of EIG
>using the Struck software are also utterly bereft of some other attributes
>that might be expected.  for example, there are no 'faces', and there are
>of course also no 'solids'.  in this respect, EIG with its minimal approach
>effectively elucidates some of the core ideas that Buckminster Fuller
>explored in quite different terms.  to go into the comparisons in detail
>would involve much more research and explanation than is fitting for this
>text, but suffice it to say that Fuller's work has been a grand inspiration
>throughout our journey through EIG.
>in order to admit the notion of a 'face' in any sense at all, one must
>first accept that the definition will have a very different origin and will
>not fully comply with existing definitions of 'faces'.   after all, we may
>only consider elastic intervals, so things like flatness,
>two-dimensionality, and impermiability are quite inaccessible.  the best
>definitions involve Fuller's idea of 'frequency' which are reminiscent of
>biological phenomena such as cell walls and osmosis.  if the density of
>things is comparatively high, a network may be less permeable than another
>with lower density.  a cell wall admits small molecules but restricts the
>passage of larger ones, due to the relative sizes of the 'holes'.  EIG is
>actually forced to go much further than this and completely negate the
>notion of face altogether, but that will require some additional explanation.
>solids, however, can perhaps be accommodated in EIG to a greater degree.  a
>configuration of connected elastic intervals does display some
>characteristics that we associate with solidity, albeit always
>incorporating flexibility and completely denying absolute rigidity.
>but there is something wrong here which goes much deeper than we initially
>perceive, and it is founded in the puzzling reflex that i mentioned
>earlier.  our conventional sense of connectivity differs in a profound way
>from that of elastic interval geometry.  we tend to quickly make the
>assumption that things in close spatial proximity to each other are always
>candidates for exerting impulses on each other.  in other words, we make
>the assumption that everything is related to everything, but it just
>happens that the relationships only become detectably manifest when
>nearness brings them into the forefront.  this may indeed be a fundamental
>property of the physical world, but such things are utterly and completely
>foreign to EIG.  the realization of this is at first shocking and
>repulsive, since it appears to relegate EIG to the realm of ideas that have
>no relationship to physical reality, but we must remember what it is we're
>doing.  we are looking for a minimal definition of structural form, and EIG
>clearly has some resonances with physical phenomenon (witness the intuitive
>response that people have when confronted with an EIG-based movie).  it
>need not fully comply with our observed physical reality.  that is simply
>not the main goal.
>if the previous passages have left you wondering what the issue is, let me
>try to state it in no uncertain terms.  EIG is a language of elastic
>intervals which are somehow click-connected together, and therefore each
>and every relationship is part of the language.  the chance spatial
>nearness of two 'nodes' which are not related in an explicit interval does
>not even in the slightest suggest that the two nodes are related.  if you
>were to travel around the elastic interval network, there would be no way
>at all to 'cross the divide' and fly between these two nodes, but rather
>you would be forced to travel along the intervals until you reached the
>other node on foot.  spatial proximity is completely meaningless in EIG,
>and the only kind of proximity is the kind explicitly defined by an
>interval itself and the other simpler kind which is the click-connection at
>which intervals meet.
>returning to 'solids' and 'faces': it is entirely possible that an elastic
>interval network consisting of a great density of intervals and somehow
>curving around and in on itself is completely unrestricted in the passage
>of one part of the network through another part.  the two superimposed
>sections of the great dense network have no perception of each other's
>existence other than through the possibly complex array of intervals that
>actually explicitly connect them.  furthermore, a dense and quite
>consistently planar network of intervals offers no resistance whatsoever to
>another network that may pass through it.  networks that are not explicitly
>connected to each other must necessarily be oblivious to each other.  these
>are some prime considerations when exploring the theoretical basis of EIG,
>and it should start to become clear that these characteristics actually
>free an elastic interval network from any particular sense of dimensionality.
>(to be continued...)
>Gerald de Jong, Beautiful Code B.V.
>Rotterdam, The Netherlands

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