struck: Part 3 - Fundamentals of Elastic Interval Geometry

Ken G. Brown kbrown at
Thu Oct 15 03:22:24 UTC 1998

Forward from:

>X-Sender: gdj at
>Date: Wed, 14 Oct 1998 12:04:31 +0200
>To: struck at
>From: Gerald de Jong <gerald at>
>Subject: struck: Fundamentals of Elastic Interval Geometry
>Mime-Version: 1.0
>Sender: owner-struck at
>Precedence: bulk
>Reply-To: struck at
>when we talk about a spatial relationship between things we are trying to
>imagine a volume within which a structural form is situated and the
>positioning of the various parts, but we are left with an open issue when
>we push the visualization further: how should we visualize the elastic
>interval itself?  if this question doesn't seem relevant, that's probably
>because an assumption has been made about how to do the visualization and
>the conclusion seems self-evident.  most often people will consider a line
>to be an effective representation of a relationship between two things, but
>a line can be very misleading.  it is generally considered to be straight
>or geodesic in some way, and lines typically themselves contain the
>contradictory assumption of zero width (or volume).  the line has some
>utility as symbol but it is not nearly a robust enough symbol to represent
>an elastic interval.
>it has been established that the 'body' of an elastic interval network is
>the intervals themselves and the simple click-connections between the
>intervals are of a very secondary importance.  a visualization should make
>this abundantly clear without any need for further explanation.
>furthermore, the connections themselves must represent 'sharability' very
>well in some way, and must indicate no influences over angle or shape other
>than the fact that they are shared by several elastic intervals.  my choice
>of visualization went through some evolution as it was forming, but ended
>up with a very pleasing result:  the ellipsoid.
>an ellipsoid has two foci, and if you were to whisper at one focus of a
>huge ellipsoid someone else could hear you loud and clear at the other
>focus and vice versa.  all rays leading out from one focus bounce off the
>inner surface and end up at the other focus, and better yet, the sum of the
>lengths of the first ray and the bounced ray are identical regardless of
>which direction was chosen.  it clearly implies an intimate relationship
>betweeen the two foci, yet the 'body' of the ellipsoid is not the foci at
>all but the zeppelin-shaped globule.  clearly an ellipsoid contains its
>foci.  i think it is appropriate to go a step further and say that an
>elastic interval is best represented by a whole family of ellipsoids, from
>the sharp double-ended-needle to the big ball nearly indistinguishable from
>a sphere, that have the same two foci.  not one, but uncountable different
>ellipsoids in the family together are required to produce an appropriate
>visualization of the elastic interval.  each member of the family
>corresponds to a certain length (sum of the two rays i mentioned).
>needless to say, for a particular rendering we will need to choose a
>certain member of the ellipsoid family, but in our imaginations we must
>maintain that any member would have been just as legitimate because it also
>has those same two foci and we must attempt to imagine them all being
>simultaneously present.
>a focus, as well, is a nice image because it is a kind of singularity.  the
>closer you look at it, the smaller it gets, but after all it's only a
>focus, not a 'thing'.  singularities are strange because the numbers go mad
>as you approach them.  few people are capable of understanding how physics
>is bent near a black hole, for example.  suppose we were to employ a
>somewhat fantastic imagery and say that the ellipsoid's focus represented a
>'portal' of some kind, leading from this 'world' (elastic interval) to the
>next one.  if one were to travel along such a path, one would experience a
>very consistent series of events:  through a portal, bounce off the inner
>surface, through a portal, bounce off the inner surface.. and so on.  your
>visit within a world would only be punctuated by one fierce encounter with
>the ellipsoid's inner surface.  with an eye on quantum electrodynamics, you
>might imagine encountering and rebounding from each and every member of the
>ellipsoid family, but that doesn't really diminish the value of the
>single-ellipsoid rendering.  the point is to achieve clarity with respect
>to the somethingness of the intervals and the nothingness of the
>connections among them through the use of the ellipsoid imagery.
>(to be continued..)
>Gerald de Jong, Beautiful Code B.V.
>Rotterdam, The Netherlands

More information about the Squeak-dev mailing list