Dimensional Lines
 
Subject: Re: Reply to "Re: Reply to Do Points Have Area?"
Author: Kirby Urner <pdx4d@teleport.com>
Date: Wed, 21 Jan 1998 16:58:06 -0800
 
 
>No matter how much sense it makes, there is nothing that states that
>if you lay a bunch of lines together you will have height! But, to
>make a cube, you need 12 line segments (which could have zero width)
>because of the angles of the line segments, the construction will be
>three dimentional! Even though lines with width makes sense, it is
>not necessary in order to (visually) construct in the 3D
> 
 
I understand you.
 
Coming from a computer background, say a ray tracing world, I'm 
used to stuff having definite dimension, as otherwise light has
nothing to bounce off, so it might as well not be there.
 
So for me, in this world, a point is a relatively tiny entity, 
too small to have its details make any difference, but there's
no quantum leap to some lower dimensional state (i.e. 0).
 
Lines and planes are likewise slender/thin, but I experience 
no intellectual pressure to alter their dimensionality relative
to any old ordinary, light-reflective substance in my ray 
traced world. A line and a cube look different, I can always
tell which is which -- but I don't go by "dimension number" 
as this is the same for both (see below).
 
So for me, points, lines and planes are all shapes with properties
(e.g. planes are "razor thin"), but don't sit on different rungs 
of the "dimension ladder". What is usually called Euclidean space
(or volume) is for me a space of "lumps" and the "point", "line" 
and "plane" characterizations still make sense, but minus the 
0D,1D,2D claptrap.
 
If I want to bring D ("dimension") into it, then I note that volume 
is containment, the logical space of things with inside/outside 
concave/convex attributes -- the space of hulls, shells, rooms, 
cells...
 
Then I do what you do, I go with thin lines (edges) and figure
out what simplest model of inside/outside I can conceive --
realizing that my lines themselves have insides/outsides (but
that doesn't mean I have to consider them my paradigm 
"containments").
 
The answer I come up with is the tetrahedral wireframe: four
windows, four corners, six edges. No shape is simpler. "Spheres"
as such turn out to be high frequency porous membranes -- just
as my planes turn out to be networks as well (mostly space).
 
I'm in Euler's world of V, F and E -- but my F is more a W (window). 
Vs are where Es cross, but they don't even have to go through each 
other exactly -- no two things occupy the same space at the same 
time.
 
So I say volume is 4D. I get my 4 from the 4 windows and 4 
corners of the tetrahedron. 0D, 1D, 2D and 3D are all undefined
in this philosophical language.
 
The aesthetics here trace to Democritus. Discontinuity, discrete,
empty space versus substance, emptiness between things, islands, 
events with novent surroundings, holes, voids... I'm not looking 
for anything to fill the holes, now that I've got them.
 
I claim I can do Euclidean geometry in this logical space. So I 
say Euclidean space is 4D, realizing this sounds all wrong, very
dissonant, to ears trained in the 1900s.
 
>I can't wait till a new geometry that makes more sense, maybe Jesse
>Yoder's circular geometry, but I don't want to disclaim things in
>Euclidean geometry yet if they still make some sense!
> 
 
I don't want to disclaim stuff in Euclidean geometry either. 
My curriculum makes use of The Elements, the kind of logic that
goes on in these proofs, but tosses some of the definitional 
beginnings. Euclidean constructions "float" in 4D space without
needing "support from below" in the form of "a bedrock of 
axioms" -- especially where this funny concept of "dimension"
is concerned.
 
Also, I'm not trying to push the old logic off stage with this 
newfangled talk (as if I could, even if I wanted to). I know the 
standard lingo and would expect kids learning my meaning of 4D 
to also learn the standard "dim talk". 
 
I say "three dimensional" just like everyone else when talking 
about volume (when in Rome) even if that's not what I'm thinking 
(I translate my thinking for backward compatibility with my peers).
 
Kirby

http://forum.swarthmore.edu/epigone/geometry-research/frangphuglang/3.0.3.32.19980121165806.032bb264@mail.teleport.com

 

 

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