|
welcome to |
|
Do Points
Have Area? 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://mathforum.org/kb/forum.jspa?forumID=130 Back to "Do Points Have Area?"
©1999-2007 |
|
|
