`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`
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` `
`>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