Re: unlearn trig!

On 18-Sep-05, at 9:53, Gerald de Jong wrote:
> Kirby, when i read that chapter of the book, talking about  
> quadrance and spread instead of distance and angle, i was struck by  
> a memory from a few years back.  do you remember when i agonized  
> over the problem of determining the volume of a tetrahedron based  
> on its six lengths?  it took me two weeks of train rides to figure  
> out how to do it, and then to avoid getting lost in the math i  
> built a program to handle all the terms.  it turned out to be  
> already discovered, of course.  no surprise that Euler had  
> expressed it at some point, although his equation didn't hilight  
> the geometry.
>
> now with quadrances, it makes even more sense.  take the 6  
> quadrances instead and the volume formula turns out to look a lot  
> nicer.  every term in the formula was a distance squared which this  
> guy now successfully argues is more primal than the distance itself!
>
> it's like this:  add the products of the quadrances of the twelve  
> open triangles, subtract the products of the quadrances of the four  
> closed triangles, and then subtract the products times the sums of  
> the quadrances of three pairs of opposite edges.  that might sound  
> a little complicated, but it's worlds less complicated than the  
> alternative.
>
> now i'm thinking that maybe there's something more primitive than  
> volume, just as the quadrance is more primitive than distance.

this is fun!  thinking further.. the "synergetic volume" we talked  
about was:

SV = square-root( (sum( qp(OT) ) - sum( qp(CT) ) - sum( qp(OE)*qs 
(OE) ) / 2 )

where OT = "open triangles", CT = "closed triangles", OE = "opposite  
edges",
and qp(X) = "quadrance product", qs(X) = "quadrance sum"

to make it briefer, define f(X) as sum(qp(X)) and g(X) as sum(qp(X)*qs 
(X)) and we get:

SV = square-root( ( f(OT) - f(CT) - g(OE) ) / 2 )

now, the only difference between "traditional" volume and  
"synergetic" volume was a linear factor (involving roots and stuff).   
the synergetic volume of a tetrahedron with all edges length one was  
one.

suppose, instead, that we're interested in some kind of volumetric  
value, but not necessarily volume directly (just as Dr. Wildberger  
considers quadrance to be more primal than distance).  suppose we  
just remove the square root and work instead with the rational value  
(half of (f(OT)-f(CT)-g(OE))).  in this case, the volume of the unit- 
edge-length tetrahedron is still 1 of course (square root has no  
effect on 1).  let's call this the QV for quadrance volume.

then it gets more fun.  take the tetrahedron with all edge lengths X  
instead of 1.  the QV is then nothing more than the product of all  
the quadrances... or X to the power of 6.  for example, a tetra with  
all edge lengths 2 gives QV = 2^6 = 64, and 3 gives QV=729.

what's the use?  i dunno yet.  maybe it's cool to have a measure of  
the regularity of a tetrahedron (TR) (perhaps analogous to  
Wildberger's measure of "spread" instead of "angle").  what would it  
be?  how about:

TR = QV/PL  where PL is the product of the edge lengths

this "regularity" value would be 1 for regular tetrahedra and less  
for all others, all the way to zero (for all "legit" tetrahedra).   
(the same is true for Wildberger's spread, 0 to 1).

it's also interesting that because you scrap the square root, you can  
get negative values for "impossible" tetrahedra (eg. lengths: 1 1 1 1  
1 2 gives TR=-1).

my suspicion is that Wildberger's insights could lead to a  
mathematical simplification of a lot of Fuller's work.  there's  
something in there, because Fuller was big on rational numbers,  
although he took dives into sin and cos when he presumed necessary.   
let's revisit that stuff!

i have a bunch of other ideas but i'll keep them to myself until they  
clear up.


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