Clifford Nelson
Jan15-09, 06:00 AM
Here is one reason to use Synergetics coordinates StoP and PtoS
orientation to the Cartesian system. Measuring area by triangles and
tetra-volume of the "side show freaks" of polyhedra that have irrational
content if you use Cartesian coordinates.
See:
http://mysite.verizon.net/cjnelson9/index.htm
Clear[a, b, c, d, e, f]
generalTriangle = {{a, b, -a - b}, {c, d, -c - d},
{e, f, -e - f}}
generalTriangle - 1/3
{{-(1/3) + a, -(1/3) + b, -(1/3) - a - b},
{-(1/3) + c, -(1/3) + d, -(1/3) - c - d},
{-(1/3) + e, -(1/3) + f, -(1/3) - e - f}}
The area in equilateral triangles is the determinant of that matrix.
FullSimplify[Det[generalTriangle - 1/3]]
b*(c - e) + d*e - c*f + a*(-d + f)
Now for tetra-volume.
Clear[a, b, c, d, e, f, g, h, i, j, k, l]
generalTetrahedron = {{a, b, c, -a - b - c},
{d, e, f, -d - e - f}, {g, h, i, -g - h - i},
{j, k, l, -j - k - l}}
generalTetrahedron - 1/4
{{-(1/4) + a, -(1/4) + b, -(1/4) + c,
-(1/4) - a - b - c}, {-(1/4) + d, -(1/4) + e,
-(1/4) + f, -(1/4) - d - e - f},
{-(1/4) + g, -(1/4) + h, -(1/4) + i,
-(1/4) - g - h - i}, {-(1/4) + j, -(1/4) + k,
-(1/4) + l, -(1/4) - j - k - l}}
The volume measured by regular tetrahedrons is the determinant of the
four by four matrix.
FullSimplify[Det[generalTetrahedron - 1/4]]
a*f*h - a*e*i - f*h*j + e*i*j - a*f*k + f*g*k +
a*i*k - d*i*k + c*((-d)*h + e*(g - j) + h*j + d*k -
g*k) + a*e*l - e*g*l - a*h*l + d*h*l +
b*((-f)*g + d*i + f*j - i*j - d*l + g*l)
If the Synergetics coordinates are rational the area and tetra-volume
are rational. If you append a column of 1/Factorial[d-1] to a d by (d-1)
matrix of d rows of Cartesian coordinates points the determinant is the
square or cubical area or volume.
Does anybody measure area and volume by anything but squares and cubes
with coordinates?
The Cayley-Menger Determinant
http://mathworld.wolfram.com/Cayley-MengerDeterminant.html
The first few coefficients for j= 0,1,... are -1, 2, -16, 288, -9216 ,
460800, ... (Sloane's A055546).
If j = 1,2,3,... = j+1 instead of Sloane's A055546 you get the area for
j +1 = 3, and volume for j+1 = 4, measured in unit edge length
simplexes, except for sign.
I mean that the square root of the quantity of (the absolute value of
the
determinant of the matrix divided by j+1) is the content measured by
unit simplexes.
Here is something from the scimath newsgroup.
In article
<b1ebb1be-7e17-4c70-815a-af329f51c2ee@g39g2000pri.googlegroups.com>,
victor_meldrew_666@yahoo.co.uk wrote:
>
> Whatever "coordinate system" you use, there will be polyhedra
> in every similarity class of every positive measure. Whatever
> "coordinate system" you use, the volumes of a reguar tetrahedron,
> a cube and a regular dodecahedron with the same side-length
> will still have pairwise irrational ratios.
Yes. So, the polyhedra whose vertices are in the IVM (closest packed
spheres lattice) have to do with Nature, and the polyhedra whose
vertices are in the closest packed cubes lattice (Cartesian) are
unnatural and often lead to irrational (awkward) constants.
Cliff Nelson
In article
<8901af8a-e8c1-402b-a91e-711f1c072e85@w39g2000prb.googlegroups.com>,
victor_meldrew_666@yahoo.co.uk wrote:
> On 9 Jan, 10:08, Clifford Nelson <cjnels...@verizon.net> wrote:
>
> > Yes. So, the polyhedra whose vertices are in the IVM (closest packed
> > spheres lattice) have to do with Nature,
>
> We can't argue with nature, sorry I mean Nature, can we?
I got a small expression of irrational numbers that version 4.1 of
Mathematica could not simplify to 0. No unknowns, just numbers. A
floating point approximation was 0.0 but not exactly zero. An exact
rational expression for the same problem using Synergetics coordinates
equalled exactly zero, 0, the integer. Stephen Wolfram said in a talk
that there are just too many ways to try to show that two things are
equal. Bucky Fuller said that men are getting along all right with the
great computers, but, I can imagine the huge expressions that can't be
simplified, just because the expressions are not rational. You're right,
they are arguing with Nature.
Cliff Nelson
Dry your tears, there's more fun for your ears,
"Forward Into The Past" 2 PM to 5 PM, Sundays, California time:
http://www.geocities.com/forwardintothepast/
Don't be a square or a blockhead; see:
http://mysite.verizon.net/cjnelson9/index.htm
http://library.wolfram.com/infocenter/search/?search_results=1;search_per
son_id=607
orientation to the Cartesian system. Measuring area by triangles and
tetra-volume of the "side show freaks" of polyhedra that have irrational
content if you use Cartesian coordinates.
See:
http://mysite.verizon.net/cjnelson9/index.htm
Clear[a, b, c, d, e, f]
generalTriangle = {{a, b, -a - b}, {c, d, -c - d},
{e, f, -e - f}}
generalTriangle - 1/3
{{-(1/3) + a, -(1/3) + b, -(1/3) - a - b},
{-(1/3) + c, -(1/3) + d, -(1/3) - c - d},
{-(1/3) + e, -(1/3) + f, -(1/3) - e - f}}
The area in equilateral triangles is the determinant of that matrix.
FullSimplify[Det[generalTriangle - 1/3]]
b*(c - e) + d*e - c*f + a*(-d + f)
Now for tetra-volume.
Clear[a, b, c, d, e, f, g, h, i, j, k, l]
generalTetrahedron = {{a, b, c, -a - b - c},
{d, e, f, -d - e - f}, {g, h, i, -g - h - i},
{j, k, l, -j - k - l}}
generalTetrahedron - 1/4
{{-(1/4) + a, -(1/4) + b, -(1/4) + c,
-(1/4) - a - b - c}, {-(1/4) + d, -(1/4) + e,
-(1/4) + f, -(1/4) - d - e - f},
{-(1/4) + g, -(1/4) + h, -(1/4) + i,
-(1/4) - g - h - i}, {-(1/4) + j, -(1/4) + k,
-(1/4) + l, -(1/4) - j - k - l}}
The volume measured by regular tetrahedrons is the determinant of the
four by four matrix.
FullSimplify[Det[generalTetrahedron - 1/4]]
a*f*h - a*e*i - f*h*j + e*i*j - a*f*k + f*g*k +
a*i*k - d*i*k + c*((-d)*h + e*(g - j) + h*j + d*k -
g*k) + a*e*l - e*g*l - a*h*l + d*h*l +
b*((-f)*g + d*i + f*j - i*j - d*l + g*l)
If the Synergetics coordinates are rational the area and tetra-volume
are rational. If you append a column of 1/Factorial[d-1] to a d by (d-1)
matrix of d rows of Cartesian coordinates points the determinant is the
square or cubical area or volume.
Does anybody measure area and volume by anything but squares and cubes
with coordinates?
The Cayley-Menger Determinant
http://mathworld.wolfram.com/Cayley-MengerDeterminant.html
The first few coefficients for j= 0,1,... are -1, 2, -16, 288, -9216 ,
460800, ... (Sloane's A055546).
If j = 1,2,3,... = j+1 instead of Sloane's A055546 you get the area for
j +1 = 3, and volume for j+1 = 4, measured in unit edge length
simplexes, except for sign.
I mean that the square root of the quantity of (the absolute value of
the
determinant of the matrix divided by j+1) is the content measured by
unit simplexes.
Here is something from the scimath newsgroup.
In article
<b1ebb1be-7e17-4c70-815a-af329f51c2ee@g39g2000pri.googlegroups.com>,
victor_meldrew_666@yahoo.co.uk wrote:
>
> Whatever "coordinate system" you use, there will be polyhedra
> in every similarity class of every positive measure. Whatever
> "coordinate system" you use, the volumes of a reguar tetrahedron,
> a cube and a regular dodecahedron with the same side-length
> will still have pairwise irrational ratios.
Yes. So, the polyhedra whose vertices are in the IVM (closest packed
spheres lattice) have to do with Nature, and the polyhedra whose
vertices are in the closest packed cubes lattice (Cartesian) are
unnatural and often lead to irrational (awkward) constants.
Cliff Nelson
In article
<8901af8a-e8c1-402b-a91e-711f1c072e85@w39g2000prb.googlegroups.com>,
victor_meldrew_666@yahoo.co.uk wrote:
> On 9 Jan, 10:08, Clifford Nelson <cjnels...@verizon.net> wrote:
>
> > Yes. So, the polyhedra whose vertices are in the IVM (closest packed
> > spheres lattice) have to do with Nature,
>
> We can't argue with nature, sorry I mean Nature, can we?
I got a small expression of irrational numbers that version 4.1 of
Mathematica could not simplify to 0. No unknowns, just numbers. A
floating point approximation was 0.0 but not exactly zero. An exact
rational expression for the same problem using Synergetics coordinates
equalled exactly zero, 0, the integer. Stephen Wolfram said in a talk
that there are just too many ways to try to show that two things are
equal. Bucky Fuller said that men are getting along all right with the
great computers, but, I can imagine the huge expressions that can't be
simplified, just because the expressions are not rational. You're right,
they are arguing with Nature.
Cliff Nelson
Dry your tears, there's more fun for your ears,
"Forward Into The Past" 2 PM to 5 PM, Sundays, California time:
http://www.geocities.com/forwardintothepast/
Don't be a square or a blockhead; see:
http://mysite.verizon.net/cjnelson9/index.htm
http://library.wolfram.com/infocenter/search/?search_results=1;search_per
son_id=607