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From: Roger Bagula on 5 Jul 2010 06:03
Q(x,y,z,t)=x2+y2+z2+t2+x*t+y*t+z*t
is the lattice form of Korkine and Zolatarreff
from
Advanced Number Theory, Harvey Cohn, Dover Books,1963, Page 89
It has basis vectors:
v1={1,0,0,0}
v2={0,1,0,0}
v3={0,0,1,0}
v4={-1,-1,-1,2}
If
f(x)=3*x2+3*x*t+t2
there is a three fold symmetry
3*Q(x,y,z,t)=f(x)+f(y)+f(z)
The interesting thing about this is that the 6 points
the f(x) solutions give in the xt,yt,and zt planes
are a 4d tetrahedron analog.
The book says the basis is not L4 ( the Lebesque space
that corresponds to quaternions I think).

Suppose we modified the basis to the BBP basis:
v1={1,0,0,0}
v2={0,1,0,0}
v3={0,0,1,0}
v4={-1,-1,-2,4}
symmetrically:
Qa(x,y,z,t)=x2+y2+z2+t2+a*(+x*t+y*t+z*t)
For
Qa(-1,-1,-2,4)=1
a=21/16
but if we wanted a zero instead
with basis:
v1={1,-1,0,0}
v2={0,1,-1,0}
v3={-1,0,1,0}
v4={-1,-1,-2,4}
a=22/16
What is interesting about this solution is the symmetrical f(x)
function:
f(x)=x2+(22/16)*x+1/3
NSolve[x2+(22/16)*x+1/3 == 0, x, 200]
{{x ->
-1.060759851399352441346798231263585251186320444243821280292248019186263135567132\
691891509793852633503027207335442503575715165977293280598497030849082383617214\
3772574038584767914544121341620522126880079}, {x ->
-0.314240148600647558653201768736414748813679555756178719707751980813736864432\
867308108490206147366496972792664557496424284834022706719401502969150917616382\
7856227425961415232085455878658379477873119921}}
which seem to confirm this geometry is related to Pi
by the second root.
Changing to at metric form:
Qa(dx,dy,dz,dt)/dt2=dx/dt2+dy/dt2+dz/dt2+(22/16)*(dx/dt+dy/dt+dz/dt)
we get the symmetrical velocity form:
f(dx/dt)=dx/dt2+(22/16)*dx/dt+1/3
>From the solution of this equation as above
we get the unlikely result
that the speed of light in
such a algebraic lattice four space would be related to Pi.

It also gives the algebraic approximation of Pi as:
NSolve[48*x2 - 660*x + 1600 == 0, x, 200]
{{x ->
3.142401486006475586532017687364147488136795557561787197077519808137368\
644328673081084902061473664969727926645574964242848340227067194015029691509176\
1638278562274259614152320854558786583794778731199213267030945398`199.\
18531771111554}, {x -> \
10.607598513993524413467982312635852511863204442438212802922480191862631355671\
326918915097938526335030272073354425035757151659772932805984970308490823836172\
1437725740385847679145441213416205221268800786732969054602`199.34429463799816}\
}

I was wondering if there were some way we could visualize
this geometry?
Respectfully, Roger L. Bagula





From: Roger Bagula on 5 Jul 2010 21:14
I used a minimal surface embedding of a sphere to get this
visualization:
x = Cos[t0]*Sin[p0]; y = Sin[t0]*Sin[p0]; z = Cos[p0];
x1 = Re[Integrate[x^2 + (22/16)*x*t + 1/3, {p0, 0, t}]];
y1 = Re[Integrate[y^2 + (22/16)*y*t + 1/3, {p0, 0, t}]];
z1 = Re[Integrate[z^2 + (22/16)*z*t + 1/3, {p0, 0, t}]];
g1 = ParametricPlot3D[{x1, y1, z1}, {t, 0, 2*Pi}, {t0, 0, 2*Pi}]
g2 = ParametricPlot3D[{x1, y1, z1}, {t, -2*Pi, 0}, {t0, 0, 2*Pi}]
g3 = ParametricPlot3D[{x1, y1, -z1}, {t, 0, 2*Pi}, {t0, -2*Pi, 0}]
g4 = ParametricPlot3D[{x1, y1, -z1}, {t, -2*Pi, 0}, {t0, -2*Pi, 0}]
Show[{g1, g2}, Boxed -> False, Axes -> False]
Show[{g3, g4}, Boxed -> False, Axes -> False]

Kind of a cludge,
but better than nothing.
Roger Bagula

From: Roger Bagula on 5 Jul 2010 21:15
I just now got a better rendering:

http://www.flickr.com/photos/fractalmusic/4764898936/
x = Cos[t0]*Sin[p0]; y = Sin[t0]*Sin[p0]; z = Cos[p0];
x1 = Re[Integrate[x^2 + (22/16)*x + 1/3, {p0, 0, t}]];
y1 = Re[Integrate[y^2 + (22/16)*y + 1/3, {p0, 0, t}]];
z1 = Re[Integrate[z^2 + (22/16)*z + 1/3, {p0, 0, t}]];
g1 = ParametricPlot3D[{x1, y1, z1, {
EdgeForm[]}}, {t, 0, 2*Pi}, {t0, 0, 2*Pi},
PlotRange -> All, PlotPoints -> {60, 60},
Boxed -> False, Axes -> False]
g2 = ParametricPlot3D[{x1, y1,
z1, {EdgeForm[]}}, {t, -2*Pi, 0}, {t0, 0, 2*Pi}, PlotRange -
> All, \
PlotPoints -> {60, 60}, Boxed -> False, Axes -> False]
g3 = ParametricPlot3D[{x1, y1, -z1, {EdgeForm[]}}, {t, 0, 2*Pi}, {t0,
-2*
Pi, 0}, PlotRange -> All, PlotPoints -> {60, 60}, Boxed ->
False, Axes -> False]
g4 = ParametricPlot3D[{x1, y1, -
z1, {EdgeForm[]}}, {t, -2*Pi, 0}, {t0, -2*Pi, 0}, PlotRange ->
All, \
PlotPoints -> {60, 60}, Boxed -> False, Axes -> False]
Show[{g1, g2}, Boxed -> False, Axes -> False]
Show[{g3, g4}, Boxed -> False, Axes -> False]

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