So far I've heard...
sqrt(5/3) - 1 (mine)
sqrt(3/2) - 1
sqrt(6)/4 - 1/2 (half the proceeding value)
And now also sqrt(11/6) - 1.
Tim
I've used a prototype version of MacSpice 3f5 2.9p34 to solve this
problem. This version 2.9p34 has a built-in optimizer that uses a
simplex method. Using default settings to minimize and n-parameter
cost function, it starts by constructing a regular n-dimensional unit
simplex with its centroid at the origin. I can extract these and solve
the problem fairly easily, using this command file:
*Gap between four spheres calculation
..control
echo
echo "Get a regular simplex with four vertices on on a unit sphere:"
echo
optimize 'let cost = 1' npar 3 report -1 maxeval 3
foreach v 0 1 2 3
print line vertices[$v]
end
echo
echo "Rescale it so each point is separated by 2 units from the
others:"
echo
let delta = (vertices[2]-vertices[3])
let separation = sqrt(length(delta)*mean(delta*delta))
let rescaled = 2*vertices/separation
foreach v 0 1 2 3
print line rescaled[$v]
end
echo
echo "The distance from the origin to any of the rescaled vertices
is:"
echo
let d = sqrt(length(rescaled[0])*mean(rescaled[0]*rescaled[0]))
print d
echo
echo "Hence the largest sphere to fit into the void has radius:"
echo
set numdgt = 15
print (d-1.0)
echo
echo "Which is the same as:"
echo
print sqrt(3/2)-1.0
..endc
Which produces the following results:
**************
*** MacSpice - 3f5 v2.9 PATCHLEVEL: 34
*** Carbon Version by CDHW
*** Date Created: Apr 7 2007
**************
Get MacSpice updates, information, userguide, tutorials from:
<
http://newton.ex.ac.uk/teaching/CDHW/MacSpice/>
Some useful Spice 3 commands:
source <filename> : loads the named file as the new circuit.
run : executes the specified analyses.
edit : edits source with helper application.
display : displays the list of output-vectors.
plot <plotargs> : draws the results.
set : displays all internal variables.
rusage all : prints the resource usage.
help : on-line help information (obsolete).
quit : quit spice.
MacSpice 1 -> source Primary:Users:cdhw
ocuments:MacSpice:4-
spheres:four-spheres.src
Get a regular simplex with four vertices on on a unit sphere:
vertices[0] = ( -8.16497e-01 -4.71405e-01 -3.33333e-01 )
vertices[1] = ( 8.164966e-01 -4.71405e-01 -3.33333e-01 )
vertices[2] = ( 0.000000e+00 9.428090e-01 -3.33333e-01 )
vertices[3] = ( 0.000000e+00 0.000000e+00 1.000000e+00 )
Rescale it so each point is separated by 2 units from the others:
rescaled[0] = ( -1.00000e+00 -5.77350e-01 -4.08248e-01 )
rescaled[1] = ( 1.000000e+00 -5.77350e-01 -4.08248e-01 )
rescaled[2] = ( 0.000000e+00 1.154701e+00 -4.08248e-01 )
rescaled[3] = ( 0.000000e+00 0.000000e+00 1.224745e+00 )
The distance from the origin to any of the rescaled vertices is:
d = 1.224745e+00
Hence the largest sphere to fit into the void has radius:
(d-1.0) = 2.247448713915894e-01
Which is the same as:
sqrt(3/2)-1.0 = 2.247448713915889e-01
MacSpice 2 ->
Happy Easter
Charles