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