SymGen
0 ? 1 : -1) : 0;
}
//PHP doesn't always get the exact value right because it can only store so many decimal places
//this cheecky function establishes if an answer is nearly right and, if it is, rounds to to the nearby correct value
function exactify($x)
{
$vals = array(1, 0, 2, 3, 0.5, sqrt(2), M_PI);
foreach ($vals as $val)
{
if(abs($x - $val) < 1e-6)
{
return $val;
}
elseif(abs($val + $x) < 1e-6)
{
return -$val;
}
}
return $x;
}
//define your symmetry operations, the HTML to render them and their type (in Schoenflies)
//identity
$type['E'] = 'identity';
$operations['1'] = array
( 1, 0, 0,
0, 1, 0,
0, 0, 1);
$operations['1']['html'] = '1';
$operations['1']['type'] = 'E';
//inversion
$type['i'] = 'inversion';
$operations['-1'] = array
(-1, 0, 0,
0, -1, 0,
0, 0, -1);
$operations['-1']['html'] = '1';
$operations['-1']['type'] = 'i';
//reflection
$type['sigma'] = 'reflection';
$planes = array (
array(1,0,0), //m_x
array(0,1,0), //m_y
array(0,0,1), //m_z
array(1,1,0), //etc
array(1,0,1),
array(0,1,1),
array(-1,1,0),
array(-1,0,1),
array(0,1,-1)
);
foreach($planes as $plane)
{
$operations['m'.$plane[0].$plane[1].$plane[2]] = array
(-pow($plane[0],2)+pow($plane[1],2)+pow($plane[2],2), -2*$plane[0]*$plane[1], -2*$plane[0]*$plane[2],
-2*$plane[1]*$plane[0], -pow($plane[1],2)+pow($plane[0],2)+pow($plane[2],2), -2*$plane[1]*$plane[2],
-2*$plane[2]*$plane[0], -2*$plane[2]*$plane[1], -pow($plane[2],2)+pow($plane[0],2)+pow($plane[1],2)
);
//go through and divide by the modulus
for($i = 0; $i < 9; $i++)
{ $operations['m'.$plane[0].$plane[1].$plane[2]][$i] = $operations['m'.$plane[0].$plane[1].$plane[2]][$i] / (pow($plane[0],2)+pow($plane[1],2)+pow($plane[2],2));
exactify($operations['m'.$plane[0].$plane[1].$plane[2]][$i]);}
$operations['m'.$plane[0].$plane[1].$plane[2]]['html'] = 'm['.($plane[0] >= 0 ? $plane[0] : ''.abs($plane[0]).'').($plane[1] >= 0 ? $plane[1] : ''.abs($plane[1]).'').($plane[2] >= 0 ? $plane[2] : ''.abs($plane[2]).'').']';
$operations['m'.$plane[0].$plane[1].$plane[2]]['type'] = 'sigma';
/* 1 / (Px2 + Py2 + Pz2)* ...
-Px2 + Pz* Pz + Py* Py - 2 * Px * Py - 2 * Px * Pz
- 2 * Py * Px -Py2 + Px*Px + Pz*Pz - 2 * Py * Pz
- 2 * Pz * Px -2 * Pz * Py -Pz2 + Py*Py + Px*Px
where Px, Py, Pz is the normal vector
(see http://www.euclideanspace.com/maths/geometry/affine/reflection/index.htm )*/
}
//proper rotation
$type['C2'] = 'proper rotation (order 2)';
$type['C3'] = 'proper rotation (order 3)';
$type['C4'] = 'proper rotation (order 4)';
$type['C6'] = 'proper rotation (order 6)';
$axes = array (
array(1,0,0), //m_x
array(0,1,0), //m_y
array(0,0,1), //m_z
array(1,1,0), //etc
array(1,0,1),
array(0,1,1),
array(-1,1,0),
array(1,1,1),
array(-1,1,1),
array(1,-1,1),
array(1,1,-1),
array(-1,-1,1),
array(-1,1,-1),
array(1,-1,-1),
array(-1,-1,-1)
);
$orders = array(2,3,4); //999 add 6 again later
foreach($orders as $order)
{
for($repeat = 1; $repeat < $order; $repeat++)
{
//angle rotated through
$a = (2*M_PI / $order) * $repeat;
$c = exactify(cos($a));
$s = exactify(sin($a));
foreach($axes as $axis)
{
//make it a unit vector u
$u[0] = $axis[0]/sqrt(pow($axis[0],2)+pow($axis[1],2)+pow($axis[2],2));
$u[1] = $axis[1]/sqrt(pow($axis[0],2)+pow($axis[1],2)+pow($axis[2],2));
$u[2] = $axis[2]/sqrt(pow($axis[0],2)+pow($axis[1],2)+pow($axis[2],2));
$operations[$order.$repeat.$axis[0].$axis[1].$axis[2]] = array
(pow($u[0],2)*(1-$c) + $c, $u[0]*$u[1]*(1-$c) - $u[2]*$s, $u[0]*$u[2]*(1-$c) + $u[1]*$s,
$u[0]*$u[1]*(1-$c) + $u[2]*$s, pow($u[1],2)*(1-$c) + $c, $u[1]*$u[2]*(1-$c) - $u[0]*$s,
$u[0]*$u[2]*(1-$c) - $u[1]*$s, $u[1]*$u[2]*(1-$c) + $u[0]*$s, pow($u[2],2)*(1-$c) + $c
);
for($i = 0; $i < 9; $i++)
{ $operations[$order.$repeat.$axis[0].$axis[1].$axis[2]][$i] = exactify($operations[$order.$repeat.$axis[0].$axis[1].$axis[2]][$i]); }
$operations[$order.$repeat.$axis[0].$axis[1].$axis[2]]['html'] = $order.($repeat > 1 ? ''.$repeat.'[' : '[').($axis[0] >= 0 ? $axis[0] : ''.abs($axis[0]).'').($axis[1] >= 0 ? $axis[1] : ''.abs($axis[1]).'').($axis[2] >= 0 ? $axis[2] : ''.abs($axis[2]).'').']';
$operations[$order.$repeat.$axis[0].$axis[1].$axis[2]]['type'] = 'C'.$order;
}
}
}
//rotoinversion
//OK, so S isn't technically rotoinversion, it's rotoreflection...but it's equivalent
$type['S3'] = 'rotoinversion (order 3)';
$type['S4'] = 'rotoinversion (order 4)';
$orders = array(3,4);
foreach($orders as $order)
{
//generate the initial ones by rotating then inverting
foreach($axes as $axis)
{
$operations['-'.$order.'1'.$axis[0].$axis[1].$axis[2]] = multiply_matrices($operations[$order.'1'.$axis[0].$axis[1].$axis[2]],$operations['-1']);
$operations['-'.$order.'1'.$axis[0].$axis[1].$axis[2]]['html'] = ''.$order.''.'['.($axis[0] >= 0 ? $axis[0] : ''.abs($axis[0]).'').($axis[1] >= 0 ? $axis[1] : ''.abs($axis[1]).'').($axis[2] >= 0 ? $axis[2] : ''.abs($axis[2]).'').']';
$operations['-'.$order.'1'.$axis[0].$axis[1].$axis[2]]['type'] = 'S'.$order;
//generate the rest by repetition
for($repeat = 2; $repeat < $order; $repeat++)
{
$operations['-'.$order.$repeat.$axis[0].$axis[1].$axis[2]] = multiply_matrices($operations['-'.$order.'1'.$axis[0].$axis[1].$axis[2]],$operations['-'.$order.($repeat-1).$axis[0].$axis[1].$axis[2]]);
$operations['-'.$order.$repeat.$axis[0].$axis[1].$axis[2]]['html'] = ''.$order.''.($repeat > 1 ? ''.$repeat.'[' : '[').($axis[0] >= 0 ? $axis[0] : ''.abs($axis[0]).'').($axis[1] >= 0 ? $axis[1] : ''.abs($axis[1]).'').($axis[2] >= 0 ? $axis[2] : ''.abs($axis[2]).'').']';
$operations['-'.$order.$repeat.$axis[0].$axis[1].$axis[2]]['type'] = 'S'.$order;
}
}
}
//to display all the element matrices generated as a load of tables, uncomment this code
/* foreach ($operations as $operation)
{
print ''.$operation['html'].' =
| '.$operation[0].' | '.$operation[1].' | '.$operation[2].' |
'.
"
".'| '.$operation[3].' | '.$operation[4].' | '.$operation[5].' |
'.
"
".'| '.$operation[6].' | '.$operation[7].' | '.$operation[8].' |
';
}*/
?>
Welcome to SymGen! This script generates multiplication tables for symmetry elements to test whether they form a complete group. Simply select the symmetry elements in your point group from those available below and then click “generate symmetry table”, or simply pick a point group which has already had its symmetry elements evaluated!
Symmetry questions sheet 1
Custom group
| | |
|---|
| '.$operations[$results[$first][$second]]['html'].'';
}
else
{
print ''.$operations[$results[$first][$second]]['html'].' | ';
}
}
else
{
print '✗ | ';
}
}
?>
|---|
- This group contains elements.
- This is an incomplete group. There are xs, where a member operated on a member creates an element not in the group.
- This group is complete.
- This group is abelian.
- This group is non-abelian. Operations which do not commute are highlighted.
">permanent link