Incidence Matrix

For a $k$-D Polytope $\Pi_k$, the incidence matrix is defined by

$$\eta_{ij}^k =\cases{ 1 & if $\Pi_{k-1}^i$\ belongs to $\Pi^j_k$\cr 0 & if $\Pi_{k-1}^i$\ does not belong to $\Pi^j_k$.\cr}$$

The $i$th row shows which $\Pi_k$s surround $\Pi^i_{k-1}$, and the $j$th column shows which $\Pi_{k-1}$s bound $\Pi_k^j$. Incidence matrices are also used to specify Projective Planes. The incidence matrices for a Tetrahedron $ABCD$ are

$\eta^0$	1	$A$	$B$	$C$
1	1	1	1	1

$\eta^1$	$AD$	$BD$	$CD$	$BC$	$AC$	$AB$
$A$	1	0	0	0	1	1
$B$	0	1	0	1	0	1
$C$	0	0	1	1	1	0
$D$	1	1	1	0	0	0

$\eta^2$	$BCD$	$ACD$	$ABD$	$ABC$
$AD$	0	1	1	0
$BD$	1	0	1	0
$CD$	1	1	0	0
$BC$	1	0	0	1
$AC$	0	1	0	1
$AB$	0	0	1	1

$\eta^3$	$ABCD$
$BCD$	1
$ACD$	1
$ABD$	1
$ABC$	1

See also Adjacency Matrix, k-Chain, k-Circuit