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Incidence Matrix

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

\begin{displaymath}
\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}
\end{displaymath}

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



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© 1996-9 Eric W. Weisstein
1999-05-26