▁which ▁represents ▁the ▁final ▁result . ▁The ▁other ▁n ▁sides ▁of ▁the ▁polygon , ▁in ▁the ▁clock wise ▁direction , ▁represent ▁the ▁matrices . ▁The ▁vertices ▁on ▁each ▁end ▁of ▁a ▁side ▁are ▁the ▁dimensions ▁of ▁the ▁matrix ▁represented ▁by ▁that ▁side . ▁ ▁With ▁n ▁matrices ▁in ▁the ▁multiplication ▁chain ▁there ▁are ▁n − 1 ▁binary ▁operations ▁and ▁C n − 1 ▁ways ▁of ▁placing ▁parentheses , ▁where ▁C n − 1 ▁is ▁the ▁( n − 1 )- th ▁Catal an ▁number . ▁The ▁algorithm ▁explo its ▁that ▁there ▁are ▁also ▁C n − 1 ▁possible ▁tri ang ulations ▁of ▁a ▁polygon ▁with ▁n + 1 ▁sides . ▁ ▁This ▁image ▁illustr ates ▁possible ▁tri ang ulations ▁of ▁a ▁regular ▁hex agon . ▁ ▁These ▁correspond ▁to ▁the ▁different ▁ways ▁that ▁parentheses ▁can ▁be ▁placed ▁to ▁order ▁the ▁multip lications ▁for ▁a ▁product ▁of ▁ 5 ▁matrices . ▁ ▁For ▁the ▁example ▁below , ▁there ▁are ▁four ▁sides : ▁A , ▁B , ▁C ▁and ▁the ▁final ▁result ▁ABC . ▁ ▁A ▁is ▁a ▁ 1 0 × 3 0 ▁matrix , ▁B ▁is ▁a ▁ 3 0 × 5 ▁matrix , ▁C ▁is ▁a ▁ 5 × 6 0 ▁matrix , ▁and ▁the ▁final ▁result ▁is ▁a ▁ 1 0 × 6 0 ▁matrix . ▁The ▁regular ▁polygon ▁for ▁this ▁example ▁is ▁a ▁ 4 - gon , ▁i . e . ▁a ▁square : ▁ ▁The ▁matrix ▁product ▁AB ▁is ▁a ▁ 1 0 x 5 ▁matrix ▁and ▁BC