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24 Charles Audet, Pierre Hansen, and Frédéric Messineof one angle and at distance one of the opposite vertex, has the property to be of maximalperimeter: 2 + 4 sin π 12∼ 3.03527618.... In order to compare, the square has a perimeter about2 √ 2 ∼ 2.82842712...Then, the next open case studied here is the octagon, see [2].Today, it seems to be unfeasible to solve directly this problem using only geometric arguments(open problem since 1922) and using only an exact global optimization tool (becausethe problem is too complicated). Our proof combines new geometric arguments and a specificinterval global optimization algorithm due to Messine et al. [9–11] which can deal withtrigonometric functions.The following summarizes the way to find the small octagon with maximal perimeter; forthe complete proof see [2].2. Bisection into 31 sub-problemsAs for the largest small octagon, [4] one uses here the decomposition in sub-problems usinglinear thrackleation graphs. A linear thrackleation graph is a graph such that there existsalways a path joining two extreme points; an edge of this graph is obtained if the distancebetween two vertices is equal to 1 (this corresponds to a binding constraints). In [2], oneproves that the solution is based on one of the 31 linear thrackleations given in [4]. One doesnot represent all these configurations but one uses the same numbering as in [4]. In Figure 1,the most important thrackleations (case 29 and case 31) are represented.•••••••••••• ••Case 29••Case 31Figure 1. Thrackleations 29 and 31.3. Bounds derived from Datta’s resultsIn [6], Datta proves that all the optimal n−gons when n ≥ 5 cannot have a side of length one.This lemma allows to eliminate 7 cases: thrackleations numbered 1, 2, 3, 4, 21, 22, 23 in [4].Datta also proves in [6], that an upper bound for the perimeter of a small n−gon is2n sin π 2n .This bounds are attained when n ≠ 2 s with s ∈ IN \ {0, 1}.Figure 2 represents a common and possible configuration inside a thrackleation. v i , v j , v j+1and v j+2 are four vertices of the octagon and of the corresponding thrackleation, c j and c j+1