Summary of area & perimeter bounds for cross-sectional polygons of cube, updating and superseding previous provisional posts --- Five distinct polygons have vertices in common with a double-unit cube: the following table shows each Cartesian vertex set, sectional plane, area A, perimeter B. Corner: x+y+z = 3 , { (1,1,1) } , A = 0 , B = 0 ; Edge: { (1,1,1), (-1,1,1) } , y+z = 2 , A = 0 , B = 4 ; Face: { (1,1,1), (-1,1,1), (-1,-1,1), (1,-1,1) } , z = 1 , A = 4 , B = 8 ; Triangle: { (-1,-1,1), (1,-1,-1), (-1,1,-1) } , x+y+z = -1 , A = 2 sqrt(3) ~ 3.46410162 , B = 6 sqrt(2) ~ 8.48528137 ; Rectangle: { (-1,-1,1), (1,-1,1), (1,1,-1), (-1,1,-1) } , y+z = 0 , A = 2 + 2 sqrt(2) ~ 5.65685425 , B = 4 + 4 sqrt(2) ~ 9.65685425 . Classified by number n of polygon vertices, corresponding extremal polygons are as follows: n = 3 : min = Corner , max = Triangle ; n = 4 : min = Edge , max = Rectangle ; n = 4 : min = Face , max = Rectangle ; n = 5 : min = Triangle , max = Rectangle ; n = 6 : min = Triangle , max = Rectangle . Notes: The same extreme polygon yields sharp bounds on both area and perimeter. The extremes are in general attainable only if coincident vertices are permitted. When n = 6 , the linear system of planes x+y+z = d yields hexagonal minimum perimeters for all |d| < 1 . When n = 4 , there are two topologically distinct families of generic quadrilaterals, splitting the 8 cube corners 6+2, 4+4 respectively. There appear to be no critical points apart from isomorphs of those identified above, either interior or constrained to its boundary, within the tetrahedral phase-space region of planes meeting the cube. This situation is reminiscent of linear programming, despite area and perimeter being no longer linear. So is there some obvious reason why these bounds should occur --- almost exclusively --- at polygons with their vertices in common with the cube? [ Such an insight would save much blundering around in a surprisingly ill-conditioned numerical multivariable constrained optimisation problem. For instance Rectangle maxima, lurking innocently at a corner of the phase region, constitute actual singularities: traps cunningly planted with maximal inconvenience and unpredictability! ] Fred Lunnon