抄録
Cutting up a cube by a plane, in generally, the cross-section presents to any one of a triangle, a quadrangle, a pentagon and a hexagon. These are a familiar sight in the mathematical text book and are frequently treated as the teaching materials at the junior high school. Here we have the problem, as the sectional figures are restricted to a hexagon, what is the hexagon with maximum area. Then one may answer that it is the regular hexagon. But, this is the conclusion due to a faulty intuition, and the correct answer is that no hexagon with maximum area exists. The first motivation for this paper was to find a solution to the above. The principal contents are the following. In Section 1, we present circumstances of the solving in mathematics that every (n-1)-dimensional cross-section of the unit cube in the n-dimensional Euclidean space R^n has volume at most √<2> and this value is best possible, and we consider the outline of the related results obtained by D. Hensley, K. Ball and S. Tanno. In Section 2, we mention the following (1)-(5) with respect to the cross-sections of the usual 3-dimensional unit cube, in current forms for the educational fields. (1) In the usual 3-dimensional Cartesian coodinates system, we carry out the classification of the cross-sections of the unit cube by treating the sectional plane with the Hesse's normal form, and we get the expression of the area for each case. (2) The maximum area of any of these cross-sections is f and its shape is attained as the rectangle through its center and containing a edge of the unit cube. (3) The supremum of these areas restricted the cross-sections to hexagons is √<2>, and there is no maximum in this case. We have also the infimum √<3>/2 in this case. (4) The supremum of these areas restricted the cross-sections to pentagons is √<2>, and there is no maximum in this case. We have also the infimum 1/2 in this case. In Section 3, I mention the views of mathematics education on the above considerations.
