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Updated: June 22, 2025


Through the water the Nevian commander darted along, steering his course accurately by means of his short, vaned tail. Through an opening in a wall he sped and along a submarine hallway, emerging upon a broad ramp. He scurried up the incline and into an elevator which lifted him to the top floor of the hexagon, directly into the office of the Secretary of Commerce of all Nevia.

It is even conceivable that an insect might, by fixing on a point at which to commence a cell, and then moving outside, first to one point, and then to five other points, at the proper relative distances from the central point and from each other, strike the planes of intersection, and so make an isolated hexagon: but I am not aware that any such case has been observed; nor would any good be derived from a single hexagon being built, as in its construction more materials would be required than for a cylinder.

My Pentagonal Sons were men of character and standing, and physicians of no mean reputation, but not great in mathematics, and, in that respect, unfit for my purpose. But it occurred to me that a young and docile Hexagon, with a mathematical turn, would be a most suitable pupil.

It is fully 56 miles in diameter, and, though generally described as nearly circular, exhibits very distinctly under high powers a polygonal outline, approximating very closely to an equilateral hexagon. There are, however, two sections of the crest of the border on the N.E. which are inflected slightly towards the centre, a peculiarity already noticed in the case of Eratosthenes.

In a square pool, the water, previously unconfined, becometh a square; in a six-sided pool it becometh a hexagon, in an eight-sided pool an octagon, and so forth. The rain itself hath no geometry, no limits, no form, but it taketh on one form or another, according to the restrictions of its vessel.

I have said that the room in which M. le Vicomte de Chagny and I were imprisoned was a regular hexagon, lined entirely with mirrors. Plenty of these rooms have been seen since, mainly at exhibitions: they are called "palaces of illusion," or some such name. But the invention belongs entirely to Erik, who built the first room of this kind under my eyes, at the time of the rosy hours of Mazenderan.

That something was evidently a soft metal washer, which fitted on to this hexagon, and which would be pressed into the grooves of the rifling, and so spin the dagger, but would drop off as soon as the weapon left the barrel. Here is such a washer, which Polton has made for us." He laid on the table a metal disc, with a hexagonal hole through it.

But the foremost, the crowning excellence of Fontainebleau forest consists in its variety. France itself, the "splendid hexagon," with its mountains, rivers and plains, is hardly more varied than this vast area of rock and woodland.

The same men were also capable of the more accurate discovery that the side of a hexagon inscribed in a circle was equal to the radius and gave us our division of sixty minutes and sixty seconds with all its advantages for calculation.

When bees have a place on which they can stand in their proper positions for working for instance, on a slip of wood, placed directly under the middle of a comb growing downwards, so that the comb has to be built over one face of the slip in this case the bees can lay the foundations of one wall of a new hexagon, in its strictly proper place, projecting beyond the other completed cells.

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