LXIII. On the division of space with minimum partitional area

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Notes

*Communicated by the Author.

†By “curvature” of a surface I mean sum of curvatures in mutually perpendicular normal sections at any point; not Gauss's “curvatura integra,” which is the product of the curvature in the two “principal normal sections,” or sections of greatest and least curvature. (See Thomson and Tait's ‘Natural Philosophy,’ part i. §§ 130, 136.)

†The rhombic dodecahedron has six tetrahedral angles and eight trihedral angles. At each tetrahedral angle the plane faces cut one another successively at 120°, while each is perpendicular to the one remote from it; and the angle between successive edges is cos⁻¹1/δ, or 70°32′. The obtuse angles (109°28′) of the rhombs meet in the trihedral angles of the solid figure. The whole figure may be regarded as composed of six square pyramids, each with its alternate slant faces perpendicular to one another, placed on six squares forming the sides of a cube. The long diagonal of each rhombic face thus made up of two sides of pyramids conterminous in the short diagonal, is times the short diagonal.

‡I see it inadvertently stated by Plateau that all the twelve films are “légèrement courbées.”

†To do for every point of meeting of twelve films what is done by blowing in the experiment of § 5.

‡The corresponding two-dimensional problem is much more easily imagined; and may probably be realized by aid of moderately simple appliances.

†This figure (but with probably indefinite extents of the truncation) is given in books on mineralogy as representing a natural crystal of red oxide of copper.

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Notes on contributors

William Thomson*

*Communicated by the Author.