The projection of the edges of the octahedronform two axes at right angles and give rise to four quadrants similar to those employed for the representation of ternary solutions (p.
For the graphic representation of systems of four components, four axes may be chosen intersecting at a point like the edges of a regular octahedron (Fig.
Combinations of the cube and octahedron are shown in figs.
Here each face of the octahedron is replaced by six scalene triangles, so that altogether there are forty-eight faces.
An octahedron thus consists of eight similar faces, and a cube is bounded by six faces all of which have the same surface characters, and parallel to each of which all the properties of the crystal are identical.
By suppressing either one or other set of alternate faces of the hexakis-octahedron two pentagonal icositetrahedra {hkl} and {khl} are derived.
Other tetrad axes of the octahedron are a2a'2 and a1a1.
This solid is bounded by twenty-four isosceles triangles, and may be considered as an octahedron with a low triangular pyramid on each of its faces.
Like the triakis-octahedron this solid is also bounded by twenty-four isosceles triangles, but here grouped in fours over the cubic faces.
A drawing of a crystal showing a combination of the cube, octahedron and rhombic dodecahedron is shown in fig.
An axis passing through the centre O and the middle points d of two opposite edges of the octahedron (fig.
Only in the last three can there be any variation in the angles: for example, the primitive octahedron of alum, nitre and sugar were determined by Rome de l'Isle to have angles of 110 deg.
Occasionally, too, the spherical form of the lattice-shell passes over into that of the regular octahedron (Pl.
Hence the regular polyhedra have as many positions of equilibrium as they have angles or sides, the icosahedron twenty, dodecahedron twelve, octahedron eight, cube six, tetrahedron four.
One would cut a piece of alum into the ordinaryoctahedron form and scrape it so as to round off the edges.
The stone, a clear yellow octahedron of about ten carats' weight, was passed from hand to hand to be admired and appraised.
Now, inscribe in the earth an icosahedron, the sphere inscribed in it will be Venus: inscribe an octahedron in Venus: the circle inscribed in it will be Mercury.
Illustration] Though we cannot really see all the sides of the octahedron at once, we can make a projection of it that suits our purpose just as well.
I will just explain that the octahedron is one of the five regular, or Platonic, bodies, and is contained under eight equal and equilateral triangles.
Among the 14 different crystalline forms of the diamond, probably the octahedron and the cube are the only ones that will give single vision.
The octahedron being thus rectified, a section is to be made parallel to the common base or girdle, so as to cut off 5 eighteenths of the whole height from the upper pyramid, and 1 eighteenth from the lower one.
Similarly, an octahedron may be inscribed in a cube, and by letting it expand a little, the faces of the cube will cut off the corners of the octahedron.
The tetrahedron was assigned to fire, the octahedron to air, the icosahedron to water, and the cube to earth, it being asserted that the smallest constituent part of each of these substances had the form here assigned to it.
The third represents a crystal formed by replacing each edge of an octahedron by a plane, it being easy to see that Euler's law still holds true.
If we think of the cube as expanding, the faces of the octahedron will cut off the corners of the cube as seen in the first figure, leaving the cube as shown in the second figure.
Although Eudemus attributes all five to Pythagoras, it is certain that the tetrahedron, cube, and octahedronwere known to the Egyptians, since they appear in their architectural decorations.
As an interesting amplification of this proposition, the centers of the faces (squares) of a cube may be connected, an inscribedoctahedron being thereby formed.
It crystallizes in the cubic system, and well-developed crystals are of common occurrence; the usual form is the cube or the cubo-octahedron (fig.
Octahedra and cubes are rare, but the six-faced octahedron occurs in some of the combinations.
They are similar to the "etched figures" produced by moistening an octahedron of alum, and have probably been produced, like them, by the action of some solvent.
The crystals often display triangular markings, either elevations or pits, upon the octahedron faces; the latter are particularly well defined and have the form of equilateral triangles (fig.
The crystals belong to the cubic system, generally assuming the form of the octahedron (fig.
The octahedron faces are usually smooth; most of the other faces are rounded (fig.
But in the cube each plane cuts 1 axis, and is parallel to 2 axes; in the dodecahedron each plane cuts 2 axes, and is parallel to a third; while in the octahedron each plane cuts the 3 axes.
Thus, the cube or hexahedron, the rhombic dodecahedron, and the octahedron all belong to the regular system, which is characterised by 3 equal axes cutting one another at right angles.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "octahedron" in a variety of sentences. We hope that you will now be able to make sentences using this word. Other words: eight; figure; octave; octet; triangle
