Home page.

An octahedron related to the Shashibo atom.
Version of Wednesday 26 February 2025.

§1. There is an octahedron suggested by the Shashibo atom. Termed the "ocho" in this report, it can be defined by giving the three-dimensional Cartesian coordinates of its six vertices, whose names are letters E through J:

E	= ( 0,	½√2,	0	)
F	= ( −½√2,	0,	0	)
G	= ( +½√2,	0,	0	)
H	= ( −½√2,	½√2,	½	)
I	= ( +½√2,	½√2,	½	)
J	= ( 0,	0,	½	)

Then the convex hull of these six points is the ocho.

These dimensions are intended to correspond as closely as possible to those of the atom.

§2. Here are some measurements of an ocho.

§2A. Lengths and dihedral angles of the edges:

edges		FG, HI	EF, EG, HJ, IJ	FH, GI	EH, EI, FJ, GJ
length	exactly	√2	1	½√3	½√3
	numerically	1.4142	1.0	0.8660	0.8660
dihedral angle		π/2 rad = 90 deg	3π/4 rad = 135 deg	π/3 rad = 60 deg	2π/3 rad = 120 deg

Edges of the ocho do have the same values as those of the atom, but it is difficult to find a more specific correspondence.

§2B. Areas of the ocho faces and the atom faces they equal:

equal faces	ocho	EFG, HIJ	EHI, FGJ	EFH, EGI, FHJ, GIJ
	atom	A	D	B?C
area	exactly	½	¼√2	¼√2
	numerically	0.5	0.3536	0.3536

Recall from the home page that "B?C" means face B or face C when they need not, or cannot, be distinguished.

§2C. Solid angles, in steradians:

vertices	E, J	F, G, H, I
angle	5π/6 sr	π/4 sr

§2D. Diagonals:

diagonals		EJ	FI, GH
length	exactly	½√3	½√11
	numerically	0.8660	1.65831

§2E. Each of the following sets of four points is coplanar:

• P₀ = {E, F, I, J}
• P₁ = {E, G, H, J}
• P₂ = {F, G, H, I}

§2F. The volume of the ocho is ¹⁄₃ ≈ 0.3333, four times the volume of an atom. However, it is not possible to dissect an ocho into four atoms by using slicing planes.

Still, an ocho can be sliced by planes P₀ and P₁. In that case, four tetrahedra result: two atoms, EFGJ and EHIJ; and two disphenoids, EFHJ and EGIJ. Any of them has a volume of ¹⁄₁₂.

If any four vertices of the ocho are selected, the volume of the tetrahedron they define is either ¹⁄₁₂ or 0. Similarly, if the center of the ocho and any three of its vertices are selected, the volume of the tetrahedron they define is either ¹⁄₂₄ or 0. Either way, a volume of 0 signifies that the four points are coplanar.

§3. Miscellaneous:

Opposite vertices are E-J, F-I, and G-H.

Opposite edges are parallel, namely EF-IJ, EG-HJ, EI-FJ, EH-GJ, FG-HI, and FH-GI.

Opposite faces are parallel, namely EFG-HIJ, EFH-GIJ, EGI-FHJ, and EHI-FGJ

Recall the Shashibo atom from the home pages, whose vertices are named A, B, C, and D. Label the respective midpoints of edges AB as I′; AC as H′; AD as J′; BC as E′; BD as G′; and CD as F′. Then the octahedron E′F′G′H′I′J′ is a miniature ocho. Specifically, it is geometrically similar at ¹⁄₆ scale.

Because the ocho has considerable symmetry, some researchers might prefer to translate it so that it is centered on the origin:

E″	= ( 0,	+¼√2,	−¼	)
F″	= ( −½√2,	−¼√2,	−¼	)
G″	= ( +½√2,	−¼√2,	−¼	)
H″	= ( −½√2,	+¼√2,	+¼	)
I″	= ( +½√2,	+¼√2,	+¼	)
J″	= ( 0,	−¼√2,	+¼	)