Octahedron

when the pyramids of the masculine Self and the feminine Self are placed end to end, they form an octahedron, an image that graphically represents the Jungian Self, which embraces both masculine and feminine qualities

Moore and Gillette propose that the octahedron — formed by joining the masculine and feminine pyramidal Self-structures — serves as the geometric emblem of the complete Jungian Self and its double quaternio.

Moore, Robert, King Warrior Magician Lover: Rediscovering the Archetypes of the Mature Masculine, 1990thesis

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to fire the pyramid, to air the octahedron, and to water the icosahedron,—according to their degrees of lightness or heaviness or power, or want of power, of penetration

Plato formally assigns the octahedron to air within his elemental cosmology, ranking it among the primary bodies by degrees of penetration and mobility.

Plato, Timaeus, -360thesis

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He begins with the construction of the equilateral triangular face which is common to the pyramid, the octahedron, and the icosahedron.

Cornford’s commentary establishes that the octahedron shares its fundamental triangular face-element with fire and water, making elemental transformation between these bodies geometrically coherent.

Plato, Plato’s cosmology the Timaeus of Plato, 1997thesis

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The twenty triangular faces of an icosahedron form the faces or sides of two regular octahedrons and of a regular pyramid (20 = 8 x 2 + 4); and therefore, according to Plato, a particle of water when decomposed is suppose

The icosahedron’s decomposition into two octahedra and a pyramid encodes Plato’s arithmetic of elemental transformation, demonstrating that water resolves into air and fire by geometric necessity.

Plato, Timaeus, -360supporting

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the 6 scalenes in the equilateral face of a pyramid can recombine, in pairs, to make three equilateral faces for pyramids or octahedra or icosahedra of the lower grade

The commentary demonstrates how the scalene triangle sub-elements of the pyramid’s face can be recombined to generate octahedral faces, formalizing the inter-elemental convertibility underlying Platonic cosmology.

Plato, Plato’s cosmology the Timaeus of Plato, 1997supporting

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varieties of each primary body, corresponding to different sizes of pyramid, octahedron, etc., and that there are indefinitely numerous combinations of all these varieties in composite bodies

Cornford notes that the octahedron, like the other primary solids, exists in multiple size-grades, enabling the composition of complex bodies from varied-scale elemental particles.

Plato, Plato’s cosmology the Timaeus of Plato, 1997supporting

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a double pyramid with its 6 faces must have less volume than an octahedron with 8 faces

A comparative volumetric remark distinguishes the double pyramid from the octahedron, clarifying their geometric relationship within the context of elemental packing and spatial interstices.

Plato, Plato’s cosmology the Timaeus of Plato, 1997supporting

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To earth let us assign the cubical figure; for of the four kinds earth is the most immobile and the most plastic of bodies.

By assigning the cube to earth on grounds of stability, Cornford’s text implicitly positions the octahedron — assigned to air — at the opposite end of the mobility spectrum among the elemental solids.

Plato, Plato’s cosmology the Timaeus of Plato, 1997aside

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every plane rectilinear figure is composed of triangles; and all triangles are originally of two kinds, both of which are made up of one right and two acute angles

Plato’s foundational reduction of all rectilinear planes to two species of right triangle establishes the geometric logic from which the octahedron, alongside the other regular solids, is derived.

Plato, Timaeus, -360aside

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