Platonic Solids

the platonic solids

Plato assumed these shapes corresponded to the properties given; in particular associating icosahedra with water (as I do in this web site).^a They are the only regular solids where all the vertices and the centers of all the faces and edges lie on spheres (the circumscribed, inscribed and mid spheres respectively) with the same center.

The properties of these solids, with edge length (el) are given in the following table:

**Trigonometric features of the platonic solids**
Name	Faces		Edges		Vertices		Surface	Volume
Name	No.	Diam., el	No.	Diam., el	No.	Diam., el	el²	el³
Tetrahedron	4 triangular	1/√6	6	√½	4	√6/2	√3	√2/12
Cube	6 square	1	12	√2	8	√3	6	1
Octahedron	8 triangular	2/√6	12	1	6	√2	2√3	√2/3
Dodecahedron	12 pentagonal	√(140+220φ)/10	30	1+φ	20	φ√3	3√(15+20φ)	(4+7φ)/2
Icosahedron	20 triangular	√(24+36φ)/6	30	φ	12	√(2+φ)	5√3	5(1+φ)/6

Name	Coordinates, el [444 ]
Tetrahedron	(-½√½,½√½,½√½)(½√½,-½√½,½√½) (½√½,½√½,-½√½)(-½√½,-½√½,-½√½)
Cube	(±½, ±½, ±½)
Octahedron	(±√½, 0, 0)(0, ±√½, 0)(0, 0, ±√½)
Dodecahedron	(0, ±½, ±½(1+φ))(±½(1+φ), 0, ±½) (±½, ±½(1+φ), 0)(±φ/2, ±φ/2, ±φ/2 )
Icosahedron	(±½, 0, ±φ/2)(±φ/2, ±½, 0)(0, ±φ/2, ±½)

where φ is the golden ratio. A rectangle with sides in the ratio 1:φ gives a similar rectangle when the square side 1 is removed:

φ = (√5+1)/2 = 1.618034....
1/φ = φ - 1= (√5-1)/2 = 0.618034....
φ² = φ + 1= (√5+3)/2 = 2.618034....

Interestingly the golden ratio also appears in aqueous chemistry as the ratio between atomic and ionic diameters. Thus the diameter of an anion (A^-) is twice its atomic diameter divided by φ and the diameter of a cation (A⁺) is twice its atomic diameter divided by φ²; with the diameter of A^- being the golden ratio times the diameter of A⁺, and simple functions of φ also relating ion-water distances to covalent radii [1091 ].

Plato would not have been wrong to connect liquid structure in general to icosahedra as spherical atoms and molecules (for example, the larger noble gases) in the liquid phase prefer icosahedral clustering which has a lower energy than crystal structures (but cannot form crystals due to the five-fold symmetry).

Shown right is an icosahedral cluster of thirteen identical spherical atoms^b as found in liquid argon, krypton, xenon and molten metals; such five-fold symmetry being optimal for short-range close packing but incompatible with long-range order and favoring amorphous structures. Its preferred formation has been shown to prevent crystallization in liquid metal melts and be the cause of their extensive supercooling [505 ].

It is clear from the evidence
presented at this site that water
may form icosahedral clusters,

so linking modern science with ancient philosophy.

Footnotes

^a The association of the dodecahedron with the Universe has also received a recent burst of interest, now somewhat subsiding [1163 ]. [Back]

^b One atom resides in the slightly-too-small cavity at the center, causing loose contact between the twelve at the vertices. Note that thirteen atoms can only fit snugly together in a cuboctahedron (with 8 triangular and 6 square faces) formed from three layers containing 3, 7 and 3 atoms and part of a hexagonal close packed arrangement. [Back]