Figure 4.
Relationship between cube and dodecahedron with quasi-equivalent contacts along their edges. (A) Left, dodecahedron (blue) with a dark gray plane through two opposite edges and the particle center; Right, cube (violet) with a light gray similar plane through two opposite edges and the particle center. (B) The dodecahedral and cubic planes of A superimposed. The assumption acube = adodecahedron ( = a) leads to Rdodeca = 1.62 Rcube as derived in the text. As can be seen at the bottom of the figure, this derivation and the requirement that the spatial relationship between the spheres and the threefold axes is the same as the relationship in the cube introduces a small error, and the spheres in the dodecahedron do not make contact. Note that geometric considerations yield μd = 41.8°, μc = 70.5°, αc = 54.7°, αd = 69.1°, and hence Δα = αd − αc = 14.4°. (C) Size of dodecahedron when the spherical subunits (with radius 1/4a) touch each other at contact points Q′d. It is easily seen that now A′dM′ = (1/4 + 1/4cosΔα) × a. Also, A′dM′ = A′dS × sin(1/2μd) = Rd × sin(1/2μd). This gives Rd × sin(1/2μd) = (1/4 + 1/4cosΔα) × a, hence Rd = 1.379a. Because in the cube Rc = 1/2a √3 = 0.866a, we obtain Rd = 1.592Rc, which is a better approximation of the observed relationship Rd = 1.561Rc than obtained in B. (D) E2p subunits are not ideal spheres, and the principle of quasi-equivalence suggests that the contact points Qc in the middle of the edge AcBc of the cube also touch each other in the case of the dodecahedron. In this case, Ad′′M′′ = A′′dQ′′d × cosΔα = 1/2a × cosΔα and Rd = A′′M′′/sin(1/2μd) = (1/2a cosΔα)/sin (1/2μd) = 1.357a. With Rc = 0.866a this yields Rd = 1.567Rc, which is quite close to the observed relationship Rd = 1.561Rc.