Ping-Pong Valence Sphere Model

Valence sphere models, a qualitative chemical bond model that includes the influence of the electron pair repulsion among valence electron pairs and attraction between positive atomic core and negative electron pairs, can be constructed with Ping-Pong balls easily. Here, I made three pairs of linked ping-pong balls and used them to create a tetrahedral sp³-hybridized AX₄ system and a octahedral d²sp³ hybridized AX₆ system.

Beautiful photo of Chern's C60@60 at the Columbia Secondary School

The superbuckyball, C60@C60, made by Chern is now on exhibition at the Columbia Secondary School, New York. This is for an event called MoSAIC—Mathematics of Science, Art, Industry, Culture—the festival, an offshoot of the annual Bridges Organization international conference dedicated to the connections between art and mathematics.

There is a nice photo of this C60@C60 in the Columbia Spectator.

Bead model of Klein's all-heptagon network

I took a picture of single tetrahedral unit (12 heptagons) of D56 bead model on the figure depicting schematically an open network consisting only of heptagons, described by Klein in his 1879 paper.

Klein, F. (1878). "Ueber die Transformation siebenter Ordnung der elliptischen Functionen" [On the order-seven transformation of elliptic functions]. Mathematische Annalen 14 (3): 428–471. Translated in Levy, Silvio, ed. (1999). The Eightfold Way. Cambridge University Press.

C20 vs C56