Schwartzite and Torus

The structure of P-Type Schwartzite may look pretty complicated at first sight. It is not easy to see the construction rule. Chuang Chern noticed that there is an interesting connection between Schwartzite and Torus. Since Schwartzite structure has no positive Gaussian curvature, but the torus has both positive and negative curbatures. In the outer-rim, a torus has five or sixe pentagons to generate the effect of positive curvature, while in the inner-rim there are the same number of heptagons.

It is not too surprising to find that the connection necks from one hole to another hole in a Schwartzite are actually the inner-rims of the torus. By symmetry, one unit cell of P-type Schwartzite can be considered as assembly of six the inner-rim of 4-fold tori. Each neck contain 4 (or 8) heptagons, which can simulate the negative Gaussiuan curvatures. So in one unit cell there are 24 heptagons.


(Picture added 4/20/2007)

Similarly, there are 3 (or 6) heptagons in each neck of the D-type Schwartzite.

However, there are some difference too. Due to the strong strain energy in the outer-rim , it is not easy to have 3- or 4-fold torus.