# Mathematical language to describe periodic strutures.

#### sph2020

##### Newbie
Hi all,

I am looking for some new vocabulary. To be more specific I am looking at unitcells of a 3d perodic lattice that are filled with material to with various fill-fractions. For example please see the attached picture of a 3d simple cubic structure. In the first depicted unicell, it is completely filled with material. In the second depiction of the unitcell, you see that at the edges, holes start to appear. However, the holes are not connected yet, in contrast to the third depiction of the unitcell, where the holes are connected throughout the lattice. In the fourth example so much material has been removed from the unit cell that the material is not connected to its neighbouring unitcells.

Is there some mathematical property and language that describes this "connectedness" of a periodic medium?

Cheers,
sph

#### timof

Topology in mathematics studies such properties as "connectedness".

But do you care about terminology only, or about physical effects caused by these properties, and about mathematical techniques that help study and understand such systems?

##### Super Moderator
Staff member
Cutouts are the intersection of 8 spheres with the cube's corners.
Sphere centers coincide (approximately) with corners of the cube.

The spheres grow larger, cutting out larger portions of the cube.
Cutouts begin to connect when their radius reaches 1/2 of a cube edge.

As cutouts get larger, sharp edges remain. These are smoothed so there are no convexities.

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