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What are lattices?

What are lattices? Here are some definitions.

Noun
  1. plural of lattice
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Crystal lattices consist of repeating units of a molecule, ion, or atom arranged in a symmetrical array.
There are only fourteen distinct types of Bravais lattices, and these are associated with seven crystal systems.
In Undecidable theories Tarski showed that group theory, lattices, abstract projective geometry, closure algebras and others mathematical systems are undecidable.
Because water molecules can bond only at certain angles, these lattices are full of empty space, and the solid takes up more room than the liquid.
Using nails or screws, connect the sides of the archways with wood or wire lattices from which the vines will grow.
Ultimately, confining logical molecules in layers or three-dimensional lattices of nanoscopic bubbles might allow a true molecular computer to be built.

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