What is the Relationship Between Lattice Points and Basis Balls?

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SUMMARY

The discussion centers on the relationship between lattice points and basis balls in a mathematical context. It establishes that while there are five lattice points, these points can indeed share basis balls, leading to a non-unique orientation. The conversation highlights that the arrangement of basis balls is not directionally significant, as the complete lattice structure encompasses all possible configurations. This insight emphasizes the importance of understanding the full lattice rather than focusing solely on a limited sample.

PREREQUISITES
  • Understanding of lattice structures in mathematics
  • Familiarity with basis balls in geometric contexts
  • Knowledge of orientation and symmetry in mathematical objects
  • Basic concepts of combinatorial geometry
NEXT STEPS
  • Research the properties of lattice points in higher dimensions
  • Explore the concept of basis balls in relation to Voronoi diagrams
  • Study the implications of shared basis balls in geometric configurations
  • Investigate combinatorial geometry and its applications in optimization problems
USEFUL FOR

Mathematicians, geometry enthusiasts, and students studying combinatorial geometry or lattice theory will benefit from this discussion.

Eth338
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Hi, take a look at the picture from my textbook, specifically the bottom part:

11.png


there are five lattice points, shouldn't that mean that there are also 5 "small basis balls"? Or can they be "shared"? If so, they are not all oriented in the same way - is that not important since there's no special direction?
 
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That small sampling of 5 lattice points is not the full lattice. When you fill it out, all of the Basis parts are there.
For example, the 5-point one could have been shown this way:
XxXx
. XxXx
XxXx

or
XxXx
xXxX
XxXx
 

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