Версия от 08:10, 26 марта 2024; EducationBot(обсуждение | вклад)(Новая страница: «{{Английская Википедия/Панель перехода}} {{Short description|Lattice group in Euclidean space whose points are integer n-tuples}} {{refimprove|date=August 2013}} thumb|Approximations of regular [[pentagrams with vertices on a square lattice with coordinates indicated]] File:Diophantine_approximation_graph.svg|thumb|[[Diophantine approximation|Rational ap...»)
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For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three-dimensional cubic lattice, we get the group of the cube, or octahedral group, of order 48.
Diophantine geometry
In the study of Diophantine geometry, the square lattice of points with integer coordinates is often referred to as the Diophantine plane. In mathematical terms, the Diophantine plane is the Cartesian product <math>\scriptstyle\mathbb{Z}\times\mathbb{Z}</math> of the ring of all integers <math>\scriptstyle\mathbb{Z}</math>. The study of Diophantine figures focuses on the selection of nodes in the Diophantine plane such that all pairwise distances are integers.
Let <math>i</math> be the number of integer points interior to the polygon, and let <math>b</math> be the number of integer points on its boundary (including both vertices and points along the sides). Then the area <math>A</math> of this polygon is:[2]
<math display=block>A = i + \frac{b}{2} - 1.</math>
The example shown has <math>i=7</math> interior points and <math>b=8</math> boundary points, so its area is <math>A=7+\tfrac{8}{2}-1=10</math> square units.
