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be the number of integer points on its boundary (including both vertices and points along the sides). Then the area
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lying on the 2-dimensional integer lattice, in terms of the number of integer points within it and on its boundary.
35:
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focuses on the selection of nodes in the
Diophantine plane such that all pairwise distances are integers.
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Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen
Vereines für Böhmen "Lotos" in Prag
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values can be mapped to points lying close to lines having gradients corresponding to the values
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462:, the square lattice of points with integer coordinates is often referred to as the
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981:. New Mathematical Library. Vol. 41. Mathematical Association of America.
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of order 8; for the three-dimensional cubic lattice, we get the group of the
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Lattice group in
Euclidean space whose points are integer n-tuples
925:(2018). "Three applications of Euler's formula: Pick's theorem".
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be the number of integer points interior to the polygon, and let
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494:{\displaystyle \scriptstyle \mathbb {Z} \times \mathbb {Z} }
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with vertices on a square lattice with coordinates indicated
248:. The two-dimensional integer lattice is also called the
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466:. In mathematical terms, the Diophantine plane is the
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398:{\displaystyle (\mathbb {Z} _{2})^{n}\rtimes S_{n}}
49:. Unsourced material may be challenged and removed.
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434:For the square lattice, this is the group of the
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931:(6th ed.). Springer. pp. 93–94.
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522:{\displaystyle \scriptstyle \mathbb {Z} }
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109:Learn how and when to remove this message
845:{\displaystyle A=7+{\tfrac {8}{2}}-1=10}
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740:{\displaystyle A=i+{\frac {b}{2}}-1.}
7:
627:in 1899, provides a formula for the
47:adding citations to reliable sources
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799:boundary points, so its area is
289:. The integer lattice is an odd
276:{\displaystyle \mathbb {Z} ^{n}}
228:{\displaystyle \mathbb {R} ^{n}}
188:{\displaystyle \mathbb {Z} ^{n}}
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886:"Geometrisches zur Zahlenlehre"
34:needs additional citations for
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329:it is given by the set of all
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285:is the simplest example of a
338:signed permutation matrices
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125:Approximations of regular
937:10.1007/978-3-662-57265-8
237:whose lattice points are
978:The Geometry of Numbers
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138:Rational approximants
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1016:Diophantine geometry
928:Proofs from THE BOOK
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773:interior points and
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702:of this polygon is:
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625:Georg Alexander Pick
509:
473:
460:Diophantine geometry
454:Diophantine geometry
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43:improve this article
792:{\displaystyle b=8}
766:{\displaystyle i=7}
531:Diophantine figures
252:, or grid lattice.
1006:Euclidean geometry
973:Davidoff, Giuliana
923:Ziegler, Günter M.
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346:semidirect product
303:automorphism group
297:Automorphism group
291:unimodular lattice
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695:{\displaystyle A}
675:{\displaystyle b}
655:{\displaystyle i}
468:Cartesian product
464:Diophantine plane
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58:"Integer lattice"
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892:. (Neue Folge).
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458:In the study of
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543:coarse geometry
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537:Coarse geometry
529:. The study of
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410:symmetric group
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633:simple polygon
621:Pick's theorem
616:Pick's theorem
614:Main article:
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553:Pick's theorem
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440:dihedral group
429:wreath product
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162:cubic lattice
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60: –
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54:Find sources:
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38:
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32:This article
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862:Regular grid
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327:matrix group
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315:permutations
300:
287:root lattice
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41:Please help
36:verification
33:
969:Lax, Anneli
965:Olds, C. D.
896:: 311–319.
882:Pick, Georg
311:congruences
164:), denoted
150:mathematics
99:August 2013
1000:Categories
902:33.0216.01
868:References
408:where the
342:isomorphic
142:irrational
127:pentagrams
69:newspapers
831:−
732:−
635:with all
483:×
438:, or the
420:acts on (
383:⋊
197:, is the
975:(2000).
884:(1899).
856:See also
637:vertices
609:− 1 = 10
325:!. As a
321:2
246:integers
607:
591:
501:of the
344:to the
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242:-tuples
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201:in the
199:lattice
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83:scholar
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436:square
152:, the
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631:of a
446:, or
319:order
307:group
90:JSTOR
76:books
983:ISBN
941:ISBN
642:Let
629:area
503:ring
444:cube
305:(or
301:The
160:(or
62:news
933:doi
898:JFM
575:= 8
566:= 7
541:In
431:).
309:of
244:of
148:In
140:of
45:by
1002::
971:;
967:;
939:.
921:;
894:19
888:.
840:10
735:1.
589:+
583:=
578:,
569:,
549:.
293:.
991:.
949:.
935::
904:.
837:=
834:1
825:2
822:8
816:+
813:7
810:=
807:A
787:8
784:=
781:b
761:7
758:=
755:i
727:2
724:b
719:+
716:i
713:=
710:A
690:A
670:b
650:i
604:2
601:/
596:b
586:i
581:A
573:b
564:i
515:Z
487:Z
479:Z
425:2
422:Z
417:n
413:S
391:n
387:S
378:n
374:)
368:2
363:Z
358:(
335:n
331:n
323:n
269:n
264:Z
240:n
221:n
216:R
181:n
176:Z
154:n
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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