1.1.................................... 3.3.................................. 4.4......................... 8.5... 10.6.................... 1.7... 14 3 16 3.1 ()........................... 16 3. 7... 17 4 18 4.1....................................... 18 4.... 19 4.3.......................... 0 4.4 1.................... 1 4.5.................... 3 5 4 5.1....................................... 4 5.... 6 6 7 6.1... 7 7 3 9 7.1... 9 7.......................... 30 8 3 31 1
9 3 9.1.................................. 33 9.................................. 34 9.3................................. 34 10 35 35 1 x y ( ) 3 1 ( 17) ( 15 19).1 1 ( ). x y xy (a) (b) (c) (d)
1: (). X S(X) S(X) =i + b 1 i = i(x) =(X ), b = b(x) =(X ) 3 (). X i(x) = 5, b(x) =7 : S(X) =i + b 1=5+7 1=15. ( 1) ( 7) 4 (). X 4 X 1 ( ) X 180 Q 180 3
3: Q P, R PR X PR PR X PQR X PR S QS X X 5 ( ). ( ) 3 4: 5:.3 3 4
6 (). X µ(x) µ X Y Z µ(x) =µ(y )+µ(z) µ 6: S S(X) =S(Y )+S(Z) µ(x) =µ(y )+µ(z) ( ) 7 (). X µ(x) =i(x)+ b(x) 1, i = i(x) =(X ), b = b(x) =(X ) µ 7: ( ) X Y Z X m i(x) =i(y )+i(z)+m, b(x) =b(y )+b(z) m 5
i(x)+ b(x) b(y )+b(z) m 1=i(Y )+i(z)+m + 1 = i(y )+i(z)+ b(y ) + b(z) ( = i(y )+ b(y ) ) ( 1 + i(z)+ b(z) ) 1 µ(x) =µ(y )+µ(z) 8 ( ). X µ(x) µ µ 3 S(X) =µ(x) (a) X µ(x) =1 (b) µ (c) X X 180 Y µ(x) =µ(y ) X S(X) =µ(x) 1 1 ( ) S(X) =µ(x) X S(X) =1 (a) S(X) =µ(x) X Y Z µ 8: S(X) =S(Y )+S(Z), µ(x) =µ(y )+µ(z) Y Z S(Y )=µ(y ) S(Z) =µ(z) S(X) =µ(x) X µ S(X) =µ(x) X X X 180 Y Z = X Y 6
9: S(X) =S(X)+S(Y ) X Y S(X) =S(Y ) = S(Z) = µ(z) Z S(Z) =µ(z) = µ(x)+µ(y ) (b) µ =µ(x) (c) µ(x) =µ(y ) S(X) =µ(x) X X Y 1, Y, Y 3 X (3 ) 10: Z = X Y 1 Y Y 3 S(X) =S(Z) S(Y 1 ) S(Y ) S(Y 3 ) = µ(z) µ(y 1 ) µ(y ) µ(y 3 ) Z Y i S(Z) =µ(z) = µ(x) (b) µ S(X) =µ(x) X 5 7
11: Y S(Y )=µ(y ) S(X) = µ(x) 3 µ S(X) =µ(x) 1: ( ( ) ) 7 µ X µ(x) =i(x)+ b(x) 1=0+ 4 1=1 X X 180 Y X Y µ(x) =µ(y ) 7 µ 8 S(X) =µ(x) =i(x)+ b(x) 1.4 Varberg 9 (). X X P P X 360 8
13: X 1 X X 1/ 90 X 1/4 X X 10 (). X µ(x) µ(x) =(X ) µ ( ) X Y Z Y Z () 14: ( ( ) ) 10 µ X µ(x) = 1 4 4=1 X X 180 Y X Y µ(x) =µ(y ) 10 µ 8 S(X) =(X ) X X i = i(x) X b = b(x) b 9
b 1 b b = b 1 + b i b / b 1 (b 1 ) 180 (b 1 )/ i + b + b 1 = i + b + b 1 1=i + b 1 S(X) =(X ) =i + b 1.5 ( ) 11 (). V E (V,E) 15: 15 V = {1,, 3, 4, 5}, E = {(1, ), (, 3), (1, 4), (, 5), (4, 5)} 15 16 16: 1 10
1. G G G 1,,...,n (i, i + 1) (1 i n 1) (1,n) 17: G ( ) 1 1 13 ( ). G () v, e, f v e + f = ( ) e e =1 18 18: e =1 v =, e = 1, f =1 (f =1) v e+f = e e 1 1 1 G ( 19) 19: 1 11
G G 1 G e 1 (v 1) (e 1) + f = v e + f = 1 G 1 G G ( 0) 0: G G 1 1 G e 1 v (e 1) + (f 1) = v e + f = 1 v e + f = G 14 ( ). ( ) 1: v e + f =.6 1/ 1
1/ () 15 ( ). 1/ ( ) X X 180 Y Y X Y Z Z Z 4 : Z Z S(Z) 1 Z P Z P r n (a, b) (a b 0 n 1 ) P n (a, b) r 1 n 1+r n S(Z) < (n 1+r) (.1) n (a, b) (a b r n 1 r ) r (.1) (.) (n 1 r) >n S(Z) (.) (n 1 r) <S(Z) < n (n 1+r) n, n (1 + r)n +(1+r) n <S(Z) < n (1 r)n +(1 r) n, (1 + r) (1 + r) (1 r) (1 r) 1 + n n <S(Z) < 1 + n n n 1 S(Z) 1, S(Z) =1 ( ( ) ) X 5 X 13
3: X i, b v, e, f v = i + b, v e + f =, 3(f 1) + b 3 f 1 3 b 3 3 1 v 3 e 3(f 1) + b = e =(i + b)+f f =i + b 1 15 1/ f 1 X S(X) S(X) =(f 1) 1 = i + b 1.7 16 ( ). h (h 0) X S(X) 14
S(X) =i + b 1+h i = i(x) =(X ), b = b(x) =(X ) ( ) X X 0 h X 1,X,...,X h X k (k =0, 1,...,h) i k = i(x k ), b k = b(x k ) 4: X i = i 0 (i 1 + i + + i h ) (b 1 + b + + b h ), b = b 0 +(b 1 + b + + b h ) (i 1 + i + + i h )+ b 1 + b + + b h ( ) S(X) =S(X 0 ) S(X 1 )+S(X )+ + S(X h ) ( = i 0 + b ) ( 0 1 i 1 + b ) ( 1 1 i + b ) 1 ( = i 0 + b ) ( 0 1 i 0 i + b ) 0 b + h = i + b 1+h = i 0 i + b 0 b ( i h + b ) h 1 15
3 3.1 () 17. (a, b) (c, d) 3 S = 1 ad bc ( ) (a, b) (c, d) θ S = 1 a + b c + d sin θ = 1 a + b c + d 1 cos θ = 1 ( ) a + b c + d ac + bd 1 a + b c + d () = 1 (a + b )(c + d ) (ac + bd) = 1 (ad bc) = 1 ad bc. 18 (). n A { a A = b } a b n. A 1 n =6 1 6, 1 5, 1 4, 1 3, 5, 1, 3 5, 3, 3 4, 4 5, 5 6, 1 1, 6 5, 5 4, 4 3, 3, 5 3, 1, 5, 3 1, 4 1, 5 1, 6 1 1 1 5 1 6 = 1 30, 3 3 5 = 1 15 ( ) n a, b (a, b) (a, b) A A 16
5: n (a, b) (c, d) (a, b) (c, d) 15 1/ ( 17) ad bc / ad bc = ±1 a/b c/d 3. 7 19 (). n 7 n ( ) n 3 C r r f r C r g r C r C r C r (C r ) G r G r v r e r G r 3 3v r e r (f r + g r + 1) e r + v r =, g r =1 f r + e r v r 17
C r f r n ( ) g r +1 3 nf r + 3(g r + 1) e r n e r 3(g r + 1) f r = e r 6g r +3g r 3 f r = e r 6(1 f r + e r v r )+3g r 3 f r = 4e r +6f r +6v r +3g r 9 f r 6f r +3g r 9 f r lim r g r /f r =0r 6 n 6 d ( ) S C r π(r d) f r S πr C r C r f r d f r + g r f r+d f r d f r π(r d) πr 1+ g r f r f r+d f r, 1+ g r π(r + d) f r π(r d) r 1 g r /f r 0 4 4.1 0 (). x y ( )+ 1 xy 18
6: 1 ( ). X S(X) S(X) =i + b i = i (X) =(X ), b = b (X) =(X ) 4., 4.3, 4.4, 4.5 4 ( ). 7 X i (X) = 6, b (X) =3 i + b =6+3 = 15 7: 4. 19
3 (). X µ(x) µ µ(x) =i (X)+ b (X) ( ) X Y Z Z X Y i (Z) i (X) i (Y ) 8: b (Z) b (X) b (Y ) G i (Z) G G i (X) i (Y ) G G b (X) b (Y ) µ ( 1) ( ( 1) ) 3 µ X µ(x) =i (X)+ b (X) =1+0=1 X X 180 Y X Y µ(x) =µ(y ) 3 µ 8 S(X) =µ(x) =i (X)+ b (X) 4.3 4 ( ). X X P P X 360 0
9: X 1 1/ X X ( 1) ( ( 1) ) µ µ(x) =(X ) µ X µ(x) =1 X X 180 Y X Y µ(x) =µ(y ) 8 S(X) =µ(x) X X i = i (X) X b = b (X) b 1/ i 1 µ(x) =i + b / S(X) =i + b 4.4 1 x y x y ( )/ ( ) 1
30: 1/ 4 4 31: 1/ 45 ( ( 1) ) h (h 0) X 16 i = i(x) =(X ), b = b(x) =(X ) S(X) =i + b 1+h (4.1) i = i (X) =(X ), b = b (X) =(X ) ( X ) =i + i, ( X ) =b + b 16
(4.1) (4.) S(X) =(i + i )+ b + b 1+h (4.) S(X) =i + b 4.5 4.4 ( ( 1) ) x m y n (m, n) 3: (m, n) (m, n) mn h (h 0) X i = i(x) =(X ), b = b(x) =(X ), i = i (X) =(X ), b = b (X) =(X ) (m, n) i (m,n) = i (m,n) (X) =((m, n) X ), b (m,n) = b (m,n) (X) =((m, n) X ) i = i (1,1), b = b (1,1) 3
i = i (,) i (1,) i (,1) + i (1,1), b = b (,) b (1,) b (,1) + b (1,1) i + b = i (,) i (1,) i (,1) + i (1,1) + b (,) b (1,) b (,1) + b (1,1) ( = i (,) + b ) ( (,) 1+h i (1,) + b ) ( (1,) 1+h i (,1) + b ) (,1) 1+h ( + i (1,1) + b ) (1,1) 1+h =4S(X) S(X) S(X)+S(X) = S(X) 5 ( 18 ) 5.1 5 ( ). X y = b (b ) x = a (a ) X X 33: X k k 4
1 1 X 34 1 34: 1 6 ( ). 0, 1, 3 ( ) 1 7 ( ). X S(X) S(X) =N 0 + N 1 N 0 = N 0 (X) =(X 0 ), N 1 = N 1 (X) =(X 1 ) 8 ( ). 35 X N 0 (X) = 4, N 1 (X) = 7, N (X) =1 N 0 (X)+ N 1 (X) =4+ 7 = 15 5
35: 5. 9 (). X µ(x) µ µ(x) =N 0 (X)+ N 1 (X) ( ) X Y Z Z 0 1 1 1 ( 36) 36: ( 0 )+( 0 )/ µ(z) = µ(x)+µ(y ) ( 7) ( ( 7) ) 9 µ X µ(x) =N 0 (X)+ N 1 (X) =1+0=1 X X 180 Y X Y µ(x) =µ(y ) 9 µ 8 S(X) =µ(x) =N 0 (X)+ N 1 (X) 6
6 ( ) 6.1 30 ( ). 31 (n ). n ( )/n n 3 (). X n (n 3 n)v (X) =(i n (X) ni 1 (X)) + b n(x) nb 1 (X) i n = i n (X) =(X n ), b n = b n (X) =(X n ) 33 (). 37 1/6 1/3 0 4 7
37: n 38 i(x) = 0, b(x) =7 38 z = 1, z =1/, z =0 X 1/ i (X) =, b (X) = ( 3) n = (8 )V (X) =( 0) + 7 V (X) =1 =6, 38: 8
7 3 7.1 34 (). 1 1 (a+1/,b+ 1/,c+1/) (a, b, c ) n 1 1/n n ((a + 1)/n, (b + 1)/n, (c + 1)/n) (a, b, c ) 35 ( ). X n (n 3 n)v (X) =(i n(x) ni 1(X)) + b n(x) nb 1(X) i n(x) =(X n ), b n(x) =(X n ) 36. 39 X i (X) = 1, b (X) =1 39 z =3/4, z =1/4 X 1/ i (X) = 7, b (X) =4 ( 35) n = (8 )V (X) =(7 1) + 4 1 V (X) =1 =6, 39: 9
7. 35 1 37 ((p, q, r) ). p, q, r x 1/p y 1/q z 1/r L (p,q,r) = {( a p, b q, c ) } a, b, c r L (p,q,r) (p, q, r) (n, n, n) n X i (p,q,r) = i (p,q,r) (X) =(X (p, q, r) ), b (p,q,r) = b (p,q,r) (X) =(X (p, q, r) ) ( 35 ) X (p, q, r) pqr pqr(n 3 n)v (X) =(i (pn,qn,rn) (X) ni (p,q,r) (X)) + 1 (b (pn,qn,rn)(x) nb (p,q,r) (X)) P 1 = {(, 1, 1), (1,, 1), (1, 1, )}, P = {(1,, ), (, 1, ), (,, 1)} i (α,β,γ) (X) =i (α,β,γ)(x) i (α,β,γ) (X), b (α,β,γ) (X) =b (α,β,γ)(x) (n 3 n)v (X) b (α,β,γ) (X), (p,q,r) P i (pα,qβ,rγ) (X)+ (p,q,r) P b (pα,qβ,rγ) (X)+ =(n 3 n)(8v (X) 1V (X)+6V (X) V (X)) (p,q,r) P 1 i (pα,qβ,rγ) (X) (p,q,r) P 1 b (pα,qβ,rγ) (X) =(i (n,n,n) (X) ni (,,) (X)) + b (n,n,n)(x) nb (,,) (X) (i (pn,qn,rn) (X) ni (p,q,r) (X)) + 1 (b (pn,qn,rn)(x) nb (p,q,r) (X)) (p,q,r) P + (i (pn,qn,rn) (X) ni (p,q,r) (X)) + 1 (b (pn,qn,rn)(x) nb (p,q,r) (X)) (p,q,r) P 1 (i (n,n,n) (X) ni (1,1,1) (X)) + b (n,n,n)(x) nb (1,1,1) (X) =(i (n,n,n) (X) ni (1,1,1) (X)) + b (n,n,n) (X) nb (1,1,1) (X) 30
8 3 38 ( ). X x = a, y = b, z = c (a, b, c ) X k k 1/n x = a/n, y = b/n, z = c/n (a, b, c ) X n 40 9 z =1 6 6 40: 39 ( ). X X (n 3 n)v (X) = t=0 t (N t n(x) nn t (X)) =(N 0 n(x) nn 0 (X)) + N 1 n(x) nn 1 (X) N 3 n(x) nn 3 (X) (N 4 n(x) nn 4 (X)) 3(N 5 n(x) nn 5 (X)). 31
N t (X) =(X t ), N t n(x) =(X t n ) 40. 41 X N (X) =, N 4 (X) = 1, N t (X) = 0(t, 4) 41 L 3 N 0 (X) =, N 1 (X) =6, N (X) =8, N 4 (X) =1, N t (X) =0(t 0, 1,, 4) ( 39) n = (8 )V (X) =( 0) + 6 0 V (X) =1 (1 1) =6, 41: 9 3 4 1 1 4 1 3
4 4 4 9.1 41 (). X R m m (m 1)m! m V (X) = j=0 ( 1) j ( m 1 j )( i m 1 j + b m 1 j ) ( +( 1) m 1 χ(x) χ( X) ) i n (X) =(X n ), b n (X) =(X n ) ( ) m 1 j X X χ(x) X 4 ( ). m = V (X) =i 1 + b 1 χ(x)+χ( X) X X χ(x) =( ) ()+() X χ( X)=( ) () h X χ(x) =1 h, χ( X)=0 33
V (X) =i 1 + b 1 1+h m =3 6V (X) = ( i + b ) ( i 1 + b ) ( 1 + χ(x) χ( X) ) X χ(x) =( ) ()+() ( ) =1, χ( X)=( ) ()+() = 6V (X) =(i i 1 )+ b b 1 9. 3 43 ( ). X R m (m 1)m! m V (X) = j=0 ( 1) j ( m 1 j )( i m 1 j + b m 1 j ) i n(x) =(X n ), b n(x) =(X n ) 9.3 3 44 ( ). X R m 3 k k n 34
45 ( ). X R m (m 1)m! m V (X) = j=0 j=0 ( 1) j ( m 1 j ) t 0 t N t m 1 j m ( )( m 1 = ( 1) j Nm 1 j 0 + 1 j N m i j 1 1 ) N m i j 3 Nm i j 4 N t n(x) =(X t n ) 10 (m, n) x m y n 3 3 ( ) 3 3 3 100% 6 1899 [1] 1969 Mathematical Snapshots 1963 1969 [1] G. A. Pick. Geometrisches zur Zahlenlehre. Lotos, Naturwissen Zeitschrift, 47:311 319, 1899. 35
[] [] M., S...,, 010. [3] [4] [4] 7 [3] R. J...,, 001. [4],..,, 011. [5] [5] Dale E. Varberg. Pick s theorem revisited. Amer. Math. Monthly, 9(8):584 587, 1985. http://www.bun-eido.co.jp/t_math/sjournal/ sj30/sj30.html [6] [6].. SIGMA JOURNAL, 30, 005. http://www.bun-eido. co.jp/t_math/sjournal/sj30/sj30194.pdf, http://www10.plala.or.jp/h-nukaga/ math/syoumei.htm [7] 3 http://surgery.matrix.jp/math/research.html [8] [7].. 5, 1993. http://www10. plala.or.jp/h-nukaga/math/nukaga001.pdf [8],. On Nukaga s theorem. (A part of) Senior Thesis, Okayama University of Science, Feb. 009. http://surgery.matrix.jp/math/ridai/008nukaga. pdf [4] [9] J. E. Reeve. On the volume of lattice polyhedra. Proc. London Math. Soc. (3), 7:378 395, 1957. [10] J. E. Reeve. A further note on the volume of lattice polyhedra. J. London Math. Soc., 34:57 6, 1959. [11] [11] I. G. Macdonald. The volume of a lattice polyhedron. Proc. Cambridge Philos. Soc., 59:719 76, 1963. 36