(, Goo Ishikawa, Go-o Ishikawa) ( ) 1
( ) ( ) ( ) G7( ) ( ) ( ) () ( ) BD = 1 DC CE EA AF FB 0 0 BD DC CE EA AF FB =1 ( ) 2
(geometry) ( ) ( ) 3
(?) (Topology) ( ) DNA ( ) 4
( ) ( ) 5
( ) H. 1 : 1+ 5 2 1, 1, 2, 3, 5, 8, 13,... 0 0 π ( ) 6
geometry geo ( ) ( ) ( ) =() 1 1 ( 1) ( 1) = 1 () 7
8
( ) () ( ) ( ) ( ) 9
( ) ABC AB < AC + CB 10
( ) ABC AB < AC + CB ( ) 180 90 R right angle, P Q P Q P Q P Q P Q P Q ( ) 11
( ) () ( ) 180 180 0 0 0 =0 0 12
( ) ( ) geo geometry geo ( J P ) ( ) 13
( ) ( ) 180 collinear R AC CB <AB A, B, C AB < AC + CB 14
AC CB <AB AC CB < AB (AC CB) <AB AC < AB + CB CB < AB + AC A, B, C AB < AC + CB 3.14 10 10 ( ) ( )1.1 0.1 1.11 1.11111 < 1.2 1.2 15
(hypothesis) (conjecture) () ( ) ( ) ( ) ( ) 16
ABC A, B, C l BC,CA, AB D, E, F BD DC CE EA AF FB = 1 1: P, Q, R AB, BC, CA ( ) CD ( ) P, Q, R AB, BC, CA AB, BC, CA 17
ABC A, B, C P A, B, C D, E, F BD DC CE EA AF FB =1 ( ) ( ) ABC DEF AD, BE, CF BC EF P CA FD Q AB DE R P, Q, R A D B E C F A, C, E m B,D,F l AB DE P BC EF Q CD FA R P, Q, R AF DE ( ) P Q ( ) Q P P (Q ) ( ) (Q ) (P ) 18
( ) P Q P ( ) Q ( ) ( ) ABC A B C AB = A B,AC = A C, ABC = A B C P ABC A B C Q AB = A B,AC = A C, ABC = A B C P Q Q P AB = A B,AC = A C, ABC = A B C ( ) () A A B A A B A B ( ) 19
ABC AB < AC + CB ABC AB < AC + CB π π e πi = 1 ( ) ( ) ( 180 ) (a 360 ) a () ( 180 ) (a 360 ) () 20
R n R n ( ) 3 2 = 6 ( ) =( 4 ) ( ) =1+ 1 2,( ) =1+2 4 1 2 + 1 3 = 2 5 ( ) ( ) ( ) 21
( ) ( ) ( ) ( ) 22
( ) 23
( ) (x 2 + y 2 z 2 =0 ) ( ) () A, B, C, D, E, F AB DE P BC EF Q CD FA R P, Q, R 0 ( ) Pa( ) ([Pa] = [N/m 2 ]). 24
( = ) = ( ) + + + ( ) 3 3 3 3 3 3 3 3 25
p p p p 150 ( ) =( ) ABCD AB CD + AD BC = AC BD A, B, C, D AB CD + AD BC = AC BD ABCD sin t, cos t ( ) (e πi = 1) ( ) () 4πr 2 4 3 πr3 r 4πr 2 θ 2πr cos θ S =2 π 2 2πr cos θrdθ =2 π 2 0 0 2πr2 cos θdθ =4πr 2. π 687 1947 2 x 2 2=0 π e 26
π 10 = lim n ( n ) (x, y, z t x 2 + y 2 + z 2 + t 2 = r 2 ) 1 3=1 0.3333 3=0.9999 1 3 0.9999 =1 0.9999 0.9, 0.99, 0.999, 0.9999, 0.99999,... 1 1 2 + 1 3 = 2 + 5 = A A 1 2 1 A 5 A 1 2 1 5 1 2 A 1 2 1 2 1 5 1 A 5 1 2 1 5 1 5 1 2 1 2 27
1 2 A A A ( ) 15, 16C ( ) () ( ) 28
( ) 2001 () 29
30
R 3 ( ) l m l, m ( ) ( ) xyz R 3 ( ) () R 3 RP 2 R 3 31
RP 2 R P R real P projective 2 ( ) ( ) ( ) ( ) 1812 32
X ( ) Y ( ) X Y X ( ) x Y ( ) y y = f(x) ( ) CAD ( ) CAD computer aided design 33
L =2πr π r ε ( ) (M. ) ( ) 1, 2, 3,... 34
() () () 1, 2, 3 ( ) ( ) ( ) 26 1 0 1 0= 1 0 0 1 1 0= 0 =1 0 1 =0 0 35
( ) ( ) NHK ( ) ( ) 36
( ) () ( ) ( ) R 3 Π (R 3 ) Π ( ) R 3 ( ) l m P l m l Q QP m R Q R () 0 l m 37
( ) ( ) () ( ) l m A l B m B A A B B l m B B P l m P BB A A B B ( ) l m l m l m 38
I, II, III, IV ( ) T T (x + x )= T (x)+t(x ),T(cx) =ct (x) ( ) ( ) ( ) ( x 2 + y 2 =0) ( ) 39
() TEX Povray = (x, y, z, t) C 2 n n ( ( ) ) 0 =1 0 1=0 0 =1 0 (0 )=(0 0) =0 =1 0 1=0 40
n n n n n n n n π 1 2 Mac Cayley ( ) (?) ( ) ( ) http//www.math.sci.hokudai.ac.jp/ 41
() (a 0 : a 1 : a 2 ) a 0 x + a 2 y + a 2 z =0 (a 0 : a 1 : a 2 ) (a 0,a 1,a 2 ) a 0 x + a 2 y + a 2 z =0 ( ) ( ) ( ) P P 42
P P P P ( ) RP 2 CP 1 RP 1 RP 2 CP 0 CP 1 ( ) ( ) () 43
( ) n r n r 1 P l m l Q PQ m n ( ) 44
( ) CB FE ABC DEF P, Q, R ( ) ( ) R 4 R 3 ( ) 45
x + iy x y C 2 x + iy, u + iv x, u y, v 100 100 n n e e =2.718 (e x ) = e x e e e = 1+1+ 1 2 + 1 6 + 1 + 24 0 ( ) ( ) ( ) ( ) 46
sin 270 = 1 270 ( ) () ( ) () ( ) Povray Shade Shade ( ) ( ) 47
( ) (?) ( ) 48
CM ( ) ( ) ( ) ax 2 + by 2 + cxy + ex + fy + d =0 ( ) ( ) (ax + by + c)(a x + b y + c )=0 49
( ) C C 2 ( CP 2 ) a =0,b=0,c=0 y = Ax + B ( ) x 2 + y2 4 =1 ax 2 + by 2 + cxy + ex + fy+ d =0 a, b, c, e, f, d 1, 1, 0, 0, 0, 1 4 n n (range, ) P, Q, R l, m (1) P (2) l (3) Q (4) 50
m (5) R OO OO O O O ( ) O ( ) O O OO OO l, m, n P l,m,n Q l l a, n m b m l c n n d m m e l n f ab, cd, ef x 2 + y 2 z 2 =0 (x, y) (ax + by + c, a x + b y + c,a x + b y + c ) xy (ax + by + c) 2 +(a x + b y + c ) 2 (a x + b y + c ) 2 =0 ( ) xy x 2 =0 x 2 + y 2 = 3 x, y ax 2 +by 2 +cxy +ex+fy+d =0 a >0.b > 0,c= e = f =0,d>0 51
l m l m l m?! ( ) ( )!! n n n 1 R n n 1 RP n 1 (2 3 4 ) R n (R n ) R n (R n ) 0 R n (n 1) () 52
sin, cos, tan N N x 2 + y 2 =4 x 2 4x + y 2 4y =16 (x 2 + y 2 4) + k(x 2 4x + y 2 4y 16) = 0 k a + bi + cj + dk (a, b, c, d ) i, j, k i 2 = 1,j 2 = 1,k 2 = 1,ij = ji = k, jk = kj = i, ki = ik = j Hamilton CG CAD (isometry) (Euler) (Gauss) () 53
1 25 1 5 54
( ) () () 55
T V Shade3D-CG Mac Shadepersonal Pro 3D Shade CG TV CM () ( ) 56
( ) ( ) ( ) ( ) t x(t),y(t) (x(t),y(t)) x(t) =a sin t, y(t) =a cos t (x (t),y (t)) (0, 0) ( ) (x(t),y(t)),a t b, (x(a),y(a)) =(x(b),y(b)), (x (t),y (t)) (0, 0) θ(t) (cos θ(t)m sin θ(t)) = (x (t),y (t)) (x (t),y (t)) 1 b 2π a θ (t)dt 0, ±1, ±2, ±3,..., ( ) ( ) ( ) (u, v) (x(u, v),y(u, v),z(u, v)) (3 2 ) 1 0 ( ) 57
( ) ( ) R 3 ax + by + cz =0 z =1 z =1 ax + by + cz =0 ax + by c =0 ax 2 + by 2 + cxy + dx + ey + f =0 ax 2 + by 2 + cxy + dxz + eyz + fz 2 =0 ( ) () ( ) 58
( ) y = x 2 xy y (y ) ( ) ( ) y = tan x y = z tan x z x 2 +y 2 = 1 x, y x = i, y =0 x 2 +y 2 = 1 curve () line () (streight line) (curved line) 59
( n ) ( ) ( ) ( ) ( ) () 60
(3+3) 2=12 13 ( (3+3+1) 2 =14 ) ( ) () ( ) CG 3D 1, 2,... 61
() ( ) (break through) CG ( ) 62
e e iπ = 1 [x] x [0.9999 ] CUBE No.1 (? ) () 100 100 ( ) () 63
( ( ) ) ( ) 64