Bianca Violet - Surfer Gallery
갤러리
Bianca Violet - Surfer Gallery
이 이미지들은 Surfer를 이용해서 만들어졌습니다. 저는 ‘Spektrum der Wissenschft’에서 이 멋진 도구에 대해 처음 알고 사용해보기 시작했고, 2008년 Surfer 대회에 참가하였습니다.
공식
- $0=x^{3}yz+x^{2}z^{2}y+5y^{3}z+5yx$
Honey Bee
SURFER 이미지 설정:
밝기 = 0, 휘도(輝度, 빛나는 정도) = 100
이 그림은 카셀(Kassel, 독일의 한 도시)에서 열린 외부인 사용자 대상 SURFER 대회에서 수상하였습니다.
저작권 CC BY-NC-ND-3.0
공식
- ((x-3.5c)^2 \cdot (y+3.5c)^2 + (y+3.5c)^2 z^2+ (x-3.5c)^2 z^2 + (x-3.5c) \cdot (y+3.5c) z) \cdot ((x+3.5c)^2 \cdot (y-3.5c)^2 + (y-3.5c)^2 z^2 + (x+3.5c)^2 z^2 + (x+3.5c) \cdot (y-3.5c) z) \cdot ((x+3.5c)^2 \cdot (y+3.5c)^2 + (y+3.5c)^2 z^2+ (x+3.5c)^2 z^2 + (x+3.5c) \cdot (y+3.5c)z) \cdot ((x-3.5c)^2 \cdot (y-3.5c)^2 + (y-3.5c)^2 z^2 + (x-3.5c)^2 z^2 + (x-3.5c) \cdot (y-3.5c) z)
- (x^2 (y+3.5c)^2 + (y+3.5c)^2 (z+3.5c)^2 + x^2 (z+3.5c)^2 + x (y+3.5c) \cdot (z+3.5c)) \cdot (x^2 (y+3.5c)^2 + (y+3.5c)^2 \cdot (z-3.5c)^2 + x^2 (z-3.5c)^2 + x (y+3.5c) \cdot (z-3.5c)) \cdot (x^2 (y-3.5c)^2 + (y-3.5c)^2 \cdot (z+3.5c)^2 + x^2 (z+3.5c)^2 + x (y-3.5c) \cdot (z+3.5c)) \cdot (x^2 (y-3.5c)^2 + (y-3.5*c)^2 \cdot (z-3.5c)^2 + x^2 (z-3.5c)^2 + x (y-3.5c) \cdot (z-3.5c))
- ((x-3.5c)^2 y^2 + y^2 (z+3.5c)^2 + (x-3.5c)^2 \cdot (z+3.5c)^2 + (x-3.5c) y (z+3.5c)) \cdot ((x+3.5c)^2 y^2 + y^2 (z-3.5c)^2 + (x+3.5c)^2 \cdot (z-3.5c)^2 + (x+3.5c) y (z-3.5c)) \cdot ((x+3.5c)^2 y^2 + y^2 (z+3.5c)^2 + (x+3.5c)^2 \cdot (z+3.5c)^2 + (x+3.5c) y (z+3.5c)) \cdot ((x-3.5c)^2 y^2 + y^2 (z-3.5c)^2 + (x-3.5c)^2 \cdot (z-3.5c)^2 + (x-3.5c) y (z-3.5c))
- (x^2 y^2 + y^2 (z+5c)^2 + x^2 (z+5c)^2 + xy (z+5c)) \cdot (x^2 y^2 + y^2 (z-5c)^2 + x^2 (z-5c)^2 + xy (z-5c))
- (x^2 (y+5c)^2 + (y+5c)^2 z^2+ x^2 z^2 + x (y+5c) z) \cdot (x^2 (y-5c)^2 + (y-5c)^2 z^2 + x^2 z^2 + x (y-5c) z)
- ((x+5c)^2 y^2 + y^2 z^2 + (x+5c)^2 z^2 + (x+5c) yz) \cdot ((x-5c)^2 y^2 + y^2 z^2 + (x-5c)^2 z^2 + (x-5c) yz)
- c = 0.11
Clinch
The formula: x² y² + y² z² + x² z² + xyz = 0
describes a shape, which was discovered by Jakob Steiner. While he stayed in Rome in 1844, he studied its geometrical properties - hence it is called Roman surface. However, it was his friend Karl Weierstraß, who first published a paper on the surface and Steiner’s results in 1863, the year of Steiner’s death.
In the picture there are 18 Roman surfaces positioned symmetrically around the origin, each surface only slightly translated off the origin along one or two of the coordinate axes, so they intersect each other. Additionally, only a small portion of the whole arrangement is shown in this picture: everything within a small spherical neighborhood of the origin. Everything outside is simply cut off.
There are four blue ones, four red ones and four yellow ones, each color within one of the three coordinate planes. Their centers lie on the diagonal lines x= +/- y, y = +/- z, and z = +/- x respectively. The centers of the two orange, two green and two purple ones lie on the vertices of a cube, each color on one diagonal of the cube, matching the three coordinate axes.
Even thogh the Roman surface is non-orientable, which means there is no outside or inside - just one side, the SURFER program assigns two colors to it. Here the first color of each surface is picked as described above, and all surfaces have the same second color, which is black. Black is also the background color.
저작권 CC BY-NC-ND-3.0
공식
- ((x-3.5c)^2 y^2 + y^2 (z+3.5c)^2 + (x-3.5c)^2 \cdot (z+3.5c)^2 + (x-3.5c) y (z+3.5c)) \cdot ((x+3.5c)^2 y^2 + y^2 (z-3.5c)^2 + (x+3.5c)^2 \cdot (z-3.5c)^2 + (x+3.5c) y (z-3.5c)) \cdot ((x+3.5c)^2 y^2 + y^2 (z+3.5c)^2 + (x+3.5c)^2 \cdot (z+3.5c)^2 + (x+3.5c) y (z+3.5c)) \cdot ((x-3.5c)^2 y^2 + y^2 (z-3.5c)^2 + (x-3.5c)^2 \cdot (z-3.5c)^2 + (x-3.5c) y (z-3.5c))
- (x^2 (y+3.5c)^2 + (y+3.5c)^2 (z+3.5c)^2 + x^2 (z+3.5c)^2 + x (y+3.5c) \cdot (z+3.5c)) \cdot (x^2 (y+3.5c)^2 + (y+3.5c)^2 \cdot (z-3.5c)^2 + x^2 (z-3.5c)^2 + x (y+3.5c) \cdot (z-3.5c))
- (x^2 (y-3.5c)^2 + (y-3.5c)^2 \cdot (z+3.5c)^2 + x^2 (z+3.5c)^2 + x (y-3.5c) \cdot (z+3.5c)) \cdot (x^2 (y-3.5c)^2 + (y-3.5*c)^2 \cdot (z-3.5c)^2 + x^2 (z-3.5c)^2 + x (y-3.5c) \cdot (z-3.5c))
- ((x+5c)^2 y^2 + y^2 z^2 + (x+5c)^2 z^2 + (x+5c) yz) \cdot ((x-5c)^2 y^2 + y^2 z^2 + (x-5c)^2 z^2 + (x-5c) yz)
- (x^2 (y+5c)^2 + (y+5c)^2 z^2+ x^2 z^2 + x (y+5c) z) \cdot (x^2 (y-5c)^2 + (y-5c)^2 z^2 + x^2 z^2 + x (y-5c) z)
- (x^2 y^2 + y^2 (z+5c)^2 + x^2 (z+5c)^2 + xy (z+5c)) \cdot (x^2 y^2 + y^2 (z-5c)^2 + x^2 (z-5c)^2 + xy (z-5c))
- ((x+3.5c)^2 \cdot (y-3.5c)^2 + (y-3.5c)^2 z^2 + (x+3.5c)^2 z^2 + (x+3.5c) \cdot (y-3.5c) z) \cdot ((x+3.5c)^2 \cdot (y+3.5c)^2 + (y+3.5c)^2 z^2+ (x+3.5c)^2 z^2 + (x+3.5c)
Diamond
저작권 CC BY-NC-ND-3.0
공식
- $0=(x^{2}+y^{2}+z^{2}-5) \cdot (x^{2}+y^{2}+z^{2}-10) \cdot (x^{2}+y^{2}+z^{2}-15) \cdot (x^{2}+y^{2}+z^{2}-20) \cdot (x^{2}+y^{2}+z^{2}-25) \cdot (x^{2}-y^{2}+z^{2}- (a-0.5)) \cdot (xyz)^{2} \cdot (x+y) \cdot (x+z) \cdot (y+z) \cdot (x-y) \cdot (x-z) \cdot (y-z) -1000$
- $0=(x^{2}+y^{2}+z^{2}-5) \cdot (x^{2}+y^{2}+z^{2}-10) \cdot (x^{2}+y^{2}+z^{2}-15) \cdot (x^{2}+y^{2}+z^{2}-20) \cdot (x^{2}+y^{2}+z^{2}-25) \cdot (x^{2}+y^{2}-z^{2}- (a-0.5)) \cdot (xyz)^{2} \cdot (x+y) \cdot (x+z) \cdot (y+z) \cdot (x-y) \cdot (x-z) \cdot (y-z) -1000 $
- $0=(x^{2}+y^{2}+z^{2}-5) \cdot (x^{2}+y^{2}+z^{2}-10) \cdot (x^{2}+y^{2}+z^{2}-15) \cdot (x^{2}+y^{2}+z^{2}-20) \cdot (x^{2}+y^{2}+z^{2}-25) \cdot (-x^{2}+y^{2}+z^{2}- (a-0.5)) \cdot (xyz)^{2} \cdot (x+y) \cdot (x+z) \cdot (y+z) \cdot (x-y) \cdot (x-z) \cdot (y-z) -1000$
Mirror Ball
SURFER 이미지 설정:
a = 0.47, 세 곡면의 색상: 빨강, 노랑, 파랑, 투명도 = 60%, 반사 = 100, 다양한 색의 빛 추가 :-)
저작권 CC BY-NC-ND-3.0
Triple Winglet
저작권 CC BY-NC-ND-3.0
공식
- $0=x^{3}-y^{3}-x^{2}z-y^{2}z-0.01$
- 0=64(1-z)^3z^3-48(1-z)^2z^2(3x^2+3y^2+2z^2) +12(1-z)z(27(x^2+y^2)^2-24z^2 (x^2+y^2) +36 \cdot 1.4142yz(y^2-3x^2)+4z^4) + (9x^2+9y^2-2z^2) \cdot (-81 (x^2+y^2)^2-72z^2 (x^2+y^2) +108 \cdot 1.4142xz (x^2-3y^2))
- 0=x^2+y^2+ (z+0.23)^2 -0.07
- 0=((x+b)^2 +y^2+ (z+0.26)^2 -0.01) \cdot ((x+a)^2 + (y+c)^2 + (z+0.26)^2-0.01)
- 0=((x+b)^2 +y^2+ (z+0.26)^2 -0.005) \cdot ((x+a)^2+ (y+c)^2 + (z+0.26) ^2-0.005)
- a=0.09 \,\, b=0.2 \,\, c=-0.18 \,\,
Fairy
저작권 CC BY-NC-ND-3.0
공식
- x^2+y^2+z^2-3b
- ( (x+c)^2 \cdot (y-c)^2 + (x+c)^2 \cdot (z+c)^2 + (y-c) ^2 \cdot (z+c)^2 -3a (x+c) \cdot (y-c) \cdot (z+c)) \cdot ( (x-c) ^2 \cdot (y+c)^2 + (x-c) ^2 \cdot (z-c)^2 + (y+c)^2 \cdot (z-c)^2 -3a (x-c) \cdot (y+c) \cdot (z-c))
- ((x+c)^2 \cdot (y+c)^2 + (x+c)^2 \cdot (z-c)^2 + (y+c)^2 \cdot (z-c)^2 + 3a(x+c) \cdot (y+c) \cdot (z-c)) \cdot ((x-c)^2 \cdot (y-c)^2 + (x-c)^2 \cdot (z+c)^2 + (y-c)^2 \cdot (z+c)^2 +3a(x-c) \cdot (y-c) \cdot (z+c))
- ((x-c) ^2 \cdot (y+c)^2 + (x-c)^2 \cdot (z+c)^2 + (y+c)^2 \cdot (z+c)^2 -3a(x-c) \cdot (y+c) \cdot (z+c)) \cdot ((x+c)^2 \cdot (y-c)^2 + (x+c)^2 \cdot (z-c)^2 + (y-c)^2 \cdot (z-c)^2 -3a(x+c) \cdot (y-c) \cdot (z-c))
- ((x+c)^2 \cdot (y+c)^2 + (x+c)^2 \cdot (z-c)^2 + (y+c)^2 \cdot (z-c)^2 -3a(x+c) \cdot (y+c) \cdot (z-c)) \cdot ( (x-c)^2 \cdot (y-c)^2 + (x-c)^2 \cdot (z+c)^2 + (y-c)^2 \cdot (z+c)^2 -3a(x-c) \cdot (y-c) \cdot (z+c))
- ((x-c)^2 \cdot (y+c)^2 + (x-c)^2) \cdot (z+c)^2 + (y+c)^2 \cdot (z+c)^2 +3a(x-c) \cdot (y+c) \cdot (z+c)) \cdot ((x+c)^2 \cdot (y-c)^2 + (x+c)^2 \cdot (z-c)^2 + (y-c)^2 \cdot (z-c)^2 +3a(x+c) \cdot (y-c) \cdot (z-c)) \cdot ( (x+c)^2 \cdot (y-c)^2 + (x+c)^2 \cdot (z+c)^2 + (y-c)^2 \cdot (z+c)^2 +3a(x+c) \cdot (y-c) \cdot (z+c)) \cdot ((x-c)^2 \cdot (y+c)^2 + (x-c)^2 \cdot (z-c)^2 + (y+c)^2 \cdot (z-c)^2 +3a(x-c) \cdot (y+c) \cdot (z-c))
- a=0.65; \,\, b=0.45; \,\, c=0.9
Flower
저작권 CC BY-NC-ND-3.0
공식
- ((x-3.5c)^2 \cdot (y+3.5c)^2 + (y+3.5c)^2 z^2+ (x-3.5c)^2 z^2 + (x-3.5c) \cdot (y+3.5c) z) \cdot ((x+3.5c)^2 \cdot (y-3.5c)^2 + (y-3.5c)^2 z^2 + (x+3.5c)^2 z^2 + (x+3.5c) \cdot (y-3.5c) z) \cdot ((x+3.5c)^2 \cdot (y+3.5c)^2 + (y+3.5c)^2 z^2+ (x+3.5c)^2 z^2 + (x+3.5c) \cdot (y+3.5c)z) \cdot ((x-3.5c)^2 \cdot (y-3.5c)^2 + (y-3.5c)^2 z^2 + (x-3.5c)^2 z^2 + (x-3.5c) \cdot (y-3.5c) z)
- (x^2 (y+3.5c)^2 + (y+3.5c)^2 (z+3.5c)^2 + x^2 (z+3.5c)^2 + x (y+3.5c) \cdot (z+3.5c)) \cdot (x^2 (y+3.5c)^2 + (y+3.5c)^2 \cdot (z-3.5c)^2 + x^2 (z-3.5c)^2 + x (y+3.5c) \cdot (z-3.5c)) \cdot (x^2 (y-3.5c)^2 + (y-3.5c)^2 \cdot (z+3.5c)^2 + x^2 (z+3.5c)^2 + x (y-3.5c) \cdot (z+3.5c)) \cdot (x^2 (y-3.5c)^2 + (y-3.5*c)^2 \cdot (z-3.5c)^2 + x^2 (z-3.5c)^2 + x (y-3.5c) \cdot (z-3.5c))
- ((x-3.5c)^2 y^2 + y^2 (z+3.5c)^2 + (x-3.5c)^2 \cdot (z+3.5c)^2 + (x-3.5c) y (z+3.5c)) \cdot ((x+3.5c)^2 y^2 + y^2 (z-3.5c)^2 + (x+3.5c)^2 \cdot (z-3.5c)^2 + (x+3.5c) y (z-3.5c)) \cdot ((x+3.5c)^2 y^2 + y^2 (z+3.5c)^2 + (x+3.5c)^2 \cdot (z+3.5c)^2 + (x+3.5c) y (z+3.5c)) \cdot ((x-3.5c)^2 y^2 + y^2 (z-3.5c)^2 + (x-3.5c)^2 \cdot (z-3.5c)^2 + (x-3.5c) y (z-3.5c))
- (x^2 y^2 + y^2 (z+5c)^2 + x^2 (z+5c)^2 + xy (z+5c)) \cdot (x^2 y^2 + y^2 (z-5c)^2 + x^2 (z-5c)^2 + xy (z-5c))
- (x^2 (y+5c)^2 + (y+5c)^2 z^2+ x^2 z^2 + x (y+5c) z) \cdot (x^2 (y-5c)^2 + (y-5c)^2 z^2 + x^2 z^2 + x (y-5c) z)
- ((x+5c)^2 y^2 + y^2 z^2 + (x+5c)^2 z^2 + (x+5c) yz) \cdot ((x-5c)^2 y^2 + y^2 z^2 + (x-5c)^2 z^2 + (x-5c) yz)
- c = 0.16
Package
저작권 CC BY-NC-ND-3.0
Blue Planet
저작권 CC BY-NC-ND-3.0
공식
- x^2+y^2+z^2-0.62
- ((x-3.5c)^2 \cdot (y+3.5c)^2 + (y+3.5c)^2 z^2+ (x-3.5c)^2 z^2 + (x-3.5c) \cdot (y+3.5c) z) \cdot ((x+3.5c)^2 \cdot (y-3.5c)^2 + (y-3.5c)^2 z^2 + (x+3.5c)^2 z^2 + (x+3.5c) \cdot (y-3.5c) z) \cdot ((x+3.5c)^2 \cdot (y+3.5c)^2 + (y+3.5c)^2 z^2+ (x+3.5c)^2 z^2 + (x+3.5c) \cdot (y+3.5c)z) \cdot ((x-3.5c)^2 \cdot (y-3.5c)^2 + (y-3.5c)^2 z^2 + (x-3.5c)^2 z^2 + (x-3.5c) \cdot (y-3.5c) z)
- (x^2 y^2 + y^2 (z+5c)^2 + x^2 (z+5c)^2 + xy (z+5c)) \cdot (x^2 y^2 + y^2 (z-5c)^2 + x^2 (z-5c)^2 + xy (z-5c))
- (x^2 (y+5c)^2 + (y+5c)^2 z^2+ x^2 z^2 + x (y+5c) z) \cdot (x^2 (y-5c)^2 + (y-5c)^2 z^2 + x^2 z^2 + x (y-5c) z)
- ((x+5c)^2 y^2 + y^2 z^2 + (x+5c)^2 z^2 + (x+5c) yz) \cdot ((x-5c)^2 y^2 + y^2 z^2 + (x-5c)^2 z^2 + (x-5c) yz)
- (x^2 (y+3.5c)^2 + (y+3.5c)^2 (z+3.5c)^2 + x^2 (z+3.5c)^2 + x (y+3.5c) \cdot (z+3.5c)) \cdot (x^2 (y+3.5c)^2 + (y+3.5c)^2 \cdot (z-3.5c)^2 + x^2 (z-3.5c)^2 + x (y+3.5c) \cdot (z-3.5c)) \cdot (x^2 (y-3.5c)^2 + (y-3.5c)^2 \cdot (z+3.5c)^2 + x^2 (z+3.5c)^2 + x (y-3.5c) \cdot (z+3.5c)) \cdot (x^2 (y-3.5c)^2 + (y-3.5*c)^2 \cdot (z-3.5c)^2 + x^2 (z-3.5c)^2 + x (y-3.5c) \cdot (z-3.5c))
- ((x-3.5c)^2 y^2 + y^2 (z+3.5c)^2 + (x-3.5c)^2 \cdot (z+3.5c)^2 + (x-3.5c) y (z+3.5c)) \cdot ((x+3.5c)^2 y^2 + y^2 (z-3.5c)^2 + (x+3.5c)^2 \cdot (z-3.5c)^2 + (x+3.5c) y (z-3.5c)) \cdot ((x+3.5c)^2 y^2 + y^2 (z+3.5c)^2 + (x+3.5c)^2 \cdot (z+3.5c)^2 + (x+3.5c) y (z+3.5c)) \cdot ((x-3.5c)^2 y^2 + y^2 (z-3.5c)^2 + (x-3.5c)^2 \cdot (z-3.5c)^2 + (x-3.5c) y (z-3.5c))
- c = 0.12
Red Planet
저작권 CC BY-NC-ND-3.0
Roman Candy
실사영평면(real projective plane)은 3차원 실수 공간 (R3)의 원점을 지나는 모든 직선들의 공간입니다. 수학자 Jakob Steiner가 로마에 체류할 때, 그는 실사영평면을 R3에 집어넣는 방법을 생각하였습니다. 그 결과물인 곡면은 자기교차하고, 이제는 로마 곡면(Roman Surface) 혹은 스타이너 곡면(Steiner Surface)라는 이름으로 불립니다.
곡면의 삼중점(triple point)인 원점에서 세 좌표평면은 곡면에 모두 접합니다. 원점을 제외하고 좌표축을 따라가는 선분들 위의 점은 모두 이중점(double point)입니다. 이 선분들은 각각 여섯 개의 뾰족점(pinch point)들에서 끝납니다.
여기서 노란색 로마 곡면은 다른 6개의 로마 곡면의 부분들에 둘러싸여 있고, 이들은 뾰족점에서 서로 만나게 됩니다. 이 그림은 로마 곡면의 고도의 대칭성을 보여줍니다.
저작권 CC BY-NC-ND-3.0
공식
- x^2+y^2+z^2-0.62
- ((x-3.5c)^2 \cdot (y+3.5c)^2 + (y+3.5c)^2 z^2+ (x-3.5c)^2 z^2 + (x-3.5c) \cdot (y+3.5c) z) \cdot ((x+3.5c)^2 \cdot (y-3.5c)^2 + (y-3.5c)^2 z^2 + (x+3.5c)^2 z^2 + (x+3.5c) \cdot (y-3.5c) z) \cdot ((x+3.5c)^2 \cdot (y+3.5c)^2 + (y+3.5c)^2 z^2+ (x+3.5c)^2 z^2 + (x+3.5c) \cdot (y+3.5c)z) \cdot ((x-3.5c)^2 \cdot (y-3.5c)^2 + (y-3.5c)^2 z^2 + (x-3.5c)^2 z^2 + (x-3.5c) \cdot (y-3.5c) z)
- (x^2 y^2 + y^2 (z+5c)^2 + x^2 (z+5c)^2 + xy (z+5c)) \cdot (x^2 y^2 + y^2 (z-5c)^2 + x^2 (z-5c)^2 + xy (z-5c))
- (x^2 (y+5c)^2 + (y+5c)^2 z^2+ x^2 z^2 + x (y+5c) z) \cdot (x^2 (y-5c)^2 + (y-5c)^2 z^2 + x^2 z^2 + x (y-5c) z)
- ((x+5c)^2 y^2 + y^2 z^2 + (x+5c)^2 z^2 + (x+5c) yz) \cdot ((x-5c)^2 y^2 + y^2 z^2 + (x-5c)^2 z^2 + (x-5c) yz)
- (x^2 (y+3.5c)^2 + (y+3.5c)^2 (z+3.5c)^2 + x^2 (z+3.5c)^2 + x (y+3.5c) \cdot (z+3.5c)) \cdot (x^2 (y+3.5c)^2 + (y+3.5c)^2 \cdot (z-3.5c)^2 + x^2 (z-3.5c)^2 + x (y+3.5c) \cdot (z-3.5c)) \cdot (x^2 (y-3.5c)^2 + (y-3.5c)^2 \cdot (z+3.5c)^2 + x^2 (z+3.5c)^2 + x (y-3.5c) \cdot (z+3.5c)) \cdot (x^2 (y-3.5c)^2 + (y-3.5*c)^2 \cdot (z-3.5c)^2 + x^2 (z-3.5c)^2 + x (y-3.5c) \cdot (z-3.5c))
- ((x-3.5c)^2 y^2 + y^2 (z+3.5c)^2 + (x-3.5c)^2 \cdot (z+3.5c)^2 + (x-3.5c) y (z+3.5c)) \cdot ((x+3.5c)^2 y^2 + y^2 (z-3.5c)^2 + (x+3.5c)^2 \cdot (z-3.5c)^2 + (x+3.5c) y (z-3.5c)) \cdot ((x+3.5c)^2 y^2 + y^2 (z+3.5c)^2 + (x+3.5c)^2 \cdot (z+3.5c)^2 + (x+3.5c) y (z+3.5c)) \cdot ((x-3.5c)^2 y^2 + y^2 (z-3.5c)^2 + (x-3.5c)^2 \cdot (z-3.5c)^2 + (x-3.5c) y (z-3.5c))
- c=0.15
Wheel of Time
저작권 CC BY-NC-ND-3.0
공식
- $0=(x^{2}+y^{2}+z^{2}-5b) \cdot (x^{2}+y^{2}+z^{2}-10b) \cdot (x^{2}+y^{2}+z^{2}-15b) \cdot (x^{2}+y^{2}+z^{2}-20b) \cdot (x^{2}+y^{2}+z^{2}-25b) \cdot (x^{2}+y^{2}-z^{2} + 0.5)) \cdot (x^{2}-y^{2}+z^{2} + 0.5)) \cdot (-x^{2}+y^{2}+z^{2} + 0.5)) -100$
Slice of Boy
하나의 닫힌 곡선으로 이루어져있는 뫼비우스 띠의 가장자리를 원판의 경계와 이어 붙이는 것을 상상해보세요! 결과적으로 얻게 되는 곡면은 여전히 뫼비우스 띠를 담고 있기 때문에 방향을 줄 수 없습니다. 다시 말하면, 곡면의 안쪽과 바깥면을 구분할 수 없다는 것입니다. 이로 인해, 우리가 살고 있는 3차원 공간에서는 이 곡면은 반드시 교차하는 부분은 갖게 됩니다. 그렇다면, 뾰족하거나 날카로운 테두리 없이 이 곡면을 만들 수 있을까요?
유명한 수학자 다비드 힐베르트(David Hilbert)는 그의 학생이었던 베르너 보이(Werner Boy)에게 뾰족하거나 날카로운 테두리 없이 이 곡면을 만드는 것이 불가능하다는 것을 증명하라고 시켰습니다. 하지만 보이는 1901년에 부드러운 곡면을 만들어내며 그의 스승을 놀라게 하였습니다. 보이는 평행한 평면과 이 곡면의 교선을 제시함으로써 이 곡면을 표현했습니다. 이 그림은 중심이 같고 반지름이 조금 다른 두 구면에 의해 잘려진 보이 곡면(Boy surface)의 단면을 보여주고 있습니다.
저작권 CC BY-NC-ND-3.0
공식
- $0=(-x^{3}-y^{2}z-z^{3}+x^{2}) \cdot (x^{3}+y^{2}z+z^{3}+x^{2})^{2}-x^{2}y^{2}z^{2}$
Yin Yang
저작권 CC BY-NC-ND-3.0
공식
- 0=(((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} +2(y+c) -1) \cdot (((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} -2(y+c) -1)^{2}-8(z+5a) ^{2}) +16(x+15b) \cdot (z+5a) \cdot ((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} -2(y+c) -1))^{2}
- $0=(((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2}+2(y+c) -1) \cdot (((x+15b)^{2} + (y+c) ^{2} + (z+5a)^{2} - 2(y+c) -1)^{2}-8(z+5a) ^{2}) +16(x+15b) \cdot (z+5a) \cdot ((x+15b) ^{2} + (y+c)^{2} + (z+5a)^{2}-2(y+c) -1) +d)^{2} \cdot (((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} +2(y+c) -1) \cdot (((x+15b) ^{2} + (y+c)^{2}+ (z+5a)^{2} -2(y+c) -1)^{2}-8(z+5a)^{2}) +16(x+15b) \cdot (z+5a) \cdot ((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} -2(y+c) -1) -d)$
- $0=(((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2}+2(y+c) -1) \cdot (((x+15b)^{2} + (y+c)^{2}+ (z+5a)^{2} - 2(y+c) -1)^{2} - 8(z+5a)^{2}) +16(x+15b) \cdot (z+5a) \cdot ((x+15b)^{2} + (y+c)^{2} + (z+5a) ^{2} - 2(y+c) -1) +2d)^{2} \cdot (((x+15b)^{2} + (y+c)^{2} + (z+5a) ^{2} + 2(y+c) -1) \cdot (((x+15b) ^{2} + (y+c)^{2} + (z+5a)^{2} - 2(y+c) -1)^{2} - 8(z+5a)^{2}) +16(x+15b) \cdot (z+5a) \cdot ((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} - 2(y+c) -1) -2d)$
- $0=(((x+15b)^{2} + (y+c)^{2} + (z+5a) ^{2} +2(y+c) -1) \cdot (((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} -2(y+c) -1)^{2} - 8(z+5a)^{2}) +16(x+15b) \cdot (z+5a) \cdot ((x+15b)^{2} + (y+c)^{2}+ (z+5a)^{2} - 2(y+c) -1) +3d)^{2} \cdot (((x+15b)^{2}+ (y+c)^{2} + (z+5a)^{2} +2(y+c) -1) \cdot (((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} - 2(y+c) -1)^{2} - 8(z+5a)^{2}) +16(x+15b) \cdot (z+5a) \cdot ((x+15b)^{2} + (y+c)^{2} + (z+5a) ^{2} - 2(y+c) -1) -3d)$
- $0=(((x+15b)^{2}+ (y+c)^{2}+ (z+5a)^{2} +2(y+c) -1) \cdot (((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} -2(y+c) -1)^{2}-8(z+5a)^{2}) +16(x+15b) \cdot (z+5a) \cdot ((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} - 2(y+c) -1) +4d)^{2} \cdot (((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} +2(y+c) -1) \cdot (((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} -2(y+c) -1)^{2} - 8(z+5a)^{2}) +16(x+15b) \cdot (z+5a) \cdot ((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} - 2(y+c) -1) -4d)$
- $0=(((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} +2(y+c) -1) \cdot (((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} - 2(y+c) -1)^{2} - 8(z+5a)^{2}) +16(x+15b) \cdot (z+5a) \cdot ((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} - 2(y+c) -1) +5d)^{2} \cdot (((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} +2(y+c) -1) \cdot (((x+15b)^{2} + (y+c)^{2} + (z+5a)^{2} - 2(y+c) -1)^{2} - 8(z+5a)^{2}) +16(x+15b) \cdot (z+5a) \cdot ((x+15b)^{2}+ (y+c)^{2} + (z+5a)^{2} - 2(y+c) -1) -5d)$
- a=b=0 \,\, c=0.47 \,\, d=2.5
Tunnel Vision
저작권 CC BY-NC-ND-3.0
Welcome
저작권 CC BY-NC-ND-3.0
