Bianca Violet - Surfer Gallery
Bu resimler SURFER’da üretildi. «Spektrum der Wissenschft» dergisinde SURFER haberini okuyup 2008’de SURFER yarışmasına katılarak bu muhteşem araçla yolculuğuma başladım.
Formül
- $0=x^{3}yz+x^{2}z^{2}y+5y^{3}z+5yx$
Honey Bee
SURFER image properties:
brightness = 0, glow = 100
This picture won the SURFR competition in Kassel, hosted by ‹Spektrum der Wissenschaft› und ‹Kasseler Sparkasse› within the category ‹Users outside of Kassel›.
Formül
- ((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.
Formül
- ((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
Formül
- $0=(x^{2}+y^{2}+z^{2}-5) \cdot (x^{2}+y^{2}+z^{2}-10) \cdot (x^{2}+y^{2}+z^{2}-15) \cdot (x^{2}+y^{2}+z^{2}-20) \cdot (x^{2}+y^{2}+z^{2}-25) \cdot (x^{2}-y^{2}+z^{2}- (a-0.5)) \cdot (xyz)^{2} \cdot (x+y) \cdot (x+z) \cdot (y+z) \cdot (x-y) \cdot (x-z) \cdot (y-z) -1000$
- $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 image properties:
a = 0.47, three surfaces colored red, yellow and blue, transparency = 60%, reflection = 100, various colorful lights :-)
Triple Winglet
Formül
- $0=x^{3}-y^{3}-x^{2}z-y^{2}z-0.01$
- 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
Formül
- 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
Formül
- ((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
Blue Planet
Formül
- 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
Roma şekerleri
Reel projektif düzlem, reel üç boyutlu uzayda (R3’te) orijinden geçen tüm doğruların birer nokta olarak temsil edildiği soyut bir topolojik uzaydır. İsviçreli matematikçi Jakob Steiner Roma’da yaşadığı günlerde, reel projektif düzlemden R3’e bir fonksiyon oluşturmayı düşündü. Oluşturduğu yüzey kendisiyle kesişir. Bu yüzeye, Roma Yüzeyi veya Steiner Yüzeyi denir.
Orijinde yüzeyin üçlü kesişim noktası vardır ve üç koordinat düzlemi de yüzeye o orijinde teğettir. Koordinat eksenlerinin orijin dışındaki noktaları yüzeyin ikili kesişim noktalarıdır ve bu eksenlerin 6 ucunda yüzey büzülerek biter.
Bu resimdeki sarı yüzey bir Roma Yüzeyi. Yüzeyin çeşitli yerlerinden kesilmiş altı kısım, yüzeye büzülme noktalarında teğet olarak yerleştirilmiş. Böylece yüzeyin yüksek simetrisini de vurgulanıyor.
Formül
- 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
Formül
- $0=(x^{2}+y^{2}+z^{2}-5b) \cdot (x^{2}+y^{2}+z^{2}-10b) \cdot (x^{2}+y^{2}+z^{2}-15b) \cdot (x^{2}+y^{2}+z^{2}-20b) \cdot (x^{2}+y^{2}+z^{2}-25b) \cdot (x^{2}+y^{2}-z^{2} + 0.5)) \cdot (x^{2}-y^{2}+z^{2} + 0.5)) \cdot (-x^{2}+y^{2}+z^{2} + 0.5)) -100$
Soap Bubble
SURFER image properties:
b = 0.38, transparency = 90%, colorful lights
Boy'dan Bir Dilim
Hem Möbius şeridinin hem de dairenin kenarı birer çember. Bu iki yüzeyi birbirine kenarından diktiğinizi düşünün! Sonucunda çıkan kapalı (kenarı olmayan) yüzey hala Möbius şeridini içeriyor ve bu yüzden de bir yönü olmayacak (yani neyin içeride neyin dışarıda olduğunu söyleyemeyeceğiz, iç-dış aynı olacak). İşte bu yüzden bu yüzeyi bizim üç boyutlu uzayımıza yerleştirdiğimizde, kendi kendini kesmek zorunda. Peki bu yerleştirme, yüzeyin hiçbir sivriliği tekillği olmadan pürüzsüz, düzgün bir biçimde de yapılabilir mi?
Ünlü Alman matematikçi David Hilbert öğrencisi Werner Boy’u, böyle bir şeyin imkansız olduğunu kanıtlaması için görevlendirdi. Boy, hocasını şaşırtarak, böyle pürüzsüz bir yüzeyi 1901 yılında inşa etti. Bu yüzeyi, birbirine paralel bir düzlemler ailesiyle kesişimlerine karşılık gelen her bir eğriyi göstererek tarif etti.
Bu resimde, düzlemler tarafından değil ama aynı merkeze ve biraz farklı yarıçaplara sahip iki küre tarafından kesilen iki tane Boy yüzeyinin dilimini görüyorsunuz (biri diğerinin ayna simetrisi).
Formül
- $0=(-x^{3}-y^{2}z-z^{3}+x^{2}) \cdot (x^{3}+y^{2}z+z^{3}+x^{2})^{2}-x^{2}y^{2}z^{2}$
Yin Yang
Formül
- 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
