Mathematical structures both natural and man-made dazzle the eye
1、著名的分形
FAMOUS FRACTAL: This is a version of one of the best known fractals, the Julia set. Fractals in general are a compelling example of how abstract mathematical forms, generated by seemingly simple algorithms in which a pattern repeats on multiple scales, are capable of intricate beauty. Nature is full of such patterns. ---Tom Beddard
Julia set ，典型的分形版本之一，由法國數學家 Gaston Julia 和 Pierre Faton 在發展了複變函數迭代的基礎理論後獲得。而 B.B.Mandelbort 為最先提出分形概念者，他在1967年《科學》雜誌上發表《英國的海岸線有多長?》的著名論文，文中把部分與整體以某種方式相似的形體稱為分形(fractal)。
2、分形的泡泡
FRACTAL BUBBLES: Richard Taylor specializes in discovering the occurrence of fractals in the world. (He has argued that fractal geometry can tell a real Jackson Pollack painting from a knock-off.) He took this picture at the edge of the pond in Sydney, Australia. The bubble outlines have a fractal dimension of 1.3, which people tend to find the most aesthetically pleasing. Taylor argues that our eyes scan a scene using a fractal pattern with a similar dimension.
3、分形的花椰菜
FRACTAL BROCCOLI: John Ostrowick, responding to our Twitter call for examples of natural mathematical beauty, suggested Romanesco broccoli. This photo by Jon Sullivan was selected by Wikipedia users as one of the most spectacular on that site. ---Jon Sullivan
John Ostrowick 在推特上響應「讓大家去找自然數學之美實例」的提議，羅馬花椰菜就是例子。
4、雙螺旋線
DOUBLE SPIRAL: Paul Nylander maintains an incredible collection of mathematical art, along with the Mathematica code to recreate it. ---Paul Nylander
5、太空中的螺旋形
HEAVENLY SPIRALS: Spiral patterns occur throughout nature, perhaps most dramatically in spiral galaxies. This pair of galaxies has particularly unusual spiral patterns that are presumably the result of the gravitational tidal forces between them. The inner spiral arms of the upper galaxy (UGC 1810) are not planar, and the outer arm may have been pulled into a ring by a direct collision with the lower one (UGC 1813). ---NASA, ESA, and the Hubble Heritage Team
6、莫比烏斯三葉形謎題
MÖBIUS TREFOIL PUZZLE: Tom Longtin is a fan of the Möbius strip and its many variants, such as this trefoil—an overhand knot with a twist in it. ---Tom Longtin
7、莫比烏斯蛋白質
MÖBIUS PROTEIN: The major component of high-density lipoprotein (HDL, sometimes known as "good cholesterol"), apolipoprotein A-I, consists of a kinked helix about 12.5 nanometers in its longest dimension. Mike Tyka of the University of Washington, a protein-folding expert and artist/tinkerer, maintains a catalog of his favorite molecules. ---Mike Tyka, University of Washington. Image was rendered using PyMol
高密度脂蛋白（HDL）的重要組成部分「阿樸脂蛋白」，由一個最大尺寸為12.5納米的螺旋結構扭結而成。華盛頓大學的 Mike Tyka ，一位蛋白質折疊專家，保存著很多這類圖片。
8、紐結理論
KNOT THEORY: The trefoil is the simplest knot in mathematicians' classification. Knot theory goes back to the 19th century, when physicists briefly thought knots might explain atoms; mathematicians continued to develop the theory for its inherent interest. Lately the theory has found use after all. It comes up in quantum field theory, which describes the fundamental particles and interactions, as well as proposed quantum theories of gravity such as string theory and loop gravity. ---Robert G. Scharein
按照數學家們的分類，三葉形是最簡單的紐結。所謂紐結，它是三維空間中不與自己相交的封閉曲線，或者說是三維空間中與圓周同胚的圖形。紐結理論要上溯到19世紀。C•F•高斯在1833年研究電動力學時引進了閉曲線之間的環繞數，這是紐結理論的基本工具之一。1880年左右出現了最早的紐結表。1910年M•W•德恩引進紐結的群的概念，1928年J•W•亞歷山大引進了紐結的多項式這個更易處理的不變數。
Originated :
The Unreasonable Beauty of Mathematics : Scientific American Slide shows
FAMOUS FRACTAL: This is a version of one of the best known fractals, the Julia set. Fractals in general are a compelling example of how abstract mathematical forms, generated by seemingly simple algorithms in which a pattern repeats on multiple scales, are capable of intricate beauty. Nature is full of such patterns. ---Tom Beddard
Julia set ，典型的分形版本之一，由法國數學家 Gaston Julia 和 Pierre Faton 在發展了複變函數迭代的基礎理論後獲得。而 B.B.Mandelbort 為最先提出分形概念者，他在1967年《科學》雜誌上發表《英國的海岸線有多長?》的著名論文，文中把部分與整體以某種方式相似的形體稱為分形(fractal)。
2、分形的泡泡
FRACTAL BUBBLES: Richard Taylor specializes in discovering the occurrence of fractals in the world. (He has argued that fractal geometry can tell a real Jackson Pollack painting from a knock-off.) He took this picture at the edge of the pond in Sydney, Australia. The bubble outlines have a fractal dimension of 1.3, which people tend to find the most aesthetically pleasing. Taylor argues that our eyes scan a scene using a fractal pattern with a similar dimension.
3、分形的花椰菜
FRACTAL BROCCOLI: John Ostrowick, responding to our Twitter call for examples of natural mathematical beauty, suggested Romanesco broccoli. This photo by Jon Sullivan was selected by Wikipedia users as one of the most spectacular on that site. ---Jon Sullivan
John Ostrowick 在推特上響應「讓大家去找自然數學之美實例」的提議，羅馬花椰菜就是例子。
4、雙螺旋線
DOUBLE SPIRAL: Paul Nylander maintains an incredible collection of mathematical art, along with the Mathematica code to recreate it. ---Paul Nylander
5、太空中的螺旋形
HEAVENLY SPIRALS: Spiral patterns occur throughout nature, perhaps most dramatically in spiral galaxies. This pair of galaxies has particularly unusual spiral patterns that are presumably the result of the gravitational tidal forces between them. The inner spiral arms of the upper galaxy (UGC 1810) are not planar, and the outer arm may have been pulled into a ring by a direct collision with the lower one (UGC 1813). ---NASA, ESA, and the Hubble Heritage Team
6、莫比烏斯三葉形謎題
MÖBIUS TREFOIL PUZZLE: Tom Longtin is a fan of the Möbius strip and its many variants, such as this trefoil—an overhand knot with a twist in it. ---Tom Longtin
7、莫比烏斯蛋白質
MÖBIUS PROTEIN: The major component of high-density lipoprotein (HDL, sometimes known as "good cholesterol"), apolipoprotein A-I, consists of a kinked helix about 12.5 nanometers in its longest dimension. Mike Tyka of the University of Washington, a protein-folding expert and artist/tinkerer, maintains a catalog of his favorite molecules. ---Mike Tyka, University of Washington. Image was rendered using PyMol
高密度脂蛋白（HDL）的重要組成部分「阿樸脂蛋白」，由一個最大尺寸為12.5納米的螺旋結構扭結而成。華盛頓大學的 Mike Tyka ，一位蛋白質折疊專家，保存著很多這類圖片。
8、紐結理論
KNOT THEORY: The trefoil is the simplest knot in mathematicians' classification. Knot theory goes back to the 19th century, when physicists briefly thought knots might explain atoms; mathematicians continued to develop the theory for its inherent interest. Lately the theory has found use after all. It comes up in quantum field theory, which describes the fundamental particles and interactions, as well as proposed quantum theories of gravity such as string theory and loop gravity. ---Robert G. Scharein
按照數學家們的分類，三葉形是最簡單的紐結。所謂紐結，它是三維空間中不與自己相交的封閉曲線，或者說是三維空間中與圓周同胚的圖形。紐結理論要上溯到19世紀。C•F•高斯在1833年研究電動力學時引進了閉曲線之間的環繞數，這是紐結理論的基本工具之一。1880年左右出現了最早的紐結表。1910年M•W•德恩引進紐結的群的概念，1928年J•W•亞歷山大引進了紐結的多項式這個更易處理的不變數。
Originated :
The Unreasonable Beauty of Mathematics : Scientific American Slide shows
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