偉大而美麗的渾然不覺
只有清晨的露水勇敢地凝視著自己的蒸發
只有清晨的露水勇敢地凝視著自己的蒸發
日出以後的第一聲鳥鳴也顯得如此平淡無奇
凡存在都是義無反顧的愛
困惑於各種數字莫名所以的起伏漲落
我們完全誤會了時間的封箱演出
雖然歷史早已悄悄散場
克羅頓海岸的幽靈仍在浪潮聲裡徘徊
—— 克羅頓(Croton克勞東 ，古希臘語Κρότων[króton]，意大利語Crotone[kroˈtoːne] 克羅拖內)，位在今天南意大利愛奧尼亞海沿岸的一個小城，曾為古希臘領地，西元前530年畢達哥拉斯在此建立他的學派會社。畢達哥拉斯的學生西伯索斯(Hippasus)提出:就邊長為1的正方形，以其對角線為斜邊所構成的直角三角形，證明了這個斜邊為無理數√2。
據說這洩露了天機，破壞了畢達哥拉斯類宗教的秘密會社的有理數信仰，導致了悲劇，西伯索斯被投海而死。
(*封箱戲、封箱演出：傳統戲曲年底的最後一場演出結束後，劇團將服裝和道具收拾好，封進箱子裡，等待新年首場演出的開啟。 它不只是舊的結束，也代表著一個新的開始即將到來。)
*請參閱《心靈DNA，靈性符號的幾何世界》“第一章第五節:畢達哥拉斯的夢想”。十多年前寫這本書的時候就想為西伯索斯的故事寫一首詩，然而故事太像希臘神話所經常流露的精神隱喻，遲遲無法動筆，十多年後的今天內心依舊感到翻騰不已，那神秘的無理數究竟從何而來又將去向何方？個體想要理解宇宙，或者只能融入於祂？
我們信仰為理所當然的，總是會存在著那麼一些理所不當然的可能性。
關於“無理數”，從靈性科學角度探討，請參閱這篇:“天人地SRT與Φeπ”
西伯索斯用幾何發現無理數的存在，底下這首歌“Ode to Hippasus”巧妙地結合數學和音樂用代數證明。
或 YouTube“Ode to Hippasus(a song about √2)”
(本文英譯最後亦附有“Ode to Hippasus”歌詞原文及中譯)
The Ghost of Croton Coast
Great and beautiful, utterly unaware,
Only the morning dew dares to gaze upon its own evaporation.
The first bird's song after sunrise seems so ordinary,
All existence is a love without hesitation.
Confused by the inexplicable fluctuation of various numbers,
We have completely misunderstood the closing performance of time.
Although history has quietly ended,
The ghost of Croton coast still wanders in the sound of waves.
—— Croton (or Crotone) is a small city on the Ionian Sea coast of southern Italy today. It was once the territory of ancient Greece. Pythagoras established his school society here in 530 BC. Hippasus, a student of Pythagoras, proposed that a square with a side length of 1 forms a right triangle with its diagonal as the hypotenuse, and proved that the hypotenuse is the irrational number √2.
It is said that this divulged the truth and destroyed the belief in rational numbers of the secret society of the Pythagoras-like religion, leading to the tragedy when Hybersus was thrown into the sea and died.
(*封箱戲、封箱演出:“Year-end Closing Performance“,or “Box-sealing opera” and “box-sealing performance”: After the final performance of traditional opera at the end of the year, the troupe packs up the costumes and props and seals them in boxes, waiting for the first performance of the new year.
It is not only the end of the old, but also represents the coming of a new beginning.)
*Please refer to "Mind DNA,The Geometric World of Spiritual Symbols" "Chapter1, Section5: Pythagoras' Dream ". When I wrote this book over ten years ago, I wanted to write a poem about the story of Hippasus. However, the story is too much like the spiritual metaphors often found in Greek myths, so I couldn't write about it, and now, more than a decade later, I still feel a stirring in my heart. Where do the mysterious irrational numbers come from and where are they going?Does the individual want to understand the universe, or can he only merge with Him?
Our beliefs are taken for granted, but there will always be some possibilities that are not taken for granted.
For a discussion of irrational numbers from a spiritual science perspective, see this article:
"SRT( 天heaven－人humankind－地earth) and Φeπ"
Hippasus used geometry to discover the existence of irrational numbers. The song "Ode to Hippasus" below cleverly combines mathematics and music to prove it algebraically.
or YouTube“Ode to Hippasus(a song about √2)”
Ode to Hippasus(a song about √2)
Lyrics by Joel David Hamkins, Barbara Gail Montero, & their daughter Hypatia.
Music by Hannah Hoffman
The diagonal of a square is incommensurable with its side
an astounding fact the Pythagoreans did hide
but Hippasus rebelled and spoke the truth
making his point with irrefutable proof
it’s absurd to suppose that the root of two
is rational, namely, p over q
square both sides and you will see
that twice q squared is the square of p
since p squared is even, then p is as well
now, if p as 2k you alternately spell
2q squared will to 4k squared equate
revealing, when halved, q’s even fate
thus, root two as fraction, p over q
must have numerator and denomerator with factors of two
to lowest terms, therefore, it can’t be reduced
root two is irrational, Hippasus deduced
as retribution for revealing this irrationality
Hippasus, it is said, was drowned in the sea
but his proof live on for the whole world to admire
a truth of elegance that will ever inspire.
正方形的對角線與其邊長無法通約
一個令人震驚的事實，畢達哥拉斯學派曾試圖隱藏
假設根號二是有理數，即p除以q
是荒謬的
但希帕索斯卻叛逆地說出了真相
用無可辯駁的證明來闡述他的觀點
假設根號二是有理數，即p除以q
是荒謬的
將兩邊平方，你就會看到
兩倍的q的平方等於p的平方
既然p的平方是偶數，那麼p也是偶數
現在，如果p等於2k，我們換種說法
2q的平方等於4k的平方
揭示出，當除以2時，q也是偶數
因此，根號二作爲分數，p除以q
必須有分子和分母都含有因子2
所以，它無法化簡到最簡形式
根號二是無理數，希帕索斯得出了這個結論
作爲揭露這個無理性的報復
據說，希帕索斯被淹死在海里
但他的證明卻流傳下來供全世界的人欣賞
一種優雅的真理，將永遠激勵着人們
