As any current or past geometry student knows, the father of geometry was Euclid, a Greek mathematician who lived in Alexandria, Egypt, around 300 B.C.E. Euclid is known as the author of a singularly influential work known as "Elements." You think your math book is long? Euclid's "Elements" is 13 volumes full of just geometry. In "Elements," Euclid structured and supplemented the work of many mathematicians that came before him, such as Pythagoras, Eudoxus, Hippocrates and others. Euclid laid it all out as a logical system of proof built up from a set of definitions, common notions, and his five famous postulates. Four of these postulates are very simple and straightforward, two points determine a line, for example. The fifth one, however, is the seed that grows our story. This fifth mysterious postulate is known simply as the parallel postulate. You see, unlike the first four, the fifth postulate is worded in a very convoluted way. Euclid's version states that, "If a line falls on two other lines so that the measure of the two interior angles on the same side of the transversal add up to less than two right angles, then the lines eventually intersect on that side, and therefore are not parallel." Wow, that is a mouthful! Here's the simpler, more familiar version: "In a plane, through any point not on a given line, only one new line can be drawn that's parallel to the original one." Many mathematicians over the centuries tried to prove the parallel postulate from the other four, but weren't able to do so. In the process, they began looking at what would happen logically if the fifth postulate were actually not true. Some of the greatest minds in the history of mathematics ask this question, people like Ibn al-Haytham, Omar Khayyam, Nasir al-Din al-Tusi, Giovanni Saccheri, János Bolyai, Carl Gauss, and Nikolai Lobachevsky. They all experimented with negating the parallel postulate, only to discover that this gave rise to entire alternative geometries. These geometries became collectively known as non-Euclidean geometries. We'll leave the details of these different geometries for another lesson. The main difference depends on the curvature of the surface upon which the lines are constructed. Turns out Euclid did not tell us the entire story in "Elements," and merely described one possible way to look at the universe. It all depends on the context of what you're looking at. Flat surfaces behave one way, while positively and negatively curved surfaces display very different characteristics. At first these alternative geometries seemed strange, but were soon found to be equally adept at describing the world around us. Navigating our planet requires elliptical geometry while the much of the art of M.C. Escher displays hyperbolic geometry. Albert Einstein used non-Euclidean geometry as well to describe how space-time becomes warped in the presence of matter, as part of his general theory of relativity. The big mystery is whether Euclid had any inkling of the existence of these different geometries when he wrote his postulate. We may never know, but it's hard to believe he had no idea whatsoever of their nature, being the great intellect that he was and understanding the field as thoroughly as he did. Maybe he did know and he wrote the postulate in such a way as to leave curious minds after him to flush out the details. If so, he's probably pleased. These discoveries could never have been made without gifted, progressive thinkers able to suspend their preconceived notions and think outside of what they've been taught. We, too, must be willing at times to put aside our preconceived notions and physical experiences and look at the larger picture, or we risk not seeing the rest of the story.
Seperti pelajar geometri kini dan dulu tahu, bapak geometri adalah Euklides. Ilmuwan Yunani di Alexandria, Mesir, yang hidup sekitar 300 tahun SM. Euklides dikenal sebagai penulis karya yang sangat berpengaruh berjudul “Elemen”. Berpikir buku matematika panjang? 13 volume “Elemen” hanya berisi tentang geometri. Dalam “Elemen”, Euklides menata dan melengkapi karya dari banyak matematikawan sebelum ia, seperti Phytagoras, Eudoxus, Hippokrates, dan lain-lainnya. Euklides memaparkan semuanya sebagai bukti sistem logika yang terdiri dari sekumpulan definisi, gagasan umum, dan kelima postulat terkenalnya. Empat diantaranya ini sederhana dan jelas, dua titik membentuk sebuah garis, misal. Namun yang kelima, adalah benih dari cerita kita ini. Postulat misterius kelima dikenal dengan postulat paralel. Tidak seperti empat postulat pertama, postulat kelima disajikan dengan cara yang sangat membelit. Versi Euklides menyatakan, “Jika 1 garis membelah dua garis lainnya maka ukuran kedua sudut pada sisi garis pembelah dijumlah kurang dari dua sudut kanan, lalu jika garis-garis itu membagi sisi itu, maka tidak paralel.” Wah, banyak sekali! Ini adalah versi yang lebih simpel dan lazimnya: “Di pesawat, suatu titik tidak diberi garis, hanya satu garis baru yang bisa digambar paralel dengan garis awal.” Banyak matematikawan berabad-abad mencoba membuktikan postulat paralel ini dari keempat lainnya, namun tetap tidak bisa. Dalam proses, mereka secara logis melihat apa yang akan terjadi jika postulat kelima sebenarnya tidak benar. Beberapa pemikir cemerlang dalam sejarah matematika menanyakan ini, orang-orang seperti Ibnu Al Haytham, Umar Khayyām, Nasir al-Din al-Tusi, Giovanni Saccheri, János Bolyai, Carl Gauss, dan Nikolai Lobachevsky. Mereka semua bereskperimen meniadakan postulat paralel, hanya untuk menemukan cikal bakal geometri alternatif. Geometri ini kemudian dikenal sebagai Geometri Non-Euklides. Kami akan membahas detail geometri ini di lain waktu. Perbedaan utama bergantung pada lengkungan permukaan pada garis mana yang dibentuk. Ternyata Euklides tidak memberitahu kita keseluruhan cerita dalam “Elemen”, dan hanya menjelaskan 1 cara yang layak untuk melihat semesta. Semua tergantung konteks tentang apa yang diamati. Permukaan datar berjalan satu arah, namun permukaan lengkung secara positif dan negatif menunjukkan ciri yang sangat berbeda. Awalnya geometri alternatif ini terkesan aneh, tapi kemudian itu sama mahirnya dalam menggambarkan dunia di sekitar kita. Menavigasi planet kita butuh geometri eliptik sementara sebagian besar seni M.C. Escher menampilkan geometri hiperbolik. Albert Einstein memakai geometri non-Euklides juga untuk menggambarkan bagaimana ruang-waktu menjadi bengkok di hadapan materi, sebagai bagian dari teori umum relativitas. Misteri besarnya adalah apakah Euklides punya firasat tentang kehadiran beragam geometri ini ketika ia menulis postulatnya. Kita mungkin tidak pernah tahu, tapi sulit dipercaya kalau dia tidak tahu sama sekali apapun sifatnya, menjadi intelek terkemuka dan memahami bidang seperti yang ia lakukan. Mungkin ia tahu dan menulis postulat sedemikian rupa agar membuat ilmuwan setelahnya menggali detailnya. Jika ya, mungkin ia senang. Penemuan ini tidak dapat ada tanpa pemikir berbakat, progresif yang meniadakan gagasan prasangka mereka dan berpikir tidak biasa dari yang diajari. Kita juga, harus bisa terkadang menghilangkan gagasan prasangka dan pengalaman fisik dan melihat gambaran besar, atau berkorban untuk tidak melihat cerita sisanya.