Euclid's puzzling parallel postulate - Jeff Dekofsky

Translator: Andrea McDonough Reviewer: Bedirhan Cinar

Translator: Mellysa Henggar Reviewer: Qomariah Najmi

As any current or past geometry student knows,

Seperti pelajar geometri kini dan dulu tahu,

the father of geometry was Euclid,

bapak geometri adalah Euklides.

a Greek mathematician who lived in Alexandria, Egypt, around 300 B.C.E.

Ilmuwan Yunani di Alexandria, Mesir, yang hidup sekitar 300 tahun SM.

Euklides dikenal sebagai penulis karya yang sangat berpengaruh

Euclid is known as the author of a singularly influential work

berjudul “Elemen”.

known as "Elements."

Berpikir buku matematika panjang?

You think your math book is long?

Euclid's "Elements" is 13 volumes full of just geometry.

13 volume “Elemen” hanya berisi tentang geometri.

In "Elements," Euclid structured and supplemented the work

Dalam “Elemen”, Euklides menata dan melengkapi karya

dari banyak matematikawan sebelum ia,

of many mathematicians that came before him,

seperti Phytagoras, Eudoxus, Hippokrates, dan lain-lainnya.

such as Pythagoras, Eudoxus, Hippocrates and others.

Euclid laid it all out as a logical system of proof

Euklides memaparkan semuanya sebagai bukti sistem logika

built up from a set of definitions,

yang terdiri dari sekumpulan definisi,

gagasan umum, dan kelima postulat terkenalnya.

common notions, and his five famous postulates.

Four of these postulates are very simple and straightforward,

Empat diantaranya ini sederhana dan jelas,

two points determine a line, for example.

dua titik membentuk sebuah garis, misal.

The fifth one, however, is the seed that grows our story.

Namun yang kelima, adalah benih dari cerita kita ini.

Postulat misterius kelima dikenal dengan postulat paralel.

This fifth mysterious postulate is known simply as the parallel postulate.

Tidak seperti empat postulat pertama,

You see, unlike the first four,

the fifth postulate is worded in a very convoluted way.

postulat kelima disajikan dengan cara yang sangat membelit.

Euclid's version states that,

Versi Euklides menyatakan,

"If a line falls on two other lines

“Jika 1 garis membelah dua garis lainnya

so that the measure of the two interior angles on the same side of the transversal

maka ukuran kedua sudut pada sisi garis pembelah

add up to less than two right angles,

dijumlah kurang dari dua sudut kanan,

then the lines eventually intersect on that side,

lalu jika garis-garis itu membagi sisi itu,

and therefore are not parallel."

maka tidak paralel.”

Wow, that is a mouthful!

Wah, banyak sekali!

Here's the simpler, more familiar version:

Ini adalah versi yang lebih simpel dan lazimnya:

"In a plane, through any point not on a given line,

“Di pesawat, suatu titik tidak diberi garis,

only one new line can be drawn that's parallel to the original one."

hanya satu garis baru yang bisa digambar paralel dengan garis awal.”

Banyak matematikawan berabad-abad mencoba membuktikan postulat paralel ini

Many mathematicians over the centuries tried to prove the parallel postulate

from the other four, but weren't able to do so.

dari keempat lainnya, namun tetap tidak bisa.

Dalam proses, mereka secara logis melihat apa yang akan terjadi

In the process, they began looking at what would happen logically

jika postulat kelima sebenarnya tidak benar.

if the fifth postulate were actually not true.

Beberapa pemikir cemerlang dalam sejarah matematika menanyakan ini,

Some of the greatest minds in the history of mathematics ask this question,

orang-orang seperti Ibnu Al Haytham, Umar Khayyām,

people like Ibn al-Haytham, Omar Khayyam,

Nasir al-Din al-Tusi, Giovanni Saccheri,

Nasir al-Din al-Tusi, Giovanni Saccheri,

János Bolyai, Carl Gauss, dan Nikolai Lobachevsky.

János Bolyai, Carl Gauss, and Nikolai Lobachevsky.

Mereka semua bereskperimen meniadakan postulat paralel,

They all experimented with negating the parallel postulate,

hanya untuk menemukan cikal bakal geometri alternatif.

only to discover that this gave rise to entire alternative geometries.

These geometries became collectively known as non-Euclidean geometries.

Geometri ini kemudian dikenal sebagai Geometri Non-Euklides.

We'll leave the details of these different geometries for another lesson.

Kami akan membahas detail geometri ini di lain waktu.

Perbedaan utama bergantung pada lengkungan permukaan

The main difference depends on the curvature of the surface

pada garis mana yang dibentuk.

upon which the lines are constructed.

Ternyata Euklides tidak memberitahu kita keseluruhan cerita dalam “Elemen”,

Turns out Euclid did not tell us the entire story in "Elements,"

dan hanya menjelaskan 1 cara yang layak untuk melihat semesta.

and merely described one possible way to look at the universe.

Semua tergantung konteks tentang apa yang diamati.

It all depends on the context of what you're looking at.

Permukaan datar berjalan satu arah,

Flat surfaces behave one way,

namun permukaan lengkung secara positif dan negatif

while positively and negatively curved surfaces

menunjukkan ciri yang sangat berbeda.

display very different characteristics.

At first these alternative geometries seemed strange,

Awalnya geometri alternatif ini terkesan aneh,

tapi kemudian itu sama mahirnya dalam menggambarkan dunia di sekitar kita.

but were soon found to be equally adept at describing the world around us.

Navigating our planet requires elliptical geometry

Menavigasi planet kita butuh geometri eliptik

while the much of the art of M.C. Escher displays hyperbolic geometry.

sementara sebagian besar seni M.C. Escher menampilkan geometri hiperbolik.

Albert Einstein used non-Euclidean geometry as well

Albert Einstein memakai geometri non-Euklides juga

to describe how space-time becomes warped in the presence of matter,

untuk menggambarkan bagaimana ruang-waktu menjadi bengkok di hadapan materi,

as part of his general theory of relativity.

sebagai bagian dari teori umum relativitas.

The big mystery is whether Euclid had any inkling

Misteri besarnya adalah apakah Euklides punya firasat

of the existence of these different geometries

tentang kehadiran beragam geometri ini

when he wrote his postulate.

ketika ia menulis postulatnya.

We may never know,

Kita mungkin tidak pernah tahu,

but it's hard to believe he had no idea whatsoever of their nature,

tapi sulit dipercaya kalau dia tidak tahu sama sekali apapun sifatnya,

being the great intellect that he was

menjadi intelek terkemuka

dan memahami bidang seperti yang ia lakukan.

and understanding the field as thoroughly as he did.

Maybe he did know and he wrote the postulate in such a way

Mungkin ia tahu dan menulis postulat sedemikian rupa

as to leave curious minds after him to flush out the details.

agar membuat ilmuwan setelahnya menggali detailnya.

If so, he's probably pleased.

Jika ya, mungkin ia senang.

These discoveries could never have been made

Penemuan ini tidak dapat ada

without gifted, progressive thinkers

tanpa pemikir berbakat, progresif

able to suspend their preconceived notions

yang meniadakan gagasan prasangka mereka

and think outside of what they've been taught.

dan berpikir tidak biasa dari yang diajari.

We, too, must be willing at times to put aside our preconceived notions

Kita juga, harus bisa terkadang menghilangkan gagasan prasangka

dan pengalaman fisik

and physical experiences

and look at the larger picture,

dan melihat gambaran besar,

or we risk not seeing the rest of the story.

atau berkorban untuk tidak melihat cerita sisanya.