Nov 30, 2012

Non-Euclidean Geometries_Transcribed by Marsigit




Non-Euclidean Geometries
Transcribed by Marsigit , Yogyakarta State University, Indonesia
Email: marsigitina@yahoo.com


The historical developments of non-Euclidean geometry were attempts to deal with the fifth axiom.

Mathematicians first tried to directly prove that the first 4 axioms could prove the fifth. However, mathematicians were becoming frustrated and tried some indirect methods.

Girolamo Saccheri (1667-1733) tried to prove a contradiction by denying the fifth axiom. He started with quadrilateral ABCD (later called the Saccheri Quadrilateral) with right angles at A and B and where AD = BC. Since he is not using the fifth axiom, he concludes there are three possible outcomes. Angles at C and D are right angles, C and D are both obtuse, or C and D are both acute. Saccheri knew that the only possible solution was right angles.

Saccheri said this was enough to claim a contradiction and he stopped. His reasoning to stop was based on faulty logic. He was going on the presumption that lines and parallel lines worked like those in flat geometry. So his contradiction was only applicable in Euclidean geometry, which was not a contradiction to what he was actually trying to prove. Of course Saccheri did not realize this at the time and he died thinking he had proved Euclid’s fifth axiom from the first four.

A contemporary of Saccheri, Johann Lambert (1728-1777), picked up where Saccheri left off and took the problem just a few steps further. Lambert considered the three possibilities that Saccheri had concluded as consequences of the first four axioms. Instead of finding a contradiction, he found two alternatives to Euclidean geometry. The first option represented Euclidean geometry and while the other two appeared silly, they could not be proven wrong.
Through time (and quite a lot of criticism), these two other possibilities were now being considered as “alternative geometries” to Euclid’s geometry. Eventually these alternate geometries were scholarly acknowledged as geometries, which could stand alone to Euclidean geometry.

The two non-Euclidean geometries were known as hyperbolic and elliptic. Hyperbolic geometry was explained by taking the acute angles for C and D on the Saccheri Quadrilateral while elliptic assumed them to be obtuse.

Let’s compare hyperbolic, elliptic and Euclidean geometries with respect to Playfair’s parallel axiom and see what role parallel lines have in these geometries:
1.)             Euclidean: Given a line L and a point P not on L, there is exactly one line passing through P, parallel to L.
2.)             Hyperbolic: Given a line L and a point P not on L, there are at least two lines passing through P, parallel to L.
3.)             Elliptic: Given a line L and a point P not on L, there are no lines passing through P, parallel to L.

Elliptic Geometry

Elliptic geometry also says that the shortest distance between two points is an arc on a great circle (the “greatest” size circle that can be made on a sphere’s surface).

As part of the revised parallel postulate for elliptic geometries, we learn that there are no parallel lines in elliptical geometry.

This means that all straight lines on the sphere’s surface intersect (specifically, they all interesect in two places).

A famous non-Euclidean geometer, Bernhard Riemann, who dealt mostly with and is credited with the development of elliptical geometries, theorized that the space (we are talking about outer space now) could be boundless without necessarily implying that space extends forever in all directions.

This theory suggests that if we were to travel one direction in space for a really long time, we would eventually come back to where we started! This theory involves the existence of four-dimensional space similar to how the surface of a sphere (which is three dimensional) represents an elliptic 2 dimensional geometry.

Hyperbolic Geometry

Recalling the corresponding Playfair’s axiom for hyperbolic geometry, we see that in hyperbolic geometry, there is more than one parallel line to L, passing through point P, not on L.

Hyperbolic geometries comes with some more restrictions about parallel lines. In Euclidean geometry, we can show that parallel lines are always equidistant, but in hyperbolic geometries, of course, this is not the case.

In hyperbolic geometries, we merely can assume that parallel lines carry only the restriction that they don’t interesect. Furthermore, the parallel lines don’t seem straight in the conventional sense. They can even approach each other in an asymptotically fashion. The surfaces on which these rules on lines and parallels hold true are on negatively curved surfaces.

In hyperbolic geometry, the triangle’s angle sum is less than 180 degrees whereas elliptic geometry has more than 180 degrees. The larger the sides of the triangle, the greater the distortion of the angle sums on both elliptic and hyperbolic geometries. Much like elliptic geometries, the area of a triangle is proportional to its angle sum and of course this implies that there are no similar triangles as well.

4 comments:

  1. Sumandri
    16709251072
    S2 Pendidikan Matematika D 2016

    Menurut saya artikel diatas cukup menyita perhatian apalagi seorang matematikawan, dalam artikel itu membahas tentang geometri non-euclidean. Dimana Perkembangan sejarah geometri non-Euclidean adalah upaya untuk menangani aksioma kelima. Matematikawan pertama kali mencoba untuk langsung membuktikan bahwa aksioma 4 sebelumnya bisa membuktikan aksioma kelima. Namun, ahli matematika yang menjadi frustrasi dan mencoba beberapa metode tidak langsung. Dua geometri non-Euclidean dikenal sebagai hiperbolik dan eliptik. geometri hiperbolik dijelaskan dengan mengambil sudut akut untuk C dan D di Saccheri Segiempat sementara eliptik diasumsikan mereka untuk menjadi tumpul. Teori Geometri elliptic menunjukkan bahwa jika kita melakukan perjalanan satu arah dalam ruang untuk waktu yang sangat lama, kita akhirnya akan kembali ke mana kita mulai! Teori ini melibatkan keberadaan ruang empat dimensi mirip dengan bagaimana permukaan bola (yang tiga dimensi) merupakan elips 2 dimensi geometri. Dalam geometri hiperbolik, kita hanya bisa berasumsi bahwa garis sejajar hanya membawa pembatasan bahwa mereka tidak interesect. Selanjutnya, garis paralel tampaknya tidak langsung dalam arti konvensional. Mereka bahkan dapat mendekati satu sama lain dalam mode asimtotik. Permukaan yang aturan-aturan ini pada baris dan paralel berlaku adalah pada permukaan negatif melengkung. Dalam geometri hiperbolik, jumlah sudut segitiga adalah kurang dari 180 derajat sedangkan geometri eliptik memiliki lebih dari 180 derajat.

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  2. Nama : Irna K.S.Blegur
    Nim : 16709251064
    kelas : PM D 2016(PPS)

    Ada 2 geometri non-Euclidean dikenal sebagai hiperbolik dan elips. Geometri hiperbolik dijelaskan dengan mengambil sudut akut untuk C dan D pada Saccheri Segiempat sementara eliptik diasumsikan mereka untuk menjadi tumpul.

    Geometri hiperbolik: Geometri yang awalnya dikembangkan oleh matematikawan Janos Bolyai dan Lobachevsky. Geometri ini mulanya mempersoalkan aksioma ke-5 dari lima aksioma geometri bidang euclid, yaitu aksioma kesejajaran. Para matematikawan mencoba untuk membuktikan bahwa aksioma kelima Euclide bukanlah aksioma , dengan menggunakan aksioma ke-1 hingga ke-4, namun usaha mereka tidaklah berhasil. Berangkat dari percobaan untuk membuktikan bahwa aksioma kesejajaran bukanlah sebagai suatu aksioma melainkan teorema, munculah inspirasi pengetahuan baru mengenai geometri hiperbolik. Janos Bolyai dan Nicholas Lobhacevsky adalah matematikawan yang layak untuk dihargai karena berhasil menemukan sesuatu yang baru tersebut, ditempat yang berbeda dan secara sendiri-sendiri.

    Geometri Non Euclid lahir setelah terpecahkannya permasalahan postulat kesejajaran Euclid oleh Bolya dan Lobachevsky. Geometri non euclid diantaranya geometri Lobachevsky dan geometri Riemann. Geometri Lobachevsky disebut geometri Hiperbolik, mengingat bahwa melalui 1 titik di luar suatu garis dapat dibuat 2 garis yang sejajar garis tersebut. Geometri Riemann disebut geometri Eliptik, mengingat tidak ada garis yang dapat dibuat sejajar garis tersebut. Sedangkan geometri Euclid disebut geometri Parabolik, mengingat bahwa hanya ada 1 garis yang sejajar garis tersebut

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  3. Saepul Watan
    16709251057
    S2 P.Mat Kelas C 2016

    Bismilahir rahmaanir rahiim..
    Assalamualaikum wr..wb...

    Geometri non Euclidean dibagi menjadi dua yaitu geometri hiperbolik dan geometri elips. Geometri hiperbolik dijelaskan dengan mengambil sudut akut pada segiempat, sementara elips diasumsikan untuk sudut tumpul. Menurut teori Bernhard Riemann tentang geometri elips, mengemukakan bahwa ruang (ruang angkasa sekarang) bisa tak terbatas tanpa perlu menyiratkan bahwa ruang meluas selama-lamanya di segala penjuru.
    Teori ini menunjukkan bahwa jika kita melakukan perjalanan satu arah di ruang untuk waktu yang sangat lama, kita akhirnya akan kembali ke mana kita mulai. Geometri hiperbolik dilengkapi dengan beberapa pembatasan lain tentang paralel baris. Dalam geometri Euclidean, kita dapat menunjukkan bahwa paralel baris selalu turutan,tetapi dalam geometri hiperbolik, tentu saja, hal ini tidak terjadi. Dalam geometri hiperbolik, kita hanya bisa berasumsi bahwa paralel baris membawahanya pembatasan bahwa mereka tidak interesect. Selain itu, garis-garis paralel tampaknya tidak langsung dalam arti konvensional. Mereka bahkan dapat mendekati satu sama lain dalam asimtotik mode. Permukaan dimana aturan-aturan ini pada baris dan paralel berlaku adalah dipermukaan yang lengkung negatif.

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  4. Wahyu Lestari
    16709251024
    PPs P.Matematika Kelas D

    dari arikel di atas, geometri alternatif ini diakui secara ilmiah sebagai geometri, yang bisa berdiri sendiri untuk geometri Euclidean. Dua geometri non-Euclidean dikenal sebagai hiperbolik dan elips. Geometri hiperbolik dijelaskan dengan mengambil sudut akut untuk C dan D pada Quadratateral Saccheri sementara elips menganggapnya tumpul.

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