GEOMETRI HIPERBOLIK Disusun untuk memenuhi salah satu tugas mata kuliah Geometri dengan dosen pengampu Prof. Dr. Dwi Juniati, M.Si
Disusun Oleh:
Fitriana Tandililing (107785004 ) Easty Kartika (107785041) Ofierenty E. Nubatonis ( 107785046)
PROGRAM STUDI PENDIDIKAN MATEMATIKA PROGRAM PASCASARJANA UNIVERSITAS NEGERI SURABAYA 2011
0
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GEOMETRI HIPERBOLIK A. Sejarah Geometri hiperbolik merupakan salah satu bentuk dari geometri nonEuclid yang muncul akibat kontroversi terhadap postulat kesejajaran euclid. Di dalam geometri euclid euclid terdapat lima postulat (aksioma/teorema) (aksioma/teorema) yang sangat terkenal. Empat postulat pertama sangat jelas dan mudah dibuktikan oleh para matematikawan pada saat itu, tetapi postulat yang kelima menimbulkan perdebatan diantara para matematikawan. Postulat kelima tersebut dikenal dengan postulat kesejajaran geometri euclid. Hal inilah yang menjadi titik tolak munculnya geometri non-euclid. Geometri hiperbolik adalah geometri yang menggunakan empat postulat geometri euclid dan mengganti postulat kesejajaran euclid dengan negasinya yaitu postulat kesejajaran hiperbolik. Akibat pergantian postulat ini terjadi sifat antara geometri euclid dan geometri hiperbolik salah satunya satunya adalah jumlah ukuran sudut segitiga. Pada geometri geometri euclid jumlah ukuran sudut segitiga adalah 180 , sedangkan pada geometri hiperbolik jumlah ukuran sudut segitiga kurang dari 180 . Para matematikawan telah berusaha untuk membuktikan postulat kelima euclid dengan asumsi negasi dan memcoba untuk menurunkan suatu kontradiksi, namun mereka gagal. Akhirnya mereka berpendapat bahwa terdapat lebih dari dua garis yang sejajar dengan suatu garis yang melalui sebuah titik tertentu di luar garis tersebut dan ukuran sudut kesejajaran untuk titik yang tidak terletak pada garis
tersebut kurang dari 90 Akibatnya jumlah ukuran sudut segitga kurang dari 180 dan jumlah ukuran sudut dalam segiempat kurang dari 360 , sehingga tidak ada persegi panjang dalam geometri hiperbolik. Segiempat Al-HaythamLambert dan segiempat Khayyam-Saccheri adalah teori pertama pada geometri hiperbolik. Dalam segiempat Lambert sudut keempat dalam segiempat ini adalah lancip, sehingga ukuran sisi yang memuat sudut lancip lebih panjang dari pada sisi yang tidak memuat sudut lancip, serta jika sudut-sudut yang bersesuaian dari dua segitiga kongruen maka dalam geometri hiperbolik dua segitiga tersebut adalah kongruen. Sekitar tahun 1700an Carl Fredrich Gauus menemukan banyak hasil
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Heinrich Lambert memperkenalkan fungsi hiperbolik dan menghitung luas segitiga hiperbolik. Pada abad ke-19, geometri hiperbolik secara luas dieksplorasi oleh Jonas Bolyai dan Nicolai Inanovich Lobachevsky. Lobachevsky pertama kali mempublikasikan idenya pada tanggal 23 februari 1826 ke departemen Fisika dan Matematika dan penelitian ini telah dicetak dalam UMA pada tahun 1829-1830. Sedangkan Bolyai menerbitkan idenya pada tahun 1832. Lobachevsky menulis paper yang berjudul “ A Concide Outline of the
Foundations of Geometry ” di publiksikan oleh Kazan Messenger tetapi di tolak
pada saat disampaikan di Akademi St Petersburg. Pada tahun 1937 Lobachevsky Lobachevsky mempublikasikan artikelnya yang berjudul “Geometrie Imaginaire” dan diterbitkan di Berlin pada tahun 1840. Beberapa ahli matematika dan sejarawan mengklaim bahwa Lobachevsky telah mencuri tentang konsep geometri non-euclid dari Gauus, tetapi hal itu tidak benar. Gauus sendiri menghargai hasil karya yang ditemukan oleh Lobachevsky, karena alasan itu maka Lobachevsky dan Bolgyai dianggap sebagai pencipta geometri hiperbolik. Setelah karya Gauus, Lobachevsky dan bolyai, muncul pertanyaan yang lain “ seperti apakah model dari geometri hiperbolik?” . Pertanyaan ini terjawab
Eugenio Beltrami tahun 1868, Dia yang pertama kali menunjukkan bahwa bidang yang berbentuk pseudosphere mempunyai kelengkungan yang sesuai untuk model sebagian ruang hiperbolik. Awalnya Lobachevsky menamakan geometri temuannya dengan sebutan “Geometrie Imaginaire” karena dia belum bisa memahami model untuk jenis
geometrinya. Geometri hiperbolik diperkenalkan oleh Felik Klein tahun 1871. Geometri hiperbolik sering jaga disebut geometri Lobachevsky, untuk memudahkan dan menandai karya lobachevsky sehingga postulatnya dikenal dengan postulat kesejajaran lobachevsky.
Tokoh-tokoh yang Berkaitan dengan Geometri Hiperbolik Biografi singkat dari tokoh-tokoh geometri hiperbolik diantaranya Gauus, Lobachevsky, dan Bolyai yang merupakan penemu dari geometri hiperbolik; Saccheri dan Lambert yang meyumbang teori segiempat dalam geometri
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1. Johann Carl Friedrich Gauss (30 April 1777 1 777
–
23 February 1855)
Lahir di Brunswick, bagian dari Lower Saxony, Jerman. Ia berasal dari keluarga miskin. Tahun 1788 Gauss memulai pendidikannya di Gymnasium High German and Latin. Pada tahun 1795 Gauss meninggalkan Brunswick untuk belajar di Universitas Göttingen. Gauss meninggalkan Göttingen tahun 1798 tanpa diploma, tetapi dia membuat penemuan penting, yaitu “ The Construction of a Regular 17-gon by
Ruler and Compasses ”. Namun, pada tahun 1799, Gauss kembali ke Brunswick
dimana dia menerima gelar. Tahun 1801, Akademi Sains St. Peterburg menunjuk Gauss menjadi direktur observatorium. Sangat disayangkan, minat matematika Gauss sempat terhenti pada usia 24 tahun. Minat terhadap matematika berubah menjadi astronomi. Hal ini tidak dapat dihindari karena tidak ada universitas yang menghargai bakat-bakat matematikanya matematikanya dan ia pun dirongrong kesulitan finansial. Gauss selalu mengalami kesulitan menjadi seorang pengajar. Cara pandangnya yang kelewat jauh membuat siswa-siswanya frustrasi. Sebaliknya, Gauss menganggap siswa-siswanya tidak pernah siap menghadapi kuliahnya. Buku karya Gauss juga sulit dipahami dimana hanya seorang yang mampu memecahkannya, ia adalah teman sekaligus murid Gauss yang bernama Peter Gustav Lejeune Dirichlet (1803 – 1859). Masa penantian diterimanya di Universitas Gottingen membawa Gauss berkenalan dengan seorang gadis cantik bernama Johanna Osthoff, anak perempuan seorang penyamak kulit yang kaya raya. Cinta pertama Gauss terjadi pada pandangan pertama. Gauss mengumpulkan uang dan keberanian sebelum menyatakan hal ini dua tahun kemudian. Mereka menikah pada tanggal 9 Oktober 1805 dan dikarunia 3 orang anak, yaitu: Joseph, Wilhelmina dan Louis. Tahun 1809, istrinya meninggal yang kemudian diikuti oleh anak ketiganya, Louis. Kematian istri dan anak ketiganya membuat Gauss depresi sehingga pada akhirnya Gauss menikah kembali dengan Friederica Wilhelmine Waldeck (Minna Waldeck), yang merupakan anak dari rekan sesama profesor di
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sebelum ia divonis terkena TBC. Masa-masa tua Gauss dihabiskan dengan setiap pagi berada di perpustakaan universitas, mengumpulan koran-koran dari seluruh daratan Eropa, mulai dari Times terbitan London sampai koran lokal yang ditumpuk dan dibaca satu per satu. Pada
usia
77
tahun,
Gauss
mengalami
pembengkakan jantung. Setiap jam 3 dini hari, ia harus bangun dan minum air soda dan susu hangat untuk meringankan sakitnya. Pada tanggal 23 Februari 1855, setelah serangan jantung, Gauss meninggal. Gauss dikebumikan di pemakaman St.
Pemakaman St. Albans
Albans di Gottingen, Hannover, Jerman. Tidak terhitung sumbangsih Gauss dalam perkembangan matematika pada umumnya dan bidang-bidang ilmu lain pada khususnya. Pemilahan dengan menggunakan sistem bilangan, statistik dan teori probabilitas lewat penemuan kurva lonceng adalah dua prestasi sangat penting yang mampu dicapai oleh Gauss. Dasar-dasar yang ditetapkan Gauss dalam matematika banyak memberi dampak bagi perkembangan matematika setelah dia meninggal. Salah satunya adalah geometri non-Euclid yang kelak mendasari teori relativitas Einstein setelah lewat sentuhan Riemann, Lobachevski dan Bolyai. Rupanya Gauss tidak puas hanya berkiprah dalam bidang matematika. Banyak disiplin ilmu-ilmu lain dijelajahi. Tujuan utamanya, sebagai jalan pintas untuk mendapatkan uang dengan menjadi pengajar atau profesi yang mendapatkan gaji rutin. Akhirnya kesinambungan “aliran“ uang ini membuat
Gauss makin intensif lagi melakukan penelitian matematika. Kolaborasi dengan Wilhelm Weber (1804-1891) menemukan telegraf elektrik dan fenomena elektromagnetik yang gagal sebelum dilanjutkan bekerjasama dengan Clark Maxwell (1831 – 1879) yang menemukan persamaan medan elektromagnetik dan dengan Friedrich Wilhelm Bessel [1784 –8146] membahas astronomi, matematika fisika untuk aplikasi dalam bidang elektrostatik, hidrodinamis, aerodinamis, orbit planet, sistem lensa, dan ditemukannya geometri geometri diferensial. Sifat perfeksionisnya terkadang mendapat kritik keras. Tanpa mau mengungkapkan penelitiannya jika belum sempurna, dianggap oleh kalangan
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membuat timbulnya pernyataan, “Apabila Gauss lebih terbuka, maka matematika akan lebih maju beberapa dasawarsa ke depan.” Meskipun jika
lebih jeli melihat permasalahan, bukan hanya hal itu penyebabnya. Kondisi perang dan tragedi yang silih berganti mendera Gauss adalah penyebab utama semua itu. 2. Janos Bolyai (15 Desember 1802-27 Januari 1860)
Janos lahir di Kota Transylvanian, Kolozsvar, Kerajaan Hungaria (sekarang Cluj-Napoca di Rumania). Ayahnya merupakan matematikawan matematikawan terkenal, yaitu Farkas Bolyai dan ibunya bernama Zsuzsanna Benkö. Janos Bolyai merupakan matematikawan Hungaria terbesar. Ia adalah penemu Geometri Absolut, Geometri Hiperbolik dan pengembang Geometri Affine. 3. Nicolai Ivannovich Lobachevsky (1 Desember 1792 12 Februari 1856) –
Lahir di Nizhny Novgorod, Russia. Ayahnya bernama Ivan Maksimovich Lobachevsky, ia bekerja sebagai pegawai di kantor
landsurveying
dan
ibunya
bernama
Praskovia
Alexandrovna Lobachevsky. Nicolai mempunyai 2 orang saudara dan mereka berasal dari keluarga miskin. Ketika umur 8 tahun (th.1800) ayah Nicolai meninggal dan mereka pun pindah ke Kazan, Rusia barat di pinggir Siberia. Di Kazan, Nicolai mendapat beasiswa dan masuk Universitas Kazan kemudian lulus pada tahun 1807. Di Universitas Kazan, profesor Johann Christian Martin Bartels (1769-1833), seorang mantan guru dan teman matematikawan Jerman, Carl Friedrich Gauss sangat tertarik pada kemampuan Nicolai dan pada tahun 1811, Nicolai menerima gelar master di bidang Fisika dan Matematika. Pada tahun 1814, Nicolai menjadi menjadi dosen di Universitas Kazan, Kazan, 1816 diangkat menjadi profesor luar biasa dan pada tahun 1822 (usia 30 tahun) Nicolai ditunjuk sebagai profesor penuh yang mengajar di bidang Matematika, Fisika dan Astronomi. Selama di Universitas Kazan, Nicolai menjabat berbagai macam
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Antara tahun 1820 dan 1825, ia diangkat menjadi dekan dari Departemen Matematika dan Fisika. Kemudian dari tahun 1825 ke 1835, Nicolai menjadi kepala perpustakaan dan menjabat sebagai ketua observatorium yang mempengaruhi kebijakan dalam universitas. Tahun 1827, Nicolai menjadi rektor universitas Kazan. Selama ia menjabat pada tahun 1830, terjadi epidema kolera dan kebakaran besar tahun
Monument Lobachevsky di Universitas Kazan
1842. Tetapi berkat keteguhannya maka ia mendapat ucapan terima kasih dari kaisar. Semasa jabatannya, ia mengajar berbagai topik yang berbeda seperti mekanik, hidrodinamika, integrasi, persamaan differensial, kalkulus variasi, fisika dan matematika. matematika. Di usia 40 tahun (1832), Nicolai menikah dengan Lady Varvara Alexeyenevna Moiseyeva yang berasal dari keluarga kaya. Mereka memiliki 18 putra, namun hanya 7 putra yang dapat bertahan hingga dewasa. Namun, Nicolai tidak beruntung dalam perkawinan. Hal ini tercantum pada buku biografinya yang menyebutkan bahwa: “Nicolai tidak pandai dalam memanage keuangan, pada saat pensiun ia
baru membeli real estate dan tidak pandai berinvestasi sehingga ia jatuh .” miskin dan diabaikan oleh pejabat lokal .”
Tahun 1846, ia dipecat dari universitas karena kesehatannya yang memburuk dan di awal tahun 1850-an ia hampir buta dan tidak mampu berjalan dan pada tahun 1856, Nicolai meninggal dalam kemiskinan. 4. Giovanni Girolamo Saccheri (5 September 1667
–
27 Januari 1733)
Saccheri lahir di San Remo, Genoa (sekarang Italia).
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Ia ditahbiskan sebagai imam pada tahun 1694 di Como, Italia. Pada tahun yang sama, Saccheri mulai mengajar filsafat di Universitas Turin sampai 1697. Selama tinggal disana, ia menerbitkan “ Logica Demonstrativa”, salah satu
karyanya yang paling penting. Pada tahun 1697 Saccheri berganti pekerjaan lagi, kali ini ia pindah ke Universitas Jesuit Pavia (juga dikenal sebagai Universitas Ticinese), di tempat itulah mana ia mengajar selama sisa hidupnya. Dua tahun kemudian, ia menjadi pimpinan di bidang matematika yang diangkat oleh Senat Milan. Milan. Pada 1708, Saccheri menerbitkan “Neo Statica”
yang berhubungan dengan mekanika.
Kemudian karya lainnya “Euclides ab Omni Naevo Vindicatus” mencoba untuk membuktikan postulat paralel Euclid. Saccheri meninggal di Milan, Italia pada tanggal 25 Oktober 1733. Dalam karyanya sintesis, Saccheri memberikan analisa lengkap tentang masalah kesejajaran dalam hal segiempat Omar Khayyam. 5. Johann Heinrich Lambert (26 Agustus 1728 - 25 September 1777)
Lambert adalah seorang matematikawan, fisikawan, dan astronom dari Swiss. Lambert lahir di kota Mulhouse, Swiss (sekarang di Alsace, Perancis). Sepulang sekolah ia terus belajar, sementara di waktu luangnya ia melakukan serangkaian pekerjaan seperti menjadi asisten ayahnya sebagai penjahit, sebagai pramuniaga di toko besi, tutor pribadi, sekretaris editor Basler Zeitung. Pada umur 20, Lambert menjadi tutor pribadi seorang anak dari pangeran di Chur. Perjalanannya ke Eropa (1756-1758) membawanya bertemu dengan matematikawan Jerman, Belanda, Perancis dan negara-negara Italia. Setelah kembali ke Chur ia menerbitkan buku pertamanya (pada optik dan kosmologi)
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geometri, terkenal dengan segiempat Lambert yang merupakan eksplorasi awal ke geometri non-Euclid. 6. Eugenio Beltrami (16 November 1835 18 Februari 1900) –
Lahir di Cremona, Lombardy, bagian kerajaan Austria dan sekarang bagian dari Italia. Beltrami memulai studi matematikanya di Universitas Pavia pada tahun 1853 sampai 1856. Pada tahun 1856 ia menghentikan studinya karena kesulitan ekonomi dan terpaksa bekerja menjadi sekretaris jalur kereta api dan berpindah ke Verona dan Milan. Pada
tahun
1861,
Beltrami
bekerja
di
bidang
matematika. Kemudian ia memulai penelitian pada tahun 1862. Pada tahun yang sama Beltrami juga ditunjuk sebagai guru besar di Universitas Bologna. Setelah dua tahun di Bologna, Beltrami menerima jabatan dari Geodesi di Unversitas Pisa pada tahun 1864 sampai 1866. Pada tahun 1866 Beltrami kembali ke Roma dan dilantik menjadi Profesor dari mekanik rasional. Setelah tiga tahun di Roma, Beltrami pindah lagi ke Pavia. Pada tahun 1891 Beltrami kembali lagi ke Roma dan menghabiskan tahun terakhirnya. Selama tahun 1898 Beltrami menjadi presiden Accademia dei Lincei dan menjadi senator di kerajaan Italia pada tahun 1899. Kontribusinya dalam matematika adalah selama tahun 1868, Beltrami menerbitkan dua memoar yang berkaitan dengan konsistensi dan interpretasi geometri non-Euclid dari Bolyai dan Lobachevsky. Dalam karyanya “ Essay on an ”, Beltrami mengusulkan bahw a Interpretation of Non-Euclidean Geometry ”,
geometri ini dapat direalisasikan pada permukaan kelengkungan negatif yang
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Untuk model setengah bidang, Beltrami mengutip sebuah catatan Liouville dalam risalah Monge pada geometri diferensial. Beltrami menunjukkan bahwa geometri Euclid n-dimensi direalisasikan pada suatu horosphere ruang hiperbolik (n+1) dimensi, sehingga hubungan logis antara kekonsistensian Euclid dan geometri non-Euclid adalah simetris. 7. Felix Klein (25 April 1849
–
22 June 1925)
Klein lahir di Dusseldorf, ayahnya seorang sekertaris kepala pemerintah. Klein hadir di Gymnasium, Düsseldorf. Setelah lulus, dia masuk Universitas Bonn. Dia belajar matematika dan fisika selama satu tahun. Dia memulai karirnya dengan tujuan untuk menjadi ahli fisika. Walaupun masih menjadi mahasiswa, ia telah ditunjuk sebagai asisten Plucker pada tahun 1866. Pada tahun 1868, Klein menerima gelar doktor dengan bimbingan Plücker dengan promotor “Über die der Transformasi allgemeinen Gleichung des
zweiten Kelas zwischen Linien-Koordinaten Auf eine kanonische Formulir ” di
bidang geometri dan aplikasi untuk mekanik. Namun pada tahun Klein menerima doktor, Plücker meninggal dunia. Klein adalah orang yang dapat menyelesaikan bagian kedua dari karya Plücker, “ Neue
Géometrie des Raumes ” dan ini menyebabkan dia bekerja dan berkenalan
dengan Clebsch. Clebsch telah dipindahkan ke Göttingen pada 1868 dan selama tahun 1869 Klein melakukan kunjungan ke Berlin, Paris dan Göttingen. Pada bulan Juli 1870 Klein berada di Paris saat Bismarck seorang rektor Prussian mempublikasikan sebuah pesan provokatif untuk pemerintahan Perancis. Perancis menyatakan perang melawan Prusia pada 19 Juli dan Klein merasa dia
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Setelah lima tahun di Technische Hochschule, Munich, Klein mendapatkan jabatan di Leipzig. Tahun 1880 ke 1886 Klein menghabiskan di Leipzig dimana ada banyak cara yang fundamental untuk mengubah hidupnya. Pada tahun 1886, Klein mendapatkan jabatan di Universitas Göttingen. Dia mengajar di Göttingen sampai ia pensiun pada tahun 1913 tetapi dia berusaha untuk kembali mendirikan Göttingen sebagai pusat penelitian matematika terkemuka di dunia. Pada tahun 1913, Klein pensiun. Namun ia melanjutkan mengajar matematika di rumahnya selama Perang Dunia I. Kontribusi Klein dalam geometri, yaitu menerbitkan dua kertas yang disebut Non-
Euclidean Geometry dimana ia menunjukkan bahwa untuk mempertimbangkan geometri Euclid dan non-Euclid sebagai sebuah kasus khusus projective dengan permukaan tertentu berbentuk kerucut. 8. Henri Poincare (29 April 1854
–
17 Juli 1912)
Henri Poincare lahir di Cite Ducale, Nancy, Meurtheet-Moselle. Ayahnya bernama Leon Poincaré dan ibunya bernama Eugénie Launois. Ayahnya seorang profesor di bidang obat-obatan di Nancy. Pada tahun 1862 Henri masuk Universitas Lycée, Nancy (sekarang berganti nama yang Lycée Henri Poincaré sebagai tanda kehormatan pada Poincare). Ia menghabiskan sebelas tahun di Lycée dan selama itu, ia menjadi salah satu siswa terbaik di setiap mata pelajaran. Pada tahun 1873, Poincaré masuk ke École Polytechnique dan lulus pada tahun 1875. Sementara ia tidak berhasil di École Polytechnique, ia mencoba untuk belajar piano. Poincaré gemar membaca dan ia mempunyai memori yang luar biasa.
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Kontribusinya di berbagai disiplin cabang, seperti: matematika, mekanik, mekanik cairan, teori relativitas dan falsafah sains. Sebelum usia 30 dia mengembangkan konsep fungsi automorphic. Poincaré meninggal pada usia 58 tahun. Pemakamannya dihadiri oleh banyak orang penting dari bidang sains dan politik, seperti: Presiden Senat dan sebagian besar anggota departemen, delegasi dari Akademi Perancis, dan banyak lembaga publik lainnya. Sang Pangeran Monako pun hadir, Bei dari Tunis diwakili oleh dua orang anak-anak, dan Pangeran Roland Bonaparte sebagai Presiden Paris Geografis Masyarakat.
B. Geometri Lobachevsky Sekarang, diperkenalkan geometri non-Euclid dari Bolyai, dan Lobachevsky, sebagai teori formal yang mendasarkan pada beberapa postulat. Teori ini dinakaman Geometri Lobachevsky untuk memudahkan dan menandai karya Lobachevsky. Geometri Lobachevsky dapat digolongkan pada geometri netral dengan memandang bahwa setiap segitiga jumlah besar sudutnya kurang dari 180o. meskipun demikian, kita lebih suka mengikuti sejarah perkembangannya dan mempelajarinya secara langsung dalam hubungannya dengan postulat kesejajaran Euclid. Jadi, untuk menggolongkan pada geometri Lobachevsky hanyalah dengan menerima semua postulat geometri Euclid dengan membuang
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seperti jarak, ketegak-lurusan, atau luas. Teorema tersebut mengenai kedudukan atau sifat garis. Teorema 1
Sebarang garis seluruhnya berada di dalam sudut tertentu. B’
A
m
P A’
B l
n
Q
Gambar 1
Bukti:
Misalkan diketahui garis l dan titik P di luar l .
Menurut postulat kesejajaran Lobachevsky maka ada garis m dan n yang melalui titik P dan sejajar l .
Garis m dan n membagi bidang tersebut menjadi 4 daerah dimana masingmasing daerah merupakan bagian dalam suatu sudut, yaitu: A’PB, A’PB’, APB’,
dan
APB
dengan P terletak diantara A dan A’ pada garis m dan
diantara B dan B’ pada garis n.
Misalkan titik Q pada l dan karena l tidak memotong m dan n maka titik Q tidak terletak pada m atau n
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Bukti:
B’
A
m
P R
A’
B l
n
Q
Gambar 2
Misalkan diketahui garis l dan titik P di luar l .
Gunakan Teorema 1 dan misalkan sebarang titik R di dalam daerah APB.
Buat garis yang melalui titik P dan R.
PR kecuali titik P seluruhnya termuat dalam daerah APB dan A’PB’ .
PR tidak memotong garis l yang termuat dalam A’PB sehingga PR // l .
Karena terdapat tak berhingga garis seperti PR maka teorema akibat terbukti.
Jadi, ada tak berhingga garis yang sejajar dengan suatu garis yang melalui suatu titik di luar garis itu. Perbedaan antara Geometri Euclid dan Geometri Lobachevsky dipandang dari sifat-sifat Non-Metrical
Berdasarkan Teorema 1 “Sebarang garis seluruhnya berada di dalam sudut
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Hal ini seharusnya tidaklah terlalu mengherankan, karena postulat kesejajaran Euclid (dalam bentuk postulat Playfair) dan postulat kesejajaran Lobachevsky memang berbeda sifat khusus grafiknya. 3. Sanggahan
Teorema 1 valid secara abstrak, tetapi tidak sesuai dengan kenyataan fisiknya. Konklusi tersebut memang secara logis diperoleh dari postulat kesejajaran Lobachevsky, tetapi asumsi itu secara fisik keliru. Yang diperlukan bagi seseorang untuk berfikir secara matematis adalah asumsi-asumsi (postulatpostulat) yang secara logis dapat menghasilkan konklusi (teorema). Viliditas argument matematis tidak bergantung pada benar atau salahnya asumsi dasar yang digunakannya. Meskipun demikian, wajarkah kita memilih asumsi yang akan menimbulkan kekeliruan jika diterapkan pada dunia nyata? Jawabannya tidak mungkin dengan “YA” atau “TIDAK” saja, tetapi harus ada beberapa penjelasan,
yaitu: 1. Ahli matematika seharusnya bebas memilih postulat dan mempelajari konsekuensinya, bebas dari pertimbangan kegunaan praktisnya maupun validitas empirisnya. 2. Proposisi matematika itu abstrak, untuk mengujinya secara empiris harus
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Dengan proses berpikir yang sama, kita yakin bahwa postulat kesejajaran Lobachevsky secara empiris adalah salah. Bisakah kita menyatakan “Jika
diketahui garis (secara fisik) l dan titik P (secara fisik) di luar l , maka ada garis m (secara fisik) yang tidak memotong l tetapi melalui P yang tidak terletak pada l ?” Bagaimana kita menguji hal itu? Apakah dengan menggunakan tali? Garis -
garis di papan tulis? atau sinar lampu? Merupakan hal yang sulit jika membuktikan secara empiris bahwa hanya ada satu garis yang seperti itu. Andaikan saja ada garis yang memenuhi sifat kesejajaran Lobachevsky, misalkan garis m. m
P
l
m’
Gambar 4
Apakah kita benar-benar tahu sifat-sifat fisik garis m sehingga dapat menunjukkan hanya ada satu garis seperti itu? Misalkan m’ adalah garis (secara fisik) yang melalui P dan membentuk sudut yang sangat kecil dengan m; dapatkah kita nyatakan bahwa secara fisik m’ pasti
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C
Bukti:
Perhatikan ABC .
Menurut Teorema Saccheri-Legendre: A
Kedua ruas dikurangi dengan C, diperoleh: A
+ B + C 180 + B
180 - C
B
A Gambar 5
Lemma tsb berlaku karena sudut luar C sama dengan 180 - C
(terbukti)
Lemma 2
Misalkan diketahui diketahui garis l, titik P di luar l, dan titik Q pada l. Misalkan diberikan sisi PQ, maka ada titik R pada l yang terletak satu pihak dengan PQ sedemikian hingga PRQ PRQ adalah sudut terkecil seperti yang diinginkan.
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Misalkan sudut luar
PQR1
diperoleh: b1+b1= 2b1 b
di Q adalah b, maka menurut Lemma 1,
….. (1)
Dengan langkah yang sama, buat segitiga baru. Perpanjang QR1 melalui R1 dan R2 sedemikian hingga R1R2=PR1.
Tarik PR2 maka PR1R2 adalah sama kaki dan R1PR2=PR2R1
= PR2Q = b2
sehingga berdasarkan Lemma 1, diperoleh: b2+b2= 2b2 b1
berarti
sesuai dengan persamaan (1) diperoleh:
Ulangi langkah sebelumnya sebanyak n kali sehingga diperoleh titik Rn pada l dan di sebelah kanan sisi PQ sedemikian hingga
.
Dengan memilih n cukup besar maka diperoleh
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P m
a
Y
90 - a
X
n
l Q
R
Gambar 9
Perhatikan Gambar 9. Misalkan X titik pada n sedemikian hingga
QPX
lancip dan Y titik pada m dan di sebelah kanan sisi PQ seperti X . Misalkan XPY = a maka QPX = 90 - a.
Misal R pada l dan berada di sebelah kanan sisi PQ, sedemikian hingga PRQ<
a (menggunakan Lemma 2)
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ada garis lain PX // l QPX < 90 . Misalkan R sebarang titik pada l di sebelah kanan PQ seperti X. Jika R menjauhi PQ sampai tak terhingga, maka seperti pada geometri Euclid. Tetapi
QPR
QRP
mendekati 0
tidak mendekati 90 , karena
QPR
selalu kurang dari QPX. Jadi, jika R cukup jauh maka jumlah besar sudut PQR kurang dari 180 . P m
X
l Q
R
Gambar 11
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5. Segiempat Saccheri dalam Geometri Hiperbolik C
D
Saccheri menarik garis yang tegak lurus pada ujungujung dua buah segmen garis yang saling sejajar. Bangun yang terbentuk ini disebut sebagai segiempat saccheri (Saccheri Quadrilateral). Pada bagian ini kita akan
B
A
mempelajari beberapa sifat dari segiempat Saccheri.
Gambar 12
Definisi 1
Segiempat saccheri adalah segiempat ABCD dengan AB sebagai alasnya, AD dan BC adalah kaki-kakinya sedemikian sehingga AD = BC. sudut siku-siku. A dan B dinamakan sudut alas dan sudut puncak
B
merupakan
C dan D
dinamakan
A
dan
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Teorema 5
Pada segiempat Saccheri, sudut-sudut atasnya lancip. Bukti:
Berdasarkan Akibat 1 Teorema 3 , yaitu jumlah besar sudut-sudut dalam segiempat kurang dari 360 maka A
+ B + C + D < 360
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7. Adakah segitiga-segitiga yang sebangun dalam Geometri Lobachevsky?
Berikut akan ditunjukkan bahwa tidak ada segitiga-segitiga yang sebangun dalam geometri hiperbolik, yang ada hanyalah segitiga-segitiga yang kongruen. Teorema 8
Dua
segitiga
dikatakan
kongruen
jika
bersesuaian sama. A
sudut-sudut
yang
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maka sudut-sudut yang bersesuaian sama besar dan oleh karena itu kedua segitiga tersebut pasti kongruen. Sehingga secara umum, dua segitiga yang sebangun pasti kongruen dan juga mempuyai ukuran ukuran yang sama. 8. Teori Luas Lobachevsky
Ukuran luas dalam geometri hiperbolik berbeda dengan geometri Euclid yang menggunakan satuan luas persegi, karena dalam geometri hiperbolik tidak
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Definisi ini juga berlaku untuk sebarang geometri netral. Dalam geometri Euclid telah kita kenal rumus luas segitiga, yaitu
yang
menghasilkan sebuah fungsi luas dengan memasangkan setiap segitiga dengan
bilangan
Selanjutnya, kita amati sifat aditif dari fungsi luas yang dapat dikembangkan sampai sejumlah suku-suku yang berhingga.
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Definisi 4 A + B + C) Defect ABC = 180 – ( C) A, B, B, dan C diambil dari besar derajat dari sudut-sudut yang
dimaksud. Jadi, defect suatu segitiga adalah bilangan real bukan bilangan derajat .
Defect suatu segitiga berlaku seperti pengukuran luas.
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