Saturday, December 3, 2011
Galapagos Versus Krakatau
Peta Terakhir Jelang Letusan Dahsyat Krakatau
KOMPAS.com - Setelah 200 tahun tertidur, pada 19 Mei 1883, Batavia (Jakarta) dikejutkan dengan dentuman keras, melebihi bunyi meriam terkeras. Kaca-kaca jendela bergetar hebat bahkan jam dinding berhenti berdetak karena sapuan gelombang kejut. Abu dan batu apung berjatuhan di Selat Sunda, menggiring orang untuk melongok ke puncak Perbuatan, salah satu puncak di pulau gunung api Krakatau, yang tiba-tiba meletus.
Namun, setelah kegaduhan itu, Krakatau kembali tenang. Pulau dengan tiga kawah itu tidur tenang, dikitari laut biru yang dalam. Setelah hari keempat berlalu dengan damai, Gubernur Jenderal Hindia Belanda Frederik s'Jacob menyimpulkan saat yang bagus untuk melihat Krakatau dari dekat, melihat apa yang terjadi, dan yang lebih penting lagi: untuk menyimpulkan apakah kejadian serupa bisa terulang kembali. Dia mengutus insinyur pertambangan, AL Schuurman, pergi ke sana.
Berbeda dengan kekhawatiran s'Jacob, perusahaan pelayaran The Netherlands Indies Steamship Company melihat Krakatau sebagai potensi besar untuk mendatangkan turis sehingga dengan sigap menyodorkan kapal wisata, Gouverneur-Generaal Loudon. "Pada Sabtu, 26 Mei, perwakilan perusahaan menempelkan pengumuman di klub Harmonie dan Concordia, mengiklankan 'wisata menyenangkan' dan mengumumkan harga yang kompetitif sebesar hanya 25 guilder," tulis Winchester.
Pada Minggu sore, kapal uap berbobot mati 1.239 ton itu terisi penuh dengan 86 penumpang dan Schuurman berada di antara mereka sebagai wakil dari pemerintah. Setelah berlayar semalaman, kapten Loudon, TH Lindeman, membuang sauh jauh dari pulau itu. Dia meminjamkan perahu kepada Schuurman. Ditemani beberapa orang yang berani dan penuh rasa ingin tahu, Schuurman mendekati pulau dengan susah payah.
"Dengan mengikuti jejak orang yang paling berani atau mungkin yang paling tolol, kami mendaki lebih jauh tanpa halangan apa pun selain abu yang ambles di bawah kaki kami. Jalannya berada di atas bukit dari mana kami bisa melihat beberapa pokok pohon yang patah mencuat dari lapisan abu, beberapa tonggak menunjukkan bahwa cabang-cabangnya direnggut dengan paksa," tulis Schuurman.
Kelompok kecil ini terus merangsek naik dengan nekad hingga mendekati dasar kawah, yang menurut Schuurman tertutup oleh "kerak buram berkilat-kilat," yang kadang-kadang membara merah dan mengeluarkan ”gulungan asap dalam gelembung-gelembung raksasa yang banyak tetapi rapat”. Schuurman akhirnya kembali ke Loudon setelah Lindeman berkali-kali membunyikan klakson.
Dua bulan kemudian Krakatau berangsur dilupakan. Hingga pada 11 Agustus, kapten angkatan darat Belanda, HJG Ferzenaar, diperintahkan menyurvei Krakatau untuk kepentingan topografi militer. Dia melewatkan dua hari di sana dan mencatat ada 14 lubang semburan di atas pulau itu. Ia membuat peta pulau itu secara detial, termasuk titik-titik berwarna merah yang menjadi pusat semburan.
Dia memberi catatan bahwa survei yang lebih rinci "harus menunggu sampai nanti, sebab pengukuran di sana masih sangat berbahaya; setidaknya, saya tidak akan suka menerima tanggung jawab mengirimkan seorang surveyor."
Namun, Krakatau tidak pernah bisa dipetakan lagi. Pada 27 Agustus 1883, pulau ini meledak dan hancur berkeping-keping. Peta Pulau Krakatau yang dibuat Ferzenaar adalah yang terakhir yang pernah dibuat.
Ledakan berkekuatan 21.574 kali bom atom (De Neve, 1984) itu tak hanya menghancurkan tubuh Pulau Krakatau. Kehancuran juga melanda pesisir Banten dan Lampung. Gelombang awan panas dan tsunami melanda, menghancurkan desa-desa di pesisir Banten dan Lampung, serta menewaskan lebih dari 36.000 jiwa.
Kengerian itu digambarkan oleh Muhammad Saleh dalam Syair Lampung Karam, satu-satunya laporan pandangan mata yang dibuat pribumi tentang letusan Krakatau. Muhammad Saleh lewat bait syairnya menggambarkan di atas langit terlihat seperti bunga api beterbangan seperti bahala yang diturunkan Tuhan dan membuat hati takut bukan kepalang. Kegelapan menyelimuti, guncangan gempa tiada henti, dan datang gelombang menghanyutkan. "Besar gelombang tidak terperi, lalulah masuk ke dalam negeri, berlarian orang ke sana kemari...," tulis Muhammad Saleh.
Petaka Krakatau itu menambah derita rakyat yang beratus tahun disengsarakan ekonomi kolonial dan priyayi pribumi yang mengisap. "Tak disangsikan lagi bahwa wabah penyakit ternak dan wabah demam, serta kelaparan yang diakibatkannya, dan letusan Gunung Krakatau yang menyusul, telah menjadi pukulan hebat bagi penduduk," tulis Sartono Kartodirdjo, dalam buku Pemberontakan Petani di Banten 1888.
Menurut sejarawan terkemuka ini, "... letusan Gunung Krakatau menyebabkan luas tanah yang tidak dapat digarap menjadi lebih besar lagi, terutama di bagian barat afdeling Caringin dan Anyer." Kondisi kesengsaraan yang kemudian bertemu dengan gerakan sosial-keagamaan ini menjadi pemantik kesadaran rakyat untuk melawan Belanda, yang dianggap sebagai pendosa dan biang dari segala kesengsaraan itu.
Dua bulan setelah letusan Krakatau, kerusuhan pecah di Serang. Seorang serdadu Belanda ditikam, pelakunya kabur di tengah keramaian. Kejadian berulang sebulan kemudian. Serentetan perlawanan terhadap Belanda terus dilakukan hingga pada Juli 1888 muncullah pemberontakan petani Banten.(Tim Penulis: Ahmad Arif, Indira Permanasari, Yulvianus Harjono, C Anto Saptowalyono. Litbang: Rustiono)
Thursday, December 1, 2011
Kenali Langkah Kecil Cegah Perubahan Iklim
JAKARTA, KOMPAS.com - Green Festival dibuka di Jakarta, Kamis (1/12/2011) di Bentara Budaya Jakarta. Pameran yang terselenggara keempat kalinya ini telah diadakan di Bandung 24-26 November, berlangsung di Jakarta hingga 4 Desember 2011 dan akan digelar di Surabaya 9-11 Desember 2011.
Datang ke Green Festival, pengunjung bisa merefleksikan masalah perubahan iklim serta mendapatkaninsight cara menyikapinya dari skala kecil. Di Green Festival, terdapat zona yang merepresentasikan persoalan pemicu perubahan iklim, yakni air, pohon, energi, sampah, dan transportasi.
Merefleksikan masalah perubahan iklim, di Zona Iklim, pengunjung bisa melihat patung es pinguin bersebelahan dengan beruang kutub. Patung yang perlahan mencair ini merefleksikan konsekuensi pemanasan global dan perubahan iklim bagi lingkungan kutub.
Di Zona Transportasi, pengunjung bisa menyadari bahwa pemakaian kendaraan bermotor bisa berdampak buruk bagi Bumi. Sepeda dipamerkan sebagai alternatif. Di Green Festival, ada pula stand Komunitas Bike to Work yang mengajak menggunakan sepeda dalam aktivitas sehari-hari.
Di Zona Air, ditunjukkan pesan lewat kartun bahwa konsumsi air yang berlebihan juga berdampak buruk bagi lingkungan. Pengunjung diajak mengkonsumsi air secara efisien. Sementara di Zona Sampah, pengunjung diajak mengurangi penggunaan sampah plastik.
Ada beberapa stand yang bisa dikunjungi di Green Festival. salah satunya adalah stand Komunitas Halaman organik yang mengajak memanfaatkan halaman yang terbatas untuk menanam tumbuhan bermanfaat. Tujuannya, mencapai kesehatan raga, jiwa, dan lingkungan.
Green Festival kali ini mengambil tema "Gaya Hidupku untuk Bumi". Nugroho F Yudho, Ketua Pelaksana Green Festival 2011 mengatakan, "Kampanye ini sekaligus mengajak masyarakat menjadikan kegiatan mengurangi efek pemanasan global sebagai gaya hidupnya sehari-hari."
Green Festival terselenggara berkat kerjasama beberapa korporat, diantaranya PT Pertamina PERSERO, Sinarmas APP, PT Tirta Investama (Danone AQUA), Tupperware, Kompas, Metro TC dan jaringan data FeMale Indonesia.
Friday, November 18, 2011
Satu Planet Terlempar dari Tata Surya?
TEXAS, KOMPAS.com — Selama ini, astronom memercayai bahwa Tata Surya memiliki 4 planet raksasa, yakni Jupiter, Saturnus, Neptunus, dan Uranus. Namun, analisis terbaru menunjukkan bahwa Tata Surya dengan 4 planet raksasa adalah janggal. Kemungkinan, Tata Surya memiliki 5 planet raksasa.
David Nesvorny dari Southwest Research Institute di San Antonio, Texas, Amerika Serikat, adalah ilmuwan yang mengungkapkan pendapat baru itu. Untuk sampai pada kesimpulannya, Nesvorny membuat 6.000 simulasi komputer yang menganalisis obyek di sekitar Neptunus dan kawah Bulan.
Berdasarkan analisis Nesvorny, Tata Surya hanya memiliki 2,5 persen kemungkinan menjadi seperti sekarang jika sejak awal hanya memiliki 4 planet raksasa. Sementara ada 10 kali lebih besar kemungkinan bagi Tata Surya menjadi seperti saat ini jika awalnya memiliki 5 planet raksasa.
Planet raksasa kelima itu dipercaya terlempar dari Tata Surya. Saat Tata Surya berusia 600 tahun, ada periode ketidakstabilan orbit planet. Ada planet yang berpindah ke Sabuk Kuiper, wilayah dekat Neptunus, dan ada yang berpindah ke dalam.
Jupiter yang memiliki pengaruh gravitasi kuat diketahui adalah salah satu biang keladinya. Orbit Jupiter bisa berubah tiba-tiba dan satu planet raksasa terlempar dari Tata Surya karenanya. Sementara planet-planet lain tetap bertahan.
Nesvorny mengatakan, "Kemungkinan Tata Surya memiliki lebih dari 4 planet raksasa dan melemparkan beberapa di antaranya, terkesan cocok dengan penemuan banyaknya planet yang ada di wilayah antarbintang, yang menunjukkan bahwa terlemparnya planet adalah hal yang umum."
Pada Space.com, Jumat (11/11/2011) lalu, Nesvorny mengatakan bahwa temuan ini memunculkan pertanyaan. Salah satunya tentang planet Mars dan planet Super Bumi, apakah mereka terbentuk di Tata Surya Luar (setelah orbit Mars) lalu tereliminasi.
Pendapat Nesvorny memang fantastis dan membuat orang tercengang. Namun, ia sendiri merasa bahwa pendapatnya masih harus diuji kebenarannya dengan serangkaian penelitian. Hasil analisis Nesvorny dipublikasikan di edisi online Astrophysical Journal Letters minggu lalu.
Monday, November 7, 2011
Logam Standar yang Disimpan
Sejak tahun 1889, kilogram telah didefinisikan sebagai massa logam standar yang disimpan International Bureau of Weight and Measure (BIPM–dalam bahasa Perancis) yang disimpan di Sevres, Paris.
Logam standar tersebut tersusun atas 10 persen iridium dan 90 persen platinum dan begitu dibanggakan. Logam itu disimpan di tiga gelas kaca di Pavilion de Breteuil.
Pada tahun 1992, logam itu ternyata berubah. Pengukuran menunjukkan bahwa massa logam tersebut telah berubah sebesar 50 mikrogram, setara dengan butir pasir berukuran 0,4 milimeter. "Sebenarnya, kami tidak yakin apakah itu bertambah atau berkurang," kata Alain Picard, Direktur Mas Department BIPM, seperti dikutip AFP, Minggu (6/11/2011).
"Perubahan mungkin disebabkan efek permukaan, kehilangan gas pada logam, atau kontaminan," kata Picard.
Perubahan telah memicu rasa penasaran di kalangan ilmuwan. Kilogram merupakan salah satu satuan standar (Satuan Internasional/SI) yang disepakati untuk mempermudah pengukuran massa suatu besaran. Satuan internasional lain misalnya meter untuk panjang.
Tak seperti SI lain, kilogram adalah satuan terakhir yang didefinisikan dengan material. SI selanjutkan didefinisikan dengan konstanta Planck yang ditemukan pada 1899. Kontansta Planck dilambangkan dengan "h". Konstanta tersebut merujuk pada paket energi terkecil, atau quanta, yang dipertukarkan oleh dua partikel.
Dengan berubahnya massa logam standar, para ilmuwan pada 21 Oktober 2011 dalam General Conference on Weights and Measures sepakat untuk mengubah acuan pengukuran kilogram.
Nantinya, kilogram tidak lagi diukur berdasarkan massa logam yang disimpan di Sevres, tetapi berdasarkan konstanta Planck. Namun, perubahan ini mungkin tak akan dilaksanakan sebelum 2014. Para ilmuwan akan melakukan evaluasi terlebih dahulu untuk memastikan akurasi pengukuran hingga 20 bagian per miliar.
Lalu, jika nanti diaplikasikan, apakah kilogram akan berubah dan kita harus mengganti timbangan yang kita punya? Ilmuwan mengatakan, dalam keseharian, tidak akan ada perubahan apa pun. "Namun, perubahan akan memberi dampak pada pengukuran yang luar biasa akurat yang dilaksanakan oleh laboratorium yang sangat terspesialisasi," demikian press release konferensi seperti dikutip AFP.
Saturday, October 29, 2011
Kehidupan Zaman Es Masih Berlangsung di China
Peneliti dari Chinese Academy of Science dan Natural History Museum Inggris menyatakan telah mengidentifikasi tujuh spesies jelatang dari Provinsi Guaxi dan Yunnan. Jelatang atau dalam bahasa Inggris disebut poison ivy adalah jenis tumbuhan berbulu halus yang bisa menyebabkan gatal di kulit.
Seharusnya di kedua provinsi tersebut terdapat tumbuhan tropis. Namun, tujuh spesies yang ditemukan tidak mirip dengan tumbuhan tropis. Jelatang tersebut ditemukan di tempat-tempat gelap, di mana jarang ada sinar matahari.
Fakta yang ada, kemungkinan jelatang untuk bisa hidup di kegelapan hanya 0,02 persen. Hal ini sangat tidak mungkin terjadi di zaman sekarang ini ehingga kemungkinan zaman es masih berlangsung di goa tersebut.
Lingkungan yang memungkinkan jelatang macam ini bisa hidup adalah pada zaman es. Goa dan jelatang ini merupakan sisa-sisa dari kehidupan zaman es yang masih ada di planet ini. Selain itu, ada kemungkinan lainnya yang memungkinkan hal ini terjadi, yaitu evolusi pada jelatang. Akan tetapi, umur dari goa tersebut hanya satu juta tahun yang berarti bahwa evolusi pada jelatang terjadi sangatlah cepat.
Peneliti mengatakan bahwa ini adalah contoh evolusi yang sangat cepat, mereka akan meneliti hal ini secepatnya.(National Geographic Indonesia/Arief Sujatmoko)
Friday, September 16, 2011
HUTCHISON EFFECT (ANTI GRAVITASI, FUSI DAN PEMANASAN ANOMALI)
Ini adalah hasil translate ane sendiri berdasarkan dari link diatas, mohon diluruskan kalo ada yang salah, thanks ..^Irfan Rasyid^
oleh Mark A. Solis
1 Timbulnya efek levitasi (daya angkat/antigravitasi) pada benda-benda berat.
2. Perpaduan/fusi dua benda yang berbeda seperti logam dengan kayu (persis seperti yang digambarkan dalam film, “Eksperimen Philadelphia”)
3. Pemanasan anomali (tidak lazim) pada logam dan tanpa menimbulkan terbakarnya benda lain yang berdekatan
4. Perekahan spontan pada logam (yang terpisah dengan bergeser secara menyamping), dan kemudian terpisah untuk sementara, dan akhirnya berubah kedalam struktur ‘kristal’ namun tetap memiliki sifat fisik sebagai ‘logam’.
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Copyright (c) 1999 oleh Mark Solis A.
illus-terasi: Jakarta Tahun 2210 :p
Tapi sebagian bilang udah ditest dan ‘berhasil’, prinsipnya memanipulasi gelombang radio dengan sinyal ponsel, batere, kartu & koin (sebagai media) dan botol pepsi:
Fibonacci Numbers in Nature & the Golden Ratio
Golden Ratio & Golden Section : : Golden Rectangle : : Golden SpiralGolden Ratio & Golden Section
In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.
Expressed algebraically:
The golden ratio is often denoted by the Greek letter phi (Φ or φ).
The figure of a golden section illustrates the geometric relationship that defines this constant. The golden ratio is an irrational mathematical constant, approximately 1.6180339887. Golden Rectangle
A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: j (one-to-phi),
that is, 1 : or approximately 1:1.618.
Golden Spiral
In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to j, the golden ratio. Specifically, a golden spiral gets wider (or further from its origin) by a factor of jfor every quarter turn it makes.
Successive points dividing a golden rectangle into squares lie on a logarithmic spiral which is sometimes known as the golden spiral. Image Source: http://mathworld.wolfram.com/GoldenRatio.html Golden Ratio in Architecture and Art
Many architects and artists have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. [Source:Wikipedia.org]
Here are few examples:
Parthenon, Acropolis, Athens. This ancient temple fits almost precisely into a golden rectangle. Source: http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm The Vetruvian Man"(The Man in Action)" by Leonardo Da Vinci We can draw many lines of the rectangles into this figure. Then, there are three distinct sets of Golden Rectangles: Each one set for the head area, the torso, and the legs. Image Source >>
Leonardo's Vetruvian Man is sometimes confused with principles of "golden rectangle", however that is not the case. The construction of Vetruvian Man is based on drawing a circle with its diameter equal to diagonal of the square, moving it up so it would touch the base of the square and drawing the final circle between the base of the square and the mid-point between square's center and center of the moved circle:
Golden Ratio in Nature
Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.[38] Zeising wrote in 1854:
Examples:
Click on the picture for animation showing more examples of golden ratio. Source: http://www.xgoldensection.com/xgoldensection.html A slice through a Nautilus shell reveals golden spiral construction principle. FIBONACCI NUMBERSAbout Fibonacci
Fibonacci Numbers
The sequence, in which each number is the sum of the two preceding numbers is known as the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ... (each number is the sum of the previous two).
The ratio of successive pairs is so-called golden section (GS) - 1.618033989 . . . . .
whose reciprocal is 0.618033989 . . . . . so that we have 1/GS = 1 + GS.
The Fibonacci sequence, generated by the rule f1 = f2 = 1 , fn+1 = fn + fn-1,
is well known in many different areas of mathematics and science. Pascal's Triangle and Fibonacci Numbers
The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui (about 500 years earlier, in fact) and the Persian astronomer-poet Omar Khayyám.
Pascal's Triangle is described by the following formula:
where is a binomial coefficient.
The "shallow diagonals" of Pascal's triangle
sum to Fibonacci numbers.
It is quite amazing that the Fibonacci number patterns occur so frequently in nature( flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal "laws of nature". Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves The Fibonacci numbers are also found in the family tree of honeybees.
Fibonacci and Nature
Plants do not know about this sequence - they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144. Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks.
Why do these arrangements occur? In the case of leaf arrangement, or phyllotaxis, some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. In the case of close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of space. This is well described in several books listed here >>
So nature isn't trying to use the Fibonacci numbers: they are appearing as a by-product of a deeper physical process. That is why the spirals are imperfect.
The plant is responding to physical constraints, not to a mathematical rule.
The basic idea is that the position of each new growth is about 222.5 degrees away from the previous one, because it provides, on average, the maximum space for all the shoots. This angle is called the golden angle, and it divides the complete 360 degree circle in the golden section, 0.618033989 . . . .
Examples of the Fibonacci sequence in nature.
Petals on flowers*
Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number:
Some species are very precise about the number of petals they have - e.g. buttercups, but others have petals that are very near those above, with the average being a Fibonacci number.
* Read the entire article here:
http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm
Related Links:
http://britton.disted.camosun.bc.ca/jbfunpatt.htm Flower Patterns and Fibonacci Numbers
Why is it that the number of petals in a flower is often one of the following numbers: 3, 5, 8, 13, 21, 34 or 55? For example, the lily has three petals, buttercups have five of them, the chicory has 21 of them, the daisy has often 34 or 55 petals, etc. Furthermore, when one observes the heads of sunflowers, one notices two series of curves, one winding in one sense and one in another; the number of spirals not being the same in each sense. Why is the number of spirals in general either 21 and 34, either 34 and 55, either 55 and 89, or 89 and 144? The same for pinecones : why do they have either 8 spirals from one side and 13 from the other, or either 5 spirals from one side and 8 from the other? Finally, why is the number of diagonals of a pineapple also 8 in one direction and 13 in the other?
Are these numbers the product of chance? No! They all belong to the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (where each number is obtained from the sum of the two preceding). A more abstract way of putting it is that the Fibonacci numbers fn are given by the formula f1 = 1, f2 = 2, f3 = 3, f4 = 5 and generally f n+2 = fn+1 + fn . For a long time, it had been noticed that these numbers were important in nature, but only relatively recently that one understands why. It is a question of efficiency during the growth process of plants.
The explanation is linked to another famous number, the golden mean, itself intimately linked to the spiral form of certain types of shell. Let's mention also that in the case of the sunflower, the pineapple and of the pinecone, the correspondence with the Fibonacci numbers is very exact, while in the case of the number of flower petals, it is only verified on average (and in certain cases, the number is doubled since the petals are arranged on two levels).
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Let's underline also that although Fibonacci historically introduced these numbers in 1202 in attempting to model the growth of populations of rabbits, this does not at all correspond to reality! On the contrary, as we have just seen, his numbers play really a fundamental role in the context of the growth of plants
THE EFFECTIVENESS OF THE GOLDEN MEAN
The explanation which follows is very succinct. For a much more detailed explanation, with very interesting animations, see the web site in the reference.
In many cases, the head of a flower is made up of small seeds which are produced at the centre, and then migrate towards the outside to fill eventually all the space (as for the sunflower but on a much smaller level). Each new seed appears at a certain angle in relation to the preceeding one. For example, if the angle is 90 degrees, that is 1/4 of a turn, the result after several generations is that represented by figure 1.
Of course, this is not the most efficient way of filling space. In fact, if the angle between the appearance of each seed is a portion of a turn which corresponds to a simple fraction, 1/3, 1/4, 3/4, 2/5, 3/7, etc (that is a simple rational number), one always obtains a series of straight lines. If one wants to avoid this rectilinear pattern, it is necessary to choose a portion of the circle which is an irrational number (or a nonsimple fraction). If this latter is well approximated by a simple fraction, one obtains a series of curved lines (spiral arms) which even then do not fill out the space perfectly (figure 2).
In order to optimize the filling, it is necessary to choose the most irrational number there is, that is to say, the one the least well approximated by a fraction. This number is exactly the golden mean. The corresponding angle, the golden angle, is 137.5 degrees. (It is obtained by multiplying the non-whole part of the golden mean by 360 degrees and, since one obtains an angle greater than 180 degrees, by taking its complement). With this angle, one obtains the optimal filling, that is, the same spacing between all the seeds (figure 3).
This angle has to be chosen very precisely: variations of 1/10 of a degree destroy completely the optimization. (In fig 2, the angle is 137.6 degrees!) When the angle is exactly the golden mean, and only this one, two families of spirals (one in each direction) are then visible: their numbers correspond to the numerator and denominator of one of the fractions which approximates the golden mean : 2/3, 3/5, 5/8, 8/13, 13/21, etc.
These numbers are precisely those of the Fibonacci sequence (the bigger the numbers, the better the approximation) and the choice of the fraction depends on the time laps between the appearance of each of the seeds at the center of the flower.
This is why the number of spirals in the centers of sunflowers, and in the centers of flowers in general, correspond to a Fibonacci number. Moreover, generally the petals of flowers are formed at the extremity of one of the families of spiral. This then is also why the number of petals corresponds on average to a Fibonacci number.
REFERENCES:
Source of the above segment:
http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html © Mathematics and Knots, U.C.N.W.,Bangor, 1996 - 2002 Fibonacci numbers in vegetables and fruit
Romanesque Brocolli/Cauliflower (or Romanesco) looks and tastes like a cross between brocolli and cauliflower. Each floret is peaked and is an identical but smaller version of the whole thing and this makes the spirals easy to see.
Brocolli/Cauliflower
© All rights reserved Image Source >>
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Human Hand
Every human has two hands, each one of these has five fingers, each finger has three parts which are separated by two knuckles. All of these numbers fit into the sequence. However keep in mind, this could simply be coincidence.
To view more examples of Fibonacci numbers in Nature explore our selection of related links>>.
Human Face
Knowledge of the golden section, ratio and rectangle goes back to the Greeks, who based their most famous work of art on them: the Parthenon is full of golden rectangles. The Greek followers of the mathematician and mystic Pythagoras even thought of the golden ratio as divine.
Later, Leonardo da Vinci painted Mona Lisa's face to fit perfectly into a golden rectangle, and structured the rest of the painting around similar rectangles.
Mozart divided a striking number of his sonatas into two parts whose lengths reflect the golden ratio, though there is much debate about whether he was conscious of this. In more modern times, Hungarian composer Bela Bartok and French architect Le Corbusier purposefully incorporated the golden ratio into their work.
Even today, the golden ratio is in human-made objects all around us. Look at almost any Christian cross; the ratio of the vertical part to the horizontal is the golden ratio. To find a golden rectangle, you need to look no further than the credit cards in your wallet.
Despite these numerous appearances in works of art throughout the ages, there is an ongoing debate among psychologists about whether people really do perceive the golden shapes, particularly the golden rectangle, as more beautiful than other shapes. In a 1995 article in the journal Perception, professor Christopher Green,
of York University in Toronto, discusses several experiments over the years that have shown no measurable preference for the golden rectangle, but notes that several others have provided evidence suggesting such a preference exists.
Regardless of the science, the golden ratio retains a mystique, partly because excellent approximations of it turn up in many unexpected places in nature. The spiral inside a nautilus shell is remarkably close to the golden section, and the ratio of the lengths of the thorax and abdomen in most bees is nearly the golden ratio. Even a cross section of the most common form of human DNA fits nicely into a golden decagon. The golden ratio and its relatives also appear in many unexpected contexts in mathematics, and they continue to spark interest in the mathematical community.
Dr. Stephen Marquardt, a former plastic surgeon, has used the golden section, that enigmatic number that has long stood for beauty, and some of its relatives to make a mask that he claims is the most beautiful shape a human face can have.
The Mask of a perfect human face
Egyptian Queen Nefertiti (1400 B.C.)
An artist's impression of the face of Jesus
based on the Shroud of Turin and corrected to match Dr. Stephen Marquardt's mask. Click here for more detailed analysis.
"Averaged" (morphed) face of few celebrities.
Related website: http://www.faceresearch.org/tech/demos/average
You can overlay the Repose Frontal Mask (also called the RF Mask or Repose Expression – Frontal View Mask) over a photograph of your own face to help you apply makeup, to aid in evaluating your face for face lift surgery, or simply to see how much your face conforms to the measurements of the Golden Ratio.
Visit Dr. Marquardt's Web site for more information on the beauty mask.
Related links:
Related websites
Fibonacci's Rabbits
The original problem that Fibonacci investigated, in the year 1202, was about how fast rabbits could breed in ideal circumstances. "A pair of rabbits, one month old, is too young to reproduce. Suppose that in their second month, and every month thereafter, they produce a new pair. If each new pair of rabbits does the same, and none of the rabbits dies, how many pairs of rabbits will there be at the beginning of each month?"
The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, etc.
The Fibonacci Rectangles and Shell Spirals
We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1).
We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles.
The next diagram shows that we can draw a spiral by putting together quarter circles, one in each new square. This is a spiral (the Fibonacci Spiral). A similar curve to this occurs in nature as the shape of a snail shell or some sea shells. Whereas the Fibonacci Rectangles spiral increases in size by a factor of Phi (1.618..) in a quarter of a turn (i.e. a point a further quarter of a turn round the curve is 1.618... times as far from the centre, and this applies to all points on the curve), the Nautilus spiral curve takes a whole turn before points move a factor of 1.618... from the centre.
A slice through a Nautilus shell
These spiral shapes are called Equiangular or Logarithmic spirals. The links from these terms contain much more information on these curves and pictures of computer-generated shells.
Here is a curve which crosses the X-axis at the Fibonacci numbers
The spiral part crosses at 1 2 5 13 etc on the positive axis, and 0 1 3 8 etc on the negative axis. The oscillatory part crosses at 0 1 1 2 3 5 8 13 etc on the positive axis. The curve is strangely reminiscent of the shells of Nautilus and snails. This is not surprising, as the curve tends to a logarithmic spiral as it expands.
Nautilus shell (cut)
© All rights reserved. Image source >> Proportion - Golden Ratio and Rule of Thirds
© R. Berdan 20/01/2004. Published with permission of the author
Proportion refers the size relationship of visual elements to each other and to the whole picture. One of the reasons proportion is often considered important in composition is that viewers respond to it emotionally. Proportion in art has been examined for hundreds of years, long before photography was invented. One proportion that is often cited as occurring frequently in design is the Golden mean or Golden ratio.
Golden Ratio: 1, 1, 2, 3, 5, 8, 13, 21, 34 etc. Each succeeding number after 1 is equal to the sum of the two preceding numbers. The Ratio formed 1:1.618 is called the golden mean - the ratio of bc to ab is the same as ab to ac. If you divide each smaller window again with the same ratio and joing their corners you end up with a logarithmic spiral. This spiral is a motif found frequently throughout nature in shells, horns and flowers (and my Science & Art logo).
The Golden Mean or Phi occurs frequently in nature and it may be that humans are genetically programmed to recognize the ratio as being pleasing. Studies of top fashion models revealed that their faces have an abundance of the 1.618 ratio.
Many photographers and artists are aware of the rule of thirds, where a picture is divided into three sections vertically and horizontally and lines and points of intersection represent places to position important visual elements. The golden ratio and its application are similar although the golden ratio is not as well known and its' points of intersection are closer together. Moving a horizon in a landscape to the position of one third is often more effective than placing it in the middle, but it could also be placed near the bottom one quarter or sixth. There is nothing obligatory about applying the rule of thirds. In placing visual elements for effective composition, one must assess many factors including color, dominance, size and balance together with proportion. Often a certain amount of imbalance or tension can make an image more effective. This is where we come to the artists' intuition and feelings about their subject. Each of us is unique and we should strive to preserve those feelings and impressions about our chosen subject that are different.
On analyzing some of my favorite photographs by laying down grids (thirds or golden ratio in Adobe Photoshop) I find that some of my images do indeed seem to correspond to the rule of thirds and to a lesser extent the golden ratio, however many do not. I suspect an analysis of other photographers' images would have similar results. There are a few web sites and references to scientific studies that have studied proportion in art and photography but I have not come across any systematic studies that quantified their results- maybe I just need to look harder (see link for more information about the use of the golden ratio:http://photoinf.com/Golden_Mean/).
In summary, proportion is an element of design you should always be aware of but you must also realize that other design factors along with your own unique sensitivity about the subject dictates where you should place items in the viewfinder. Understanding proportion and various elements of design are guidelines only and you should always follow your instincts combined with your knowledge. Never be afraid to experiment and try something drastically different, and learn from both your successes and failures. Also try to be open minded about new ways of taking pictures, new techniques, ideas - surround yourself with others that share an open mind and enthusiasm and you will improve your compositional skills quickly.
35 mm film has the dimensions 36 mm by 24 mm (3:2 ratio) - golden mean ration of 1.6 to 1 Points of intersection are recommended as places to position important elements in your picture.
Note: The above segment is part of the article COMPOSITION & the ELEMENTS of VISUAL DESIGN by Robert Berdan ( http://www.scienceandart.org/ )
© R. Berdan 20/01/2004 Published with permission of the author.
The entire article can be found here (PDF): http://www.scienceandart.org/photography/elementsofdesign.pdf |