iMath

A Trillion Triangles

Ghelvinny — Tue, 01 Dec 2009 00:57:35 +0000

September 22, 2009 — Mathematicians from North America, Europe, Australia, and South America have resolved the first one trillion cases of an ancient mathematics problem. The advance was made possible by a clever technique for multiplying large numbers. The numbers involved are so enormous that if their digits were written out by hand they would stretch to the moon and back. The biggest challenge was that these numbers could not even fit into the main memory of the available computers, so the researchers had to make extensive use of the computers’ hard drives.According to Brian Conrey, Director of the American Institute of Mathematics, “Old problems like this may seem obscure, but they generate a lot of interesting and useful research as people develop new ways to attack them.”

The problem, which was first posed more than a thousand years ago, concerns the areas of right-angled triangles. The surprisingly difficult problem is to determine which whole numbers can be the area of a right-angled triangle whose sides are whole numbers or fractions. The area of such a triangle is called a “congruent number.” For example, the 3-4-5 right triangle which students see in geometry has area 1/2 × 3 × 4 = 6, so 6 is a congruent number. The smallest congruent number is 5, which is the area of the right triangle with sides 3/2, 20/3, and 41/6.

The 3-4-5 triangle has area 6.

The first few congruent numbers are 5, 6, 7, 13, 14, 15, 20, and 21. Many congruent numbers were known prior to the new calculation. For example, every number in the sequence 5, 13, 21, 29, 37, …, is a congruent number. But other similar looking sequences, like 3, 11, 19, 27, 35, …., are more mysterious and each number has to be checked individually.

The calculation found 3,148,379,694 of these more mysterious congruent numbers up to a trillion.

Consequences, and future plans

Team member Bill Hart noted, “The difficult part was developing a fast general library of computer code for doing these kinds of calculations. Once we had that, it didn’t take long to write the specialized program needed for this particular computation.” The software used for the calculation is freely available, and anyone with a larger computer can use it to break the team’s record or do other similar calculations.

In addition to the practical advances required for this result, the answer also has theoretical implications. According to mathematician Michael Rubinstein from the University of Waterloo, “A few years ago we combined ideas from number theory and physics to predict how congruent numbers behave statistically. I was very pleased to see that our prediction was quite accurate.” It was Rubinstein who challenged the team to attempt this calculation. Rubinstein’s method predicts around 800 billion more congruent numbers up to a quadrillion, a prediction that could be checked if computers with a sufficiently large hard drive were available.

History of the problem

The congruent number problem was first stated by the Persian mathematician al-Karaji (c.953 – c.1029). His version did not involve triangles, but instead was stated in terms of the square numbers, the numbers that are squares of integers: 2=1, 2²=4, 3²=9, 5²=25, –> 1, 4, 9, 16, 25, 36, 49, …, or squares of rational numbers: 25/9, 49/100, 144/25, etc. He asked: for which whole numbers n does there exist a square a² so that a²-n and a²+n are also squares? When this happens, n is called a congruent number. The name comes from the fact that there are three squares which are congruent modulo n. A major influence on al-Karaji was the Arabic translations of the works of the Greek mathematician Diophantus (c.210 – c.290) who posed similar problems.

Al-Fakhri fi’l-jabr wa’l-muqabala, by al-Karaji.

A small amount of progress was made in the next thousand years. In 1225, Fibonacci (of “Fibonacci numbers” fame) showed that 5 and 7 were congruent numbers, and he stated, but did not prove, that 1 is not a congruent number. That proof was supplied by Fermat (of “Fermat’s last theorem” fame) in 1659. By 1915 the congruent numbers less than 100 had been determined, and in 1952 Kurt Heegner introduced deep mathematical techniques into the subject and proved that all the prime numbers in the sequence 5, 13, 21, 29,… are congruent. But by 1980 there were still cases smaller than 1000 that had not been resolved.

Modern results

In 1982 Jerrold Tunnell of Rutgers University made significant progress by exploiting the connection (first used by Heegner) between congruent numbers and elliptic curves, mathematical objects for which there is a well-established theory. He found a simple formula for determining whether or not a number is a congruent number. This allowed the first several thousand cases to be resolved very quickly. One issue is that the complete validity of his formula (therefore also the new computational result) depends on the truth of a particular case of one of the outstanding problems in mathematics known as the Birch and Swinnerton-Dyer Conjecture. That conjecture is one of the seven Millennium Prize Problems posed by the Clay Math Institute with a prize of one million dollars.

The computations

Results such as these are sometimes viewed with skepticism because of the complexity of carrying out such a large calculation and the potential for bugs in either the computer or the programming. The researchers took particular care to verify their results, doing the calculation twice, on different computers, using different algorithms, written by two independent groups. The team of Bill Hart (Warwick University, in England) and Gonzalo Tornaria (Universidad de la Republica, in Uruguay) used the computer Selmer at the University of Warwick. Selmer is funded by the Engineering and Physical Sciences Research Council in the UK. Most of their code was written during a workshop at the University of Washington in June 2008.

The team of Mark Watkins (University of Sydney, in Australia), David Harvey (Courant Institute, NYU, in New York) and Robert Bradshaw (University of Washington, in Seattle) used the computer Sage at the University of Washington. Sage is funded by the National Science Foundation in the US. The team’s code was developed during a workshop at the Centro de Ciencias de Benasque Pedro Pascual in Benasque, Spain, in July 2009. Both workshops were supported by the American Institute of Mathematics through a Focused Research Group grant from the National Science Foundation.

Math In The Movies. “Mathematicians To Thank For Great Graphics”

Ghelvinny — Tue, 01 Dec 2009 00:52:41 +0000

100 powerful supercomputers perform geometrical, algebraic and calculus-based calculations to animate Pixar’s characters. The laws of physics that inform the dynamics of fabric movement are most used in the computations.

Most students in high school dread their math classes and wonder when they will ever use the information in “real life.” Now, with so much work being done on computers, the algebra and trigonometry learned in high school is actually being put to good use.

The animation industry is one that can be a math teacher’s best friend. It is high school math that can actually help bring animated movies to life. Tony DeRose, a computer scientist at Pixar Animation Studios, realized his love of mathematics could transfer into a real world, real interesting job by bringing the pretend world of animation to life. He told DBIS, “Without mathematics, we wouldn’t have these visually rich environments, and visually rich characters.”

Advances in math can lead to advances in animation. Earlier math techniques show simple, hard, plastic toys. Now, advances in math help make more human-like characters and special effects. DeRose explains the difference a few years can make, “You didn’t see any water in Toy Story, whereas by the time we got to Finding Nemo, we had the computer techniques that were needed to create all the splash effects.”

How exactly do the high school math classes help with the animation? Trigonometry helps rotate and move characters, algebra creates the special effects that make images shine and sparkle and calculus helps light up a scene. DeRose encourages people to stick with their math classes. He says, “I remember as a mathematics student thinking, ‘Well, where am I ever going to use simultaneous equations?’ And I find myself using them every day, all the time now.”

The American Mathematical Society and the Mathematical Association of America contributed to the information contained in the TV portion of this report.

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BACKGROUND: Pixar Animation Studios is undergoing a digital revolution thanks to advances in areas such as computer technology, computational physics, and approximation theory. Tony Derose provided a behind-the-scenes look at the role that geometry plays in the revolution using examples drawn from Pixar’s feature films, such as Toy Story I and II. Upcoming movie characters will be animated using a new advancement in geometry recently developed at Pixar.

ABOUT ANIMATION: The term animation refers generally to graphical displays in which a sequence of images with gradual differences results in the same effect as a photographed movie. Computer generated animations are getting more and more common, replacing hand drawn images and other special techniques. There are several ways to generate dynamic changes in computer graphics. Geometry animation is the most complex, and requires changing the geometric elements of a scene dynamically. This is also what most people generally refer to when using the term “animation,” evidenced by motion pictures like “Toy Story” and “A Bug’s Life.”

HOW PIXAR DOES IT: Perhaps the most difficult aspect of animation is making people and clothing look real. Pixar’s software is based on complex studies of how cloth moves when draped on a character, based on the laws of physics. For instance, drape a bedsheet between two points, and the center will hang downward, adjusting itself until it comes to rest in a state of pure tension. The animators begin with drawings of the characters, which they use to build computer puppets, later adding digital “strings” that correspond to various geometric points on the puppet. These strings serve as animation controls, ensuring that as each string is “pulled,” the puppet’s movements reflect what would occur in real life. Color and lighting effects are added last before the puppet is “animated.” Pixar uses 100 powerful supercomputers that run 24 hours a day, seven days a week. It still takes the computers five to six hours to render a single frame lasting 1/24th of a second. For every second of film, it takes the computer six days.

WHAT IS GEOMETRY? Geometry is the field of mathematical knowledge dealing with spatial relationships. The earliest written records — dating from Egypt and Mesopotamia about 3100 BC — demonstrate that ancient peoples had already begun to devise mathematical rules and techniques useful for surveying land areas, constructing buildings, and measuring storage containers. Beginning about the 6th century BC, the Greeks gathered and extended this practical knowledge and from it generalized the abstract subject now known as geometry, from the combination of the Greek words geo (“Earth”) and metron (“measure”) for the measurement of the Earth.

Fakta Unik Angka di Indonesia

Ghelvinny — Tue, 01 Dec 2009 00:41:44 +0000

Setiap negara bangsa, negara dan daerah pasti memiliki penyebutan sendiri untuk angka-angka dari satu, dua sampai dengan sepuluh. Misalnya angka tiga kita menyebutnya di Indonesia tapi di negara lain ada yang menyebutnya tri, three, san, tolu dan lain sebagainya.

Bahkan bila ada yang masih ingat angka-angka tersebut dalam bahasa daerah teman-teman masing-masing dari satu sampai sepuluh maka kadang ada angka yang penyebutannya sama dan ada pula yang berbeda dengan Bahasa Indonesia. Mungkin tergantung dari enaknya di lidah atau di telinga.

Langsung saja. Di sini saya bukan mengajarkan Anda berhitung tapi coba perhatikan deretan angka-angka di bawah ini.

1 = Satu
2 = Dua
3 = Tiga
4 = Empat
5 = Lima
6 = Enam
7 = Tujuh
8 = Delapan
9 = Sembilan

Ternyata setiap bilangan mempunyai saudara ditandai dengan huruf awal yang sama. Bila kedua saudara ini dijumlahkan angkanya, maka hasilnya pasti sepuluh. Contohnya Satu dan Sembilan.. Mempunyai huruf awal yaitu S dan bila diumlahkan satu dan sembilan hasilnya adalah sepuluh.

Begitu juga dengan Dua dan Delapan, Tiga dan Tujuh kemudian Empat dan Enam. Terurut sampai dengan angka Lima. Lima dijumlah dengan dirinya sendiri juga hasilnya sepuluh.

Tidak sampai disitu, ternyata huruf awalnya juga punya peranan penting terbentuknya bilangan itu. Misalnya Satu dan Sembilan sama-sama huruf awalnya adalah S yang secara kebetulan berada pada urutan 19 dalam alpabet. Bila angka satu dan sembilan dijumlahkan kemudian dibagi dua untuk mencari rata-ratanya maka hasilnya adalah 5. Bentuk angka 5 sangat identik dengan huruf S.

Kemudian Dua dan Delapan. Huruf awalnya adalah D yang urutan keempat. Bila delapan dibagi dua maka hasilnya adalah empat (pembenaran).

Selanjutnya Empat dan Enam. Huruf awalnya adalah E yang urutan kelima. Lima berada diantara Empat dan Enam (pembenaran lagi).

Sedangkan angka Lima huruf awalnya adalah L. Dimana L digunakan untuk simbol angka lima puluh dalam perhitungan Romawi (pembenaran yang masih nyambung).

Lalu bagaimana dengan Tiga dan Tujuh? Ternyata susah cari pembenarannya. Ditambah, dikurang, dibagi dan dikali ternyata belum juga ketemu. Tiga dikali tujuh hasilnya 21, kurang satu angka dengan huruf T yang urutan ke 20. Tapi simbol V digunakan untuk menunjukkan angka tujuh dalam perhitungan Arabic. Dan V diurutan ke-22.

Ternyata, tidak pake matematika. Cukup ditulis saja dikertas kosong kemudian pasti bisa ketemu hubungannya. Coba tulis huruf T kecil (t) di sebuah kertas. Kemudian putar kertasnya 180 derajat maka kamu bisa lihat angka tujuh dengan jelas. Lalu bagaimana dengan angka tiga? Juga sama. Tulis huruf T besar di kertas pake font Times New Roman kemudian putar 90 derajat ke kanan searah jarum jam. Tada…. Kamu pasti bisa lihat angka tiga dengan jelas. Tapi sedikit mancung. (pembenaran yang juga dipakasakan sekali).

Pola unik ini mungkin hanya bisa ditemukan di Indonesia. Lalu bagaimana dengan di Malaysia yang juga memakai bahasa yang sama? Ternyata di Malaysia angka 8 tidak disebut sebagai Delapan tapi Lapan. Jadi pola ini hanya milik Indonesia. Jangan sampai diklaim juga sama mereka.

sumber :http://icang69.blogspot.com/