Medieval Muslims Made Stunning Math Breakthrough
Magnificently sophisticated geometric patterns in medieval Islamic architecture indicate their designers achieved a mathematical breakthrough 500 years earlier than Western scholars, scientists said on Thursday.
By the 15th century, decorative tile patterns on these masterpieces of Islamic architecture reached such complexity that a small number boasted what seem to be "quasicrystalline" designs, Harvard University's Peter Lu and Princeton University's Paul Steinhardt wrote in the journal Science.
Click here for more information.
Tuesday, March 20, 2007
Monday, March 19, 2007
Lie Groups - E8
Mathematicians Map E8
Mathematicians have mapped the inner workings of one of the most complicated structures ever studied: the object known as the exceptional Lie group E8. This achievement is significant both as an advance in basic knowledge and because of the many connections between E8 and other areas, including string theory and geometry.
Click here for more information.
Mathematicians have mapped the inner workings of one of the most complicated structures ever studied: the object known as the exceptional Lie group E8. This achievement is significant both as an advance in basic knowledge and because of the many connections between E8 and other areas, including string theory and geometry.
Click here for more information.
Proof Without Words
The sum of the first n odd numbers is n2.
Mathematically, $\sum_{i=1}^{n} (2i - 1) = n^2$.
Mathematically, $\sum_{i=1}^{n} (2i - 1) = n^2$.
Friday, March 16, 2007
Starship Dimensions
Okay, this isn't a mathematical post per se, but it does make use of dimensions and exponents.
Note: The images on that site are all exactly to scale, 10 pixels to a meter. Internet Explorer users may Click and Drag the starships to compare them as you like. At the bottom of the image are some contemporary vehicles and buildings for reference. Have fun!
Note: The images on that site are all exactly to scale, 10 pixels to a meter. Internet Explorer users may Click and Drag the starships to compare them as you like. At the bottom of the image are some contemporary vehicles and buildings for reference. Have fun!
GNU MP Bignum Library
The GNU Multiple Precision (MP) Bignum Library is a free library for arbitrary precision arithmetic, operating on signed integers, rational numbers, and floating point numbers. There is no practical limit to the precision except the ones implied by the available memory in the machine GMP runs on. GMP has a rich set of functions, and the functions have a regular interface.
Compare it to MIRACL for functionality and speed.
Compare it to MIRACL for functionality and speed.
Thursday, March 15, 2007
Tuesday, March 13, 2007
Wednesday, February 21, 2007
Folding Paper in Half 12 Times
Britney Gallivan has solved the Paper Folding Problem. This well known challenge was to fold paper in half more than seven or eight times, using paper of any size or shape. Click here for the rest of the article.
Friday, January 19, 2007
Tupper's Self-Referential Formula
J. Tupper concocted the amazing formula
$1/2 < \lfloor mod(\lfloor y/(17) \rfloor 2^(-17 \lfloor x \rfloor -mod(\lfloor y \rfloor,17)),2) \rfloor$,
where $\lfloor x \rfloor$ is the floor function and mod(b, m) is the mod function, which, when graphed over 0 ≤ x ≤ 105 and n ≤ y ≤ n + 16 with
n = $960,939,379,918,958,884,971,672,962,127,$
$852,754,715,004,339,660,129,306,651,505,519,$
$271,702,802,395,266,424,689,642,842,174,350,$
$718,121,267,153,782,770,623,355,993,237,280,$
$874,144,307,891,325,963,941,337,723,487,857,$
$735,749,823,926,629,715,517,173,716,995,165,$
$232,890,538,221,612,403,238,855,866,184,013,$
$235,585,136,048,828,693,337,902,491,454,229,$
$288,667,081,096,184,496,091,705,183,454,067,$
$827,731,551,705,405,381,627,380,967,602,565,$
$625,016,981,482,083,418,783,163,849,115,590,$
$225,610,003,652,351,370,343,874,461,848,378,$
$737,238,198,224,849,863,465,033,159,410,054,$
$974,700,593,138,339,226,497,249,461,751,545,$
$728,366,702,369,745,461,014,655,997,933,798,$
$537,483,143,786,841,806,593,422,227,898,388,$
$722,980,000,748,404,719$,
gives the self-referential "plot" illustrated above.
Tuesday, January 16, 2007
Latest, Greatest Twins in Their Prime
The Twin Internet Prime Search and PrimeGrid distributed computation projects have recently discovered the largest known twin primes, which is a pair of prime numbers separated by two. The pair discovered on January 15th are $2,003,663,613 \times 2^{195,000} ± 1$. The two primes are 58,711 digits long. The discoverer was Eric Vautier from France.
Thursday, December 28, 2006
Sunday, December 10, 2006
Math + JavaScript = Cool
Check out ASCIIMathML and ASCIIsvg to learn how to display math equations and graphs in your website using nothing but JavaScript. You've probably seen ASCIIMathML in action in some of my other posts.
Saturday, December 02, 2006
Golomb Ruler
In mathematics, the term "Golomb Ruler" refers to a set of non-negative integers such that no two distinct pairs of numbers from the set have the same difference. Conceptually, this is similar to a ruler constructed in such a way that no two pairs of marks measure the same distance. An Optimal Golomb Ruler (OGR) is the shortest Golomb Ruler possible for a given number of marks. However, finding (and proving) OGR's becomes exponentially more difficult as the number of marks increases, and it is for this reason that we have turned to the web for help in finding the OGR's with 24 and more marks.
For more information, check out the distributed.net: Project OGR page and more info at MathWorld and Wikipedia.
For more information, check out the distributed.net: Project OGR page and more info at MathWorld and Wikipedia.
Thursday, November 30, 2006
Antikythera Mechanism
The Antikythera mechanism is an ancient mechanical analog computer (as opposed to digital computer) designed to calculate astronomical positions. It was discovered in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, and has been dated to about 150-100 BC. Click here for more information.
Sunday, November 05, 2006
InstaCalc Online Calculator
InstaCalc is a vision for a fast, simple and powerful calculator. You can embed it in your website or blog.
Wednesday, November 01, 2006
Wednesday, September 13, 2006
44th Known Mersenne Prime Found!
Less than a year after their last discovery, the Central Missouri State University (CMSU) team, led by professors Curtis Cooper and Steven Boone, has broken their own record for the largest known prime number: 232,582,657-1.
Tuesday, August 22, 2006
Great Mathematical Books
Sunday, July 09, 2006
The Mathematics Genealogy Project
The intent of this project is to compile information about ALL the mathematicians of the world. We earnestly solicit information from all schools who participate in the development of research level mathematics and from all individuals who may know desired information.
Please notice: Throughout this project when we use the word "mathematics" or "mathematician" we mean that word in a very inclusive sense. Thus, all relevant data from statistics, or computer science or operations research is welcome.
In the following paragraphs we shall try to outline our goals and our underlying philosophy for the GENEALOGY PROJECT. It is our goal to list all individuals who have received a doctorate in mathematics. For each individual we plan to show the following:
Please notice: Throughout this project when we use the word "mathematics" or "mathematician" we mean that word in a very inclusive sense. Thus, all relevant data from statistics, or computer science or operations research is welcome.
In the following paragraphs we shall try to outline our goals and our underlying philosophy for the GENEALOGY PROJECT. It is our goal to list all individuals who have received a doctorate in mathematics. For each individual we plan to show the following:
- The complete name of the degree recipient.
- The name of the university which awarded the degree.
- The year in which the degree was awarded.
- The complete title of the dissertation.
- The complete name(s) of the advisor(s).
Hilbert's 23 Problems (#10)
Definition: Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.
What this basically says is that is there a way for us to give a boolean answer to whether any such equation is solvable or not without necessarily knowing the solution?
The answer turns out to be no. Get the book Hilbert's Tenth Problem by Yuri V. Matiyasevich to understand why.
Although I've started reading this book, I must say that it is a little above my level of understanding. I've already read the first chapter twice just to get some basic understanding of his definitions. However, the commentary at the end of Chapter 3 was absolutely remarkable.
In my number theory and abstract algebra classes in college (university doesn't have that nice ring to it), we learned that there does not exist any polynomial that returns only primes for all integer arguments, but rather only for some (Euler's $f(x) = x^2 + x + 41$).
Julia Robinson proved in 1952 that the binomial coefficients and the factorial are exponential Diophantine, and gave an exponential Diophantine representation for the set of all prime numbers. In 1960, Putnam noted that a Diophantine set is the positive part of the range of a polynomial. Thus, it became clear (to someone) that if exponentiation were established to be Diophantine, it would become possible to construct a polynomial such that the positive values it assumed would coincide precisely with the prime numbers. In 1971a, Matiyasevich gave the first upper bound of 24 variables in his Russian article, which was later reduced to 21 in the appendix of the English translation.
In 1976, Jones, Sato, Wada and Wiens exhibited the following polynomial:
$(k + 2){1 - [wz + h + j - q]^2 - [(gk + 2g + k + 1)(h + j) + h - z]^2$ - $[2n + p + q + z - e]^2$ - $[16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2]^2$ - $[e^3(e + 2)(a + 1)^2 + 1 - o^2]^2$ - $[(a^2 - 1)y^2 + 1 - x^2]^2$ - $[n + l + v - y]^2$ - $[((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2]^2$ - $[(a^2 - 1)t^2 + 1 - m^2]^2$ - $[q + y(a - p -1) + s(2ap + 2a - p^2 - 2p - 2) - x]^2$ - $[z + pl(a - p) + t(2ap - p^2 - 1) - p\m]^2$ - $[ai + k + 1 - l -i]^2$ - $[p + l(a - n - 1) + b(2an +2a - n^2 - 2n - 2) - m]^2\}$
This polynomial contains 26 variables (all the letters of the English alphabet), and the set of its positive values is exactly the set of all prime numbers. Note: The polynomial written above, representing only primes, is itself the product of two polynomials.
Later, the bound was further reduced to 12 variables by Wada in 1975, and by Jones, Sato, Wada and Wiens in 1976. Currently, the record is at 10 variables, achieved by Matiyasevich in 1977a.
What this basically says is that is there a way for us to give a boolean answer to whether any such equation is solvable or not without necessarily knowing the solution?
The answer turns out to be no. Get the book Hilbert's Tenth Problem by Yuri V. Matiyasevich to understand why.
Although I've started reading this book, I must say that it is a little above my level of understanding. I've already read the first chapter twice just to get some basic understanding of his definitions. However, the commentary at the end of Chapter 3 was absolutely remarkable.
In my number theory and abstract algebra classes in college (university doesn't have that nice ring to it), we learned that there does not exist any polynomial that returns only primes for all integer arguments, but rather only for some (Euler's $f(x) = x^2 + x + 41$).
Julia Robinson proved in 1952 that the binomial coefficients and the factorial are exponential Diophantine, and gave an exponential Diophantine representation for the set of all prime numbers. In 1960, Putnam noted that a Diophantine set is the positive part of the range of a polynomial. Thus, it became clear (to someone) that if exponentiation were established to be Diophantine, it would become possible to construct a polynomial such that the positive values it assumed would coincide precisely with the prime numbers. In 1971a, Matiyasevich gave the first upper bound of 24 variables in his Russian article, which was later reduced to 21 in the appendix of the English translation.
In 1976, Jones, Sato, Wada and Wiens exhibited the following polynomial:
$(k + 2){1 - [wz + h + j - q]^2 - [(gk + 2g + k + 1)(h + j) + h - z]^2$ - $[2n + p + q + z - e]^2$ - $[16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2]^2$ - $[e^3(e + 2)(a + 1)^2 + 1 - o^2]^2$ - $[(a^2 - 1)y^2 + 1 - x^2]^2$ - $[n + l + v - y]^2$ - $[((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2]^2$ - $[(a^2 - 1)t^2 + 1 - m^2]^2$ - $[q + y(a - p -1) + s(2ap + 2a - p^2 - 2p - 2) - x]^2$ - $[z + pl(a - p) + t(2ap - p^2 - 1) - p\m]^2$ - $[ai + k + 1 - l -i]^2$ - $[p + l(a - n - 1) + b(2an +2a - n^2 - 2n - 2) - m]^2\}$
This polynomial contains 26 variables (all the letters of the English alphabet), and the set of its positive values is exactly the set of all prime numbers. Note: The polynomial written above, representing only primes, is itself the product of two polynomials.
Later, the bound was further reduced to 12 variables by Wada in 1975, and by Jones, Sato, Wada and Wiens in 1976. Currently, the record is at 10 variables, achieved by Matiyasevich in 1977a.
Subscribe to:
Posts (Atom)
