Four color theorem Example of a four-colored map A four-coloring of a map of the states of the United States (ignoring lakes). In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions.[1] For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a point that also belongs to Arizona and Colorado, are not. Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers. The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. Precise formulation of the theorem[edit] History[edit]
A mathematical bug shows us why the 3D universe leads to Murphy's Law Let's also not forget that unlike, a path, the movement of any string no matter how thin is at least partially governed by the slight recoiling that occurs at the bends and curves. I think analogies are wonderful for explaining complex systems to simple folk like myself, but I hate it when "scientists" try to prove a mathematical system with an insufficient metaphor. Exactly what I was thinking. Like if you have a spiral, and you imagine this bugs walk is from the top spiralling down, this bug cannot actually walk directly downwards because another part of the spiral is already there, no matter how much you try to randomise it with shaking.
A mathematical bug shows us why the 3D universe carries the possibility of despair. Really. For N bug-steps, there are two things to consider: how many total possible paths of N steps the bug has available to it, and how many of those N-step paths lead home. For example, let's look at 1D. After just 2 steps (N=2), where the bug could only go left or right each step, the bug has had 4 possible paths available to it: LR - Home RL - Home Of those, 2 lead home, giving it a home-probability of 50% after 2 steps. Now let's look at 4 steps (still in 1D). LLRR - Home LRLL - Home (earlier) LRLR - Home (twice) LRRL - Home (twice) LRRR - Home (earlier) RLLL - Home (earlier) RLLR - Home (twice) RLRL - Home (twice) RLRR - Home (earlier) RRLL - Home Out of 16 possible 4-step paths, 10 of them led back home. Now let's look at 2D, where the bug has 4 choices each step: up, down, left, or right. 4 steps leaves the bug with 256 possible paths, 84 of which lead home at either 2 or 4 steps. That's why the bug isn't guaranteed to make it back in 3D.
Uncovering Da Vinci's Rule of the Trees As trees shed their foliage this fall, they reveal a mysterious, nearly universal growth pattern first observed by Leonardo da Vinci 500 years ago: a simple yet startling relationship that always holds between the size of a tree's trunk and sizes of its branches. A new paper has reignited the debate over why trees grow this way, asserting that they may be protecting themselves from wind damage. "Leonardo's rule is an amazing thing," said Kate McCulloh of Oregon State University, a scientist specializing in plant physiology. "Until recently, people really haven't tested it." Da Vinci wrote in his notebook that "all the branches of a tree at every stage of its height when put together are equal in thickness to the trunk." In other words, if a tree’s branches were folded upward and squeezed together, the tree would look like one big trunk with the same thickness from top to bottom. Once the skeleton was completed, Eloy put it to the test in a virtual wind tunnel.
The nature of nothingness Zilch… Naught… Nada… It’s easy to dismiss the concept of nothing as, well, nothing. In fact, nothing is everything to science – understanding the intangible voids has lead to breakthroughs we could never have imagined possible. Read on to find out why nothing is more important than nothing… Nothingness: Zero, the number they tried to ban Every schoolchild knows the concept of zero – so why did it take so long to catch on? Read more: "The nature of nothingness" I USED to have seven goats. This is not a trick question. This is a tangled story of two zeroes: zero as a symbol to represent nothing, and zero as a number that can be used in calculations and has its own mathematical properties. Zero the symbol was in fact the first of the two to pop up by a long chalk. It is through such machinations that the string of digits "2012" comes to have the properties of a number with the value equal to 2 × 103 + 0 × 102 + 1 × 101 + 2. This particular number has nothing missing. Facing the void
Bifurcation diagram In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values (equilibria/fixed points or periodic orbits) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable solutions with a solid line and unstable solutions with a dotted line. Bifurcations in 1D discrete dynamical systems[edit] Logistic map[edit] Animation showing the formation of bifurcation diagram An example is the bifurcation diagram of the logistic map: The bifurcation parameter r is shown on the horizontal axis of the plot and the vertical axis shows the possible long-term population values of the logistic function. The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. Real quadratic map[edit] The map is Symmetry breaking in bifurcation sets[edit] In a dynamical system such as which is structurally stable when , if a bifurcation diagram is plotted, treating , the case See also[edit]
Mathematics as the raw material for art Kat Austen, Culture Lab editor (Image: Hiroshi Sugimoto) In a new Paris exhibition, prizewinning mathematicians team up with artists to inspire works that bring intangible concepts to life Mathematics - A Beautiful Elsewhere, Fondation Cartier, Paris, until 18 March IF YOU think of cosmology, you picture colourful nebulae; with neurology, intricate brain scans. The product of the collaboration is the exhibition Mathematics - A Beautiful Elsewhere, at the Fondation Cartier in Paris, France. Ambitious perhaps, but the team has impressive credentials: the mathematical line-up boasts three Fields medal winners, including 2010 recipient Cédric Villani. Although some of the works might also fit well in a science museum, they are very much at home in the airy halls of one of Paris's premier contemporary art spaces. One example of artistic licence has made it into the show. But it is a tangible realisation of impossibilities that awaits in the exhibition's final room.
Ralph Abraham Ralph H. Abraham (b. July 4, 1936, Burlington, Vermont) is an American mathematician. He has been a member of the mathematics department at the University of California, Santa Cruz since 1968. Life and work[edit] Ralph Abraham earned his Ph.D. from the University of Michigan in 1960, and held positions at UC Santa Cruz, Berkeley, Columbia, and Princeton. He founded the Visual Math Institute[1] at UC Santa Cruz in 1975, at that time it was called the "Visual Mathematics Project". Abraham has been involved in the development of dynamical systems theory in the 1960s and 1970s. Another interest of Abraham's concerns alternative ways of expressing mathematics, for example visually or aurally. Abraham developed an interest in "Hip" activities in Santa Cruz in the 1960s and set up a website gathering information on the topic.[3] He credits his use of the psychedelic drug DMT for "swerv[ing his] career toward a search for the connections between mathematics and the experience of the Logos".[4]
Los socorridos números de Fibonacci La secuencia de Fibonacci es una sucesión infinita de números naturales. Se denominan así los números que permiten contar los elementos de un conjunto. Son el primer grupo de números que fueron usados por los seres humanos para contar cosas. El uno, el dos, el cinco, por nombrar algunos, son números naturales. En la secuencia de Fibonacci cada número viene dado por la suma de los dos anteriores. Fue Leonardo de Pisa, un matemático italiano del siglo XIII, conocido como Fibonacci quien dio nombre a esta sucesión en Liber Abaci, un libro publicado en el año 1202 . Cierto hombre tenía una pareja de conejos juntos en un lugar cerrado y desea saber cuántos son creados a partir de este par en un año cuando es su naturaleza parir otro par en un simple mes, y en el segundo mes los nacidos parir también Esta secuencia numérica se ha hecho muy conocida en la cultura popular, podemos encontrar referencias a la misma en libros, canciones, películas y series de televisión. Series de televisión Libros
11/11/11 Palindrome Date | The Amazing Mathematics of Nov. 11, 2011 | Weird News Today's date, 11/11/11, is a once-in-a-century occurrence, adding to a November has been a very fun month for recreational mathematicians. Last week, a rare eight-digit palindrome date — 11/02/2011, which reads the same frontward and backward — was found to have other mathematical qualities that made it a once-in-10,000-years date. Aziz Inan, a professor of electrical engineering at the University of Portland, Oregon, crunched the numbers and found that when the date was expressed as a number, 11,022,011, it has very special properties. "It is the product of 7 squared times 11 cubed times 13 squared. That is impressive because those are three consecutive prime numbers. No other palindrome date up to A.D. 10,000 is like that," Inan said. A once-in-10,000-years date is hard to top, but 11/11/11 is no slouch. After today, 11/11/11 will next occur 100 years down the road, on Nov. 11, 2111. Then, split 36 into three consecutive numbers that add up to 36 (11, 12 and 13).
Known Amicable Pairs This is an attempt to collect all known amicable pairs. I would appreciate receiving any kind of updates and corrections. Comments are welcome too. The table below contains the number of amicable pairs arranged according to the number of digits in the smallest member. Update History: 28-Sep-2007 Now 11,994,387 pairs. 20-Nov-2006 Now 11,446,960 pairs. 05-Jun-2006 Now 11,222,079 pairs. 29-Dec-2005 Now 10,410,218 pairs. 04-Oct-2005 Major update -- now 10,306,909 pairs. Thanks to: Herman te Riele, David Moews, Mariano Garcia, David Einstein, Derek Ball, Frank Zweers, Yasutoshi Kohmoto, Patrick Costello, Stefan Battiato, Anatoly Gubanov, Andrew Walker, Walter Borho, Axel vom Stein, Sergey Chernych, Harvey Dubner, Paul Jobling, Sally Knight, Michel Marcus, Ren Yuanhua. Parts of the database software is/has been based on the Numbers library created by Ivo Düntsch, and the MIRACL library created by Shamus Software Ltd. Last update: 28-Sep-2007
5 Seriously Mind-Boggling Math Facts by Natalie Wolchover | January 25, 2013 07:19am ET Credit: public domain Mathematics is one of the only areas of knowledge that can objectively be described as "true," because its theorems are derived from pure logic. Some people find math boring.
Hartverdrahtet – Infinite complexity in 4096 Bytes - winner of the PC 4kB intro comp Worthy winner of the PC 4kB intro competition at Revision 2012 and latest example of the compact-coding tradition exercised within the demoscene, Hartverdrahtet by Akronyme Analogiker is a three minute long audio-visual trip into a procedural fractalverse, compressed into a minuscule piece of software. No bigger than 4096 bytes – less than an empty Word document, as demoscene activists like to point out – the executable file contains all the mathematics needed to generate the unfolding visual complexity and audible ambience upon a double-click. A solo effort by a talented coder who calls himself Demoscene Passivist, Hartverdrahtet reveals a mesmerizing cosmos observed through what could be an electron microscope – ethereal, greenish and a little eerie. The real magic is in the lingo: “It’s a mix of shere-tracing, fake ambient occlusion and a lot of post-processing. Hartverdrahtet runs on Windows XP/Vista/7 (fast CPU/GPU adviced) and can be downloaded here.