Wang tile - Wikipedia Square tiles with a color on each edge The basic question about a set of Wang tiles is whether it can tile the plane or not, i.e., whether an entire infinite plane can be filled this way. The next question is whether this can be done in a periodic pattern. Domino problem[edit] In 1961, Wang conjectured that if a finite set of Wang tiles can tile the plane, then there also exists a periodic tiling, which, mathematically, is a tiling that is invariant under translations by vectors in a 2-dimensional lattice. This can be likened to the periodic tiling in a wallpaper pattern, where the overall pattern is a repetition of some smaller pattern. The Domino Problem deals with the class of all domino sets. In other words, the domino problem asks whether there is an effective procedure that correctly settles the problem for all given domino sets. Aperiodic sets of tiles[edit] The smallest set of aperiodic tiles was discovered by Emmanuel Jeandel and Michael Rao in 2015, with 11 tiles and 4 colors.
Integrating Knowledge With Needs Chris Lucas "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."Albert Einstein (1879-1955) "Yet we also stress that truth is not the only aim of science.We want more than mere truth: what we look for is interesting truth...what we look for are answers to our problems." Introduction Scientists, in their attempts to maintain a detached 'objectivity' have always rejected the consideration of subjects, of values, of teleology, of purpose. To repair this long-running erroneous worldview we must first realise that science is about people - no people, no science. What Is It Like to be An Amoeba ? Rather than starting with humans, let us instead follow the path of evolution and start with that simple lifeform studied in high school biology. This purpose isn't a single dimensional one, even for such a simple animal. The Meaning of Knowledge Metascience as Integrator To do this we ask ourselves three simple questions:
Platonic Solid -- from Wolfram MathWorld The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are exactly five such solids (Steinhaus 1999, pp. 252-256): the cube, dodecahedron, icosahedron, octahedron, and tetrahedron, as was proved by Euclid in the last proposition of the Elements. The Platonic solids are sometimes also called "cosmic figures" (Cromwell 1997), although this term is sometimes used to refer collectively to both the Platonic solids and Kepler-Poinsot solids (Coxeter 1973). The Platonic solids were known to the ancient Greeks, and were described by Plato in his Timaeus ca. 350 BC. In this work, Plato equated the tetrahedron with the "element" fire, the cube with earth, the icosahedron with water, the octahedron with air, and the dodecahedron with the stuff of which the constellations and heavens were made (Cromwell 1997). If 1. all lie on a sphere. 2. 3. 4. 5. Let (sometimes denoted (or , edges . . and
Chinese meditation boosts brain activity U. OREGON (US)—Just 11 hours of learning a Chinese meditation technique boosts efficiency in a part of the brain that helps a person regulate behavior, according to new research. The technique—integrative body-mind training (IBMT)—has been the focus of scrutiny by researchers led by Yi-Yuan Tang of Dalian University of Technology in collaboration with University of Oregon psychologist Michael Posner. IBMT was adapted from traditional Chinese medicine in the 1990s in China, where it is practiced by thousands of people. It is now being taught to undergraduates involved in research. The new research involves 45 students (28 males and 17 females); 22 subjects received IBMT while 23 participants were in a control group that received the same amount of relaxation training. Details are published online ahead of regular publication in the Proceedings of the National Academy of Sciences. The changes in connectivity began after six hours of training and became clear by 11 hours of practice.
Mangahigh.com - Play maths, love maths Mr Goodfish : good for the sea, good for you Music Math Harmony -- Math Fun Facts It is a remarkable(!) coincidence that 27/12 is very close to 3/2. Why? Harmony occurs in music when two pitches vibrate at frequencies in small integer ratios. For instance, the notes of middle C and high C sound good together (concordant) because the latter has TWICE the frequency of the former. Well, almost! In the 16th century the popular method for tuning a piano was to a just-toned scale. So, the equal-tempered scale (in common use today), popularized by Bach, sets out to "even out" the badness by making the frequency ratios the same between all 12 notes of the chromatic scale (the white and the black keys on a piano). So to divide the ratio 2:1 from high C to middle C into 12 equal parts, we need to make the ratios between successive note frequencies 21/12:1. What a harmonious coincidence! The Math Behind the Fact: It is possible that our octave might be divided into something other than 12 equal parts if the above coincidence were not true!
Making Lunch a Social Networking Game - Bits Blog 4food.comThe 4food ordering page allows customers to customize a burger. Nick Bilton/The New York TimesThe 4Food leaderboard shows the most-ordered burgers. Thursday night I had dinner in the future, and the hamburgers were shaped like doughnuts. Before I get to these strange hamburger shapes, let me explain why they were from the future: my burger was created on the Internet and broadcast to Twitter, Foursquare and Facebook, part of a multiplayer online game, and my order was checked in by a receptionist with an iPad. It’s the result of a new, hyper-connected, “healthy fast food” restaurant called 4Food, which will open in New York City on 40th and Madison Avenue next month. Customers start by going to 4food.com, where they can build a burger. This is also why the burgers are shaped like doughnuts: customers are asked to pick a “scoop,” which goes into the middle of the burger, from options like avocado and chili mango, baked beans or mac and cheese.
Odd Numbers in Pascal's Triangle -- Math Fun Facts Pascal's Triangle has many surprising patterns and properties. For instance, we can ask: "how many odd numbers are in row N of Pascal's Triangle?" For rows 0, 1, ..., 20, we count: row N: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 odd #s: 1 2 2 4 2 4 4 8 2 4 04 08 04 08 08 16 02 04 04 08 04 It appears the answer is always a power of 2. In fact, the following is true: THEOREM: The number of odd entries in row N of Pascal's Triangle is 2 raised to the number of 1's in the binary expansion of N. Presentation Suggestions: Prior to the class, have the students try to discover the pattern for themselves, either in HW or in group investigation. The Math Behind the Fact: Our proof makes use of the binomial theorem and modular arithmetic. (1+x)N = SUMk=0 to N (N CHOOSE k) xk. If we reduce the coefficients mod 2, then it's easy to show by induction on N that for N >= 0, (1+x)2^N = (1+x2^N) [mod 2]. Thus: (1+x)10 = (1+x)8 (1+x)2 = (1+x8)(1+x2) = 1 + x2 + x8 + x10 [mod 2].