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Uncoiling the spiral: Maths and hallucinations

Uncoiling the spiral: Maths and hallucinations
December 2009 You can read a more accessible version of this article here. Think drug-induced hallucinations, and the whirly, spirally, tunnel-vision-like patterns of psychedelic imagery immediately spring to mind. Computer generated representations of form constants. Geometric hallucinations were first studied systematically in the 1920s by the German-American psychologist Heinrich Klüver. In the 1970s the mathematicians Jack D. How the cortex got its stripes... In humans and mammals the first area of the visual cortex to process visual information is known as V1. An object or scene in the visual world is projected as a two-dimensional image on the retina of each eye, so what we see can also be treated as flat sheet: the visual field. To translate visual patterns to neural activity, what is needed is a coordinate map, a rule which links each point in the visual field to a point on the flat model of V1. So how does this retino-cortical map transform Klüver's geometric patterns? and (where Related:  Sensory MechanismsScience!

Science Mysteries, Fibonacci Numbers and Golden section in Nature Golden Ratio & Golden Section : : Golden Rectangle : : Golden Spiral Golden 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). 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. Successive points dividing a golden rectangle into squares lie on a logarithmic spiral which is sometimes known as the golden spiral. Golden Ratio in Architecture and Art Here are few examples: Parthenon, Acropolis, Athens. Detailed explanation about geometrical construction of the Vitruvian Man by Leonardo da Vinci >> Examples: where

Interactive Ear tool showing how the ear works by Amplifon The ear is the organ which controls hearing and balance, allowing us to understand our surroundings and position ourselves correctly. It is split into three parts: outer, middle and inner. This guide will take you through each part of the ear in turn, answering those essential questions – what are the parts, what do they do, and how? Pinna Helix Antihelix Concha Antitragus Lobe Cartilage Temporal Muscle (Temporalis) Temporal Bone Semicircular Canals Ganglia of the Vestibular Nerve Facial Nerve Ear Canal (External acoustic meatus) Mastoid Process Internal Jugular Vein Styloid Process Internal Cartoid Artery Eardrum (Tympanic Membrane) Auditory Tube (Eustachian Tube) Outer Ear – Welcome to the Interactive Ear! This is the part of the ear that people can see, and funnels sound into your ear canal. The rim of the pinna. A curved panel of cartridge. Bowl-shaped part of pinna. The small, hard bump above your ear lobe. The earlobe contains a large blood supply, helping to keep the ears warm.

Unveiling the Mandelbrot set September 2006 Back in the 1970s and 1980s, mathematicians working in an area called dynamical systems made use of the ever-advancing computing power to draw computer images of the objects they were working on. What they saw blew their minds: fractal-like structures whose beauty and complexity is only rivalled by Nature itself. At the heart of them lay the Mandelbrot set, which today has achieved fame even outside the field of dynamics. This article describes where it comes from and explores its infinite intricacies. Iteration The Mandelbrot set is generated by what is called iteration, which means to repeat a process over and over again. To iterate x2 + c, we begin with a seed for the iteration. x1 = x02 + c. Now, we iterate using the result of the previous computation as the input for the next. x2 = x12 + c x3 = x22 + c x4 = x32 + c x5 = x42 + c and so forth. The theory of iterated functions is motivated by questions from real life. Let's begin with a few examples. x0 = 0 x1 = 02 + 1 = 1

Wireless Brain Implant Aims To Give Paralyzed Power Over Their Limbs Prosthetic limbs that users control with their minds aren’t yet widely available, but several have been shown to work. Soldiers returning from the wars in Iraq and Afghanistan have made amputeeism a much more prevalent disability, and one with enough funding to drive innovative solutions. Those who are paralyzed have remained more difficult to help because human nerves and muscles require more intricate forms of control than the simplified prosthetic devices. But BrainGate, a program that pools research from several universities, is moving ever closer to giving paralyzed patients use of their limbs by using the same technology developed to drive computerized prosthetics to drive the paralyzed limbs. BrainGate is developing a system in which a patient’s mental signal to move an arm is recorded, filtered through a computer and sent as a command to an electric stimulation device that activates the patient’s muscles. The BrainGate research is, so far, limited to single-arm movement.

Math and Programs you might be interested in. Math and Programs you might be interested in. ^That I found to be really intriguing. It's like not only did Turing invent the concept of the modern computer, but he also created math on paper that is remarkably similar to programmed shaders. ^^Context free. ^Structure Synth is like a 3d version of this. Here's a piece of code for that program to start off with and or remix... #define HA 1.2262415695892090741876149601471 #define HB 0.12262415695892090741876149601471 #define TILESIZE 0.9 #define YOFFSET 3.4641012081519749420331123133873 set maxdepth 1000 r3 shape2astart rule hexO { { x 1.0 s HB HA HB } box { rz 60 x 1.0 s HB HA HB } box { rz 120 x 1.0 s HB HA HB } box { rz 180 x 1.0 s HB HA HB } box { rz 240 x 1.0 s HB HA HB } box { rz 300 x 1.0 s HB HA HB } box } rule start md 20{ {hue 560}hexO {rx 120 ry 120 rz 120 h -0.2}start } Anyway.

Van der Pol oscillator Evolution of the limit cycle in the phase plane. Notice the limit cycle begins as circle and, with varying μ, become increasingly sharp. An example of a Relaxation oscillator. In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second order differential equation: History[edit] Two dimensional form[edit] Liénard's Theorem can be used to prove that the system has a limit cycle. , where the dot indicates the time derivative, the Van der Pol oscillator can be written in its two-dimensional form:[8] Another commonly used form based on the transformation is leading to Results for the unforced oscillator[edit] Relaxation oscillation in the Van der Pol oscillator without external forcing. Two interesting regimes for the characteristics of the unforced oscillator are:[9] When μ = 0, i.e. there is no damping function, the equation becomes: When μ > 0, the system will enter a limit cycle. Popular culture[edit] See also[edit]

Human Senses Nearly everyone has experienced a moment when a faint fragrance brings a memory of a long-lost moment in time crashing back to the forefront of their minds.Often we will have forgotten about the event completely, yet it transpires our unfathomable minds have filed it neatly in some unreachable corner of the brain, primed for instant retrieval. Nigel Marven goes in search of the most disgusting and the most attractive smells, sets out to discover why we are excellent at seeing some things, but sometimes miss what’s right in front of our eyes, the biological reasons why humans eat such a diverse range of foods, from rotten raw ducks eggs to a sweaty blue cheese, demonstrates that when it comes to our sense of touch humans are similar to elephants, tracks down the sounds which have the most powerful emotional effects on us and joins stunt co-ordinator Marc Cass for a dramatic drive and experiences how the balance organs let us know how we’re being yanked around and even turned upside down.

Interesting numbers - Imaginary and complex numbers Interesting numbers --- zero --- one --- complex --- root 2 --- golden ratio --- e --- pi --- googol --- infinity What is i? The square of a number is itself multiplied by itself. It's written with a little 2 above and after the number. So 32 = 9 means three squared is nine. i2 = -1 -i2 = -1 √(-1) = ±i √(-4) = ±2i These are called imaginary numbers. What is √i? You may wonder what √i is. This shows how complex number arithmetic works. The complex plane Complex numbers can be plotted on the complex plane. Mandelbrot Set The Mandelbrot set is a mathematical formula which uses the complex plane to make stunning pictures. This is an incredibly complicated picture. The formula is zn+1 = zn2 + c c is a point on the complex plane. z0, z1, z2, ..., zn, zn+1 are a sequence of values of z. So we do all this 'feeding back into the formula' (iterations). However, if the number of iterations is very large, and z still hasn't reached 2, it probably never will. © Jo Edkins 2007 - Return to Numbers index

Residents of Poor Nations Have a Greater Sense of Meaning in Life Than Residents of Wealthy Nations Shigehiro Oishi, Department of Psychology, University of Virginia, P. O. Box 400400, Charlottesville, VA 22904-4400 E-mail: Author Contributions E. Abstract Using Gallup World Poll data, we examined the role of societal wealth for meaning in life across 132 nations. Article Notes Declaration of Conflicting Interests The authors declared that they had no conflicts of interest with respect to their authorship or the publication of this article.

Science After Sunclipse » Blog Archive » “Is Algebra Necessary?” Are You High? “This room smells of mathematics! Go out and fetch a disinfectant spray!” —A.H. Trelawney Ross, Alan Turing’s form master It’s been a while since I’ve felt riled enough to blog. But now, the spirit moves within me once more. First, I encourage you to read Andrew Hacker’s op-ed in The New York Times, “Is Algebra Necessary?” I will try to gather a few observations here which I haven’t seen made elsewhere, for the most part. Towards the end, Hacker’s reasoning gets just bizarre. Quantitative literacy clearly is useful in weighing all manner of public policies, from the Affordable Care Act, to the costs and benefits of environmental regulation, to the impact of climate change. OK, so how are we supposed to teach where “numbers come from” and “what they actually convey” when the students can’t manipulate algebraic formulas? It’s also nice how he skips past the serious problems we have with infrastructure. Philosophers don’t need to know mathematics? (by Oliver Byrne) Oh, poor Euclid. (by Eugene)

9 Mental Math Tricks Math can be terrifying for many people. This list will hopefully improve your general knowledge of mathematical tricks and your speed when you need to do math in your head. 1. Multiplying by 9, or 99, or 999 Multiplying by 9 is really multiplying by 10-1. So, 9×9 is just 9x(10-1) which is 9×10-9 which is 90-9 or 81. Let’s try a harder example: 46×9 = 46×10-46 = 460-46 = 414. One more example: 68×9 = 680-68 = 612. To multiply by 99, you multiply by 100-1. So, 46×99 = 46x(100-1) = 4600-46 = 4554. Multiplying by 999 is similar to multiplying by 9 and by 99. 38×999 = 38x(1000-1) = 38000-38 = 37962. 2. To multiply a number by 11 you add pairs of numbers next to each other, except for the numbers on the edges. Let me illustrate: To multiply 436 by 11 go from right to left. First write down the 6 then add 6 to its neighbor on the left, 3, to get 9. Write down 9 to the left of 6. Then add 4 to 3 to get 7. Then, write down the leftmost digit, 4. So, 436×11 = is 4796. Let’s do another example: 3254×11. 3. 4. 5.

Humans Have a Lot More Than Five Senses Today I found out humans have a lot more than five senses. It turns out, there are at least nine senses and most researchers think there are more like twenty-one or so. Just for reference, the commonly held definition of a “sense” is “any system that consists of a group of sensory cell types that respond to a specific physical phenomenon and that corresponds to a particular group of regions within the brain where the signals are received and interpreted.” The commonly held human senses are as follows: Sight: This technically is two senses given the two distinct types of receptors present, one for color (cones) and one for brightness (rods).Taste: This is sometimes argued to be five senses by itself due to the differing types of taste receptors (sweet, salty, sour, bitter, and umami), but generally is just referred to as one sense. If you liked this article, you might also enjoy subscribing to our new Daily Knowledge YouTube channel, as well as: Bonus Facts: