Type of Spirals: A spiral is a curve in the plane or in the space, which runs around a centre in a special way.
Different spirals follow. Most of them are produced by formulas:The radius r(t) and the angle t are proportional for the simplest spiral, the spiral of Archimedes. Therefore the equation is:
(3) Polar equation: r(t) = at [a is constant].
From this follows
(2) Parameter form: x(t) = at cos(t), y(t) = at sin(t),
(1) Central equation: x²+y² = a²[arc tan (y/x)]².
You can make a spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. Both motions start at the same point.
(1) The uniform motion on the left moves a point to the right. - There are nine snapshots.
(2) The motion with a constant angular velocity moves the point on a spiral at the same time. - There is a point every 8th turn.
(3) A spiral as a curve comes, if you draw the point at every turn(Image).
Figure 1: (1) Archimedean spiral - (2) Equiangular Spiral (Logarithmic Spiral, Bernoulli’s Spiral).
Figure 2 : (1) Clothoide (Cornu Spiral) - (2) Golden spiral (Fibonacci number).
More Spirals: If you replace the term r(t)=at of the Archimedean spiral by other terms, you get a number of new spirals. There are six spirals, which you can describe with the functions f(x)=x^a [a=2,1/2,-1/2,-1] and f(x)=exp(x), f(x)=ln(x). You distinguish two groups depending on how the parameter t grows from 0.
Figure 4: If the absolute modulus of a function r(t) is increasing, the spirals run from inside to outside and go above all limits. The spiral 1 is called parabolic spiral or Fermat’s spiral.
Figure 5: If the absolute modulus of a function r(t) is decreasing, the spirals run from outside to inside. They generally run to the centre, but they don’t reach it. There is a pole. Spiral 2 is called the Lituus (crooked staff).
Figure 7: Spirals Made of Line Segments.
Source: Spirals by Jürgen Köller.
See more on Wikipedia: Spiral, Archimedean spiral, Cornu spiral, Fermat’s spiral, Hyperbolic spiral, Lituus, Logarithmic spiral,
Fibonacci spiral, Golden spiral, Rhumb line, Ulam spiral,
Hermann Heights Monument, Hermannsdenkmal.
Spiral compulsion. But this is a handy reference.
Music and Harmonics Theory - 432 HZ - Sacred Musical Scales
Whale song art
These exquisite images are the visual
representations of songs sung by whales and
dolphins. The sounds were recorded by US
engineer Mark Fischer and transformed into
visuals by clever mathematics. But these are
not just pretty pictures - the patterns reveal
tantalising clues to how these majestic
animals communicate through song.
© 2008 SCIENCE PHOTO LIBRARY
Dysfunction in dopamine signaling profoundly changes the activity level of about 2,000 genes in the brain’s prefrontal cortex and may be an underlying cause of certain complex neuropsychiatric disorders, such as schizophrenia, according to UC Irvine scientists.
This epigenetic alteration of gene activity in brain cells that receive this neurotransmitter showed for the first time that dopamine deficiencies can affect a variety of behavioral and physiological functions regulated in the prefrontal cortex.
The study, led by Emiliana Borrelli, a UCI professor of microbiology & molecular genetics, appears online in the journal Molecular Psychiatry.
K Brami-Cherrier, A Anzalone, M Ramos, I Forne, F Macciardi, A Imhof, E Borrelli. Epigenetic reprogramming of cortical neurons through alteration of dopaminergic circuits. Molecular Psychiatry, 2014; DOI: 10.1038/mp.2014.67
Image via Resverlogix
Narrowly Avoiding Global Blackout: Earth Just Missed the Most Powerful Solar Flare Ever Recorded
According to a new report from NASA, the Earth barely escaped a massive solar storm that could have knocked “modern civilization back to the 18th century.” On July 23, 2012, two giant plasma clouds, known as coronal mass ejections (CMEs), erupted from the sun to form an unusually large solar storm. At the time, the earth was facing away from the blast of the CMEs. However, if the flares had occurred one week earlier, the electromagnetic ejections would have caused trillions of dollars of damage to the planet. NASA reported: “Analysts believe that a direct hit by an extreme CME such as the one that missed Earth in July 2012 could cause widespread power blackouts, disabling everything that plugs into a wall socket. Most people wouldn’t even be able to flush their toilet because urban water supplies largely rely on electric pumps.” Physicists were particularly surprised by the strength of the CME given that the sun is in its weakest solar cycle in a century.
“Organs-on-a-chip” don’t look like much: They are very thin clear pieces of plastic, but when they are filled with cells, they take on a life of their own and mimic human systems far more effectively than simple petri dish cell cultures. - The Coming Human Body On A Chip That Will Change How We Make Drugs
No more animal testing and no more guesswork about whether drugs that work on animals might also work on humans. Scientists are making an entire electonic set of organs that can test our drugs quickly and easily.
i have no idea….
"People are scared to empty their minds fearing that they will be engulfed by the void. What they don’t realize is that their own mind is the void."
~ Huang Po
"Brand-New Day" - Vogue US May 2014