Heartening news of the day: Stanford’s **Maryam Mirzakhani** becomes the first woman to win the Fields Medal, the “Nobel Prize of mathematics.” In some distant galaxy, Maria Mitchell’s heart is bursting with joy.

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**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) *

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.

**Image: **I shared at Spirals by Jürgen Köller - Ferns by Margaret Oomen & Ferns by Rocky.

(via blazinuzumaki: / mathhombre: / spring-of-mathematics:)

Every shape has an infinite number of lines which divide its area in some ratio, and there is one dividing line for each direction. Above are the lines that divide a semicircle in ratio 1 : 9. They envelope a curve that approaches the border as the ratio gets smaller.

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**BEAUTY OF MATHEMATICS** (by PARACHUTES)

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without the gorgeous trappings of painting or music." —Bertrand Russell

"Mathematical knowledge is unlike any other knowledge. Its truths are objective, necessary and timeless."

—

**Edward Frenkel**, author of the fantastic *Love and Math: The Heart of a Hidden Reality*, considers whether the universe is a simulation.

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Another interesting property of the logarithmic spiral is revealed if you roll it along a horizontal line. This animation shows the curves traced by points on the spiral, and note that the very centre follows the path of a straight line. The angle between this line and the horizontal is called the pitch of the spiral, and for our spiral galaxy the pitch is around 12 degrees. [more] [code]

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"Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can’t be. They are objective, persistent , necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere — no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what’s also amazing is that we own all of them. No one can patent a mathematical formula, it’s ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It’s too precious to be given away to the “initiated few.” It belongs to all of us."

—

*Love and Math* – a beautiful read on the whimsy of math and how it serves as an equalizer for humanity

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Awesome. To be seen in full screen on high definition. But to be accurate, I have to say that all the things featured in this video actually are *Applied Mathematics*, a crucial branch of *Pure Mathematics*, but not “just” *Mathematics*.

By Yann Pineill & Nicolas Lefaucheux

Seen in *Gizmodo*

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The physics of beauty requires math. The sunflower has spirals of 21, 34, 55, 89, and - in very large sunflowers - 144 seeds. Each number is the sum of the two preceding numbers. This pattern seems to be everywhere: in pine needles and mollusk shells, in parrot beaks and spiral galaxies. After the fourteenth number, every number divided by the next highest number results in a sum that is the length-to-width ratio of what we call *the golden mean*, the basis for the Egyptian pyramids and the Greek Parthenon, for much of our art and even our music. In our own spiral-shaped inner ear’s cochlea, musical notes vibrate at a similar ratio.

The patterns of beauty repeat themselves, over and over. Yet the physics of beauty is enhanced by a self, a unique, self-organizing system. Scientists now know that a single flower is more responsive, more individual, than they had ever dreamed. Plants react to the world. Plants have ways of seeing, touching, tasting, smelling, and hearing.

Rooted in soil, a flower is always on the move. Sunflowers are famous for turning toward the sun, east in the morning, west in the afternoon. Light-sensitive cells in the stem “see” sunlight, and the stem’s growth orients the flower. Certain cells in a plant see the red end of the spectrum. Other cells see blue and green. Plants even see wavelengths we cannot see, such as ultraviolet.

Most plants respond to touch. The Venus’s-flytrap snaps shut. Stroking the tendril of a climbing pea will cause it to coil. Brushed by the wind, a seedling will thicken and shorten its growth. Touching a plant in various ways, at various times, can cause it to close its leaf pores, delay flower reproduction, increase metabolism, or produce more chlorophyll.

Plants are touchy-feely. They taste the world around them. Sunflowers use their roots to “taste” the surrounding soil as they search for nutrients. The roots of a sunflower can reach down eight feet, nibbling, evaluating, growing toward the best sources of food. The leaves of some plants can taste a caterpillar’s saliva. They “sniff” the compounds sent out by nearby damaged plants. Research suggests that some seeds taste or smell smoke, which triggers germination.

The right sound wave may also trigger germination. Sunflowers, like pea plants, seem to increase their growth when they hear sounds similar to but louder than the human speaking voice.

In other ways, flowers and pollinators find each other through sound. A tropical vine, pollinated by bats, uses a concave petal to reflect the bat’s sonar signal. The bat calls to the flower. The flower responds.

*Sharman Apt Russell* | **Anatomy of A Rose**: * Exploring the Secret Life of Flowers* [x]

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**Five great mathematicians and their contributions, in minimalist posters – the best thing since those minimalist posters celebrating pioneering women in science and philosophy’s major movements distilled in minimalist graphics.**

Pair with 17 equations that changed the world.

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**String Theory** is probably the best candidate for a *Theory of Everything*, including the so far elusive *Quantum Gravity*. I have to say, that as viewer of the the matter from the outside, I’ve gone from a deep skepticism, mostly because of the lack of empirical validation, to believe that is the theory with most likely to thrive. The absence of reasonable alternatives, the internal consistency of the theory itself, and the historical background of other theories that emerged from mere theoretical considerations (eg the *Standard Model*), is behind of this personal evolution.

We’ll see, but meanwhile, this site does a good job of popularizing the theory itself, as well as an excellent review of almost all *Theoretical Physics*. From the page:

This site provides a brief and entertaining introduction to string theory for the general public. Topics include quantum gravity, string physics, current research, future prospects, history and news. Kindly supported by *The Royal Society* and *Oxford Physics*.

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If you like math check this out (via TED Blog)

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"By the time you get to the frontiers of math, the words to describe the concepts don’t really exist yet. Communicating these ideas is a bit like trying to explain a vacuum cleaner to someone who has never seen one, except you’re only allowed to use words that are four letters long or shorter."

— Fantastic read for non-mathematicians on what learning advanced math is like.

Work by **Jason Padgett**, a man with Acquired Savant Syndrome who now sees all of reality as mathematical fractals describable by equations.

About:

The beauty of numbers and their connection to the pure geometry of space time and the universe is shown in his fractal diagrams…He is currently studying how all fractals arise from limits and how E=MC2 is itself a fractal. When he first started drawing he had no traditional math training and could only draw what he saw as maths. Eventually a physicist saw his drawings and helped him get traditional mathematics training to be able to describe in equations the complex geometry of his drawings. He is currently a student studying mathematics in Washington state where he is learning traditional mathematics so he can better describe what he sees in a more traditional form. Many of the captions were written before he had any traditional maths training. His drawing of E=MC^2 is based on the structure of space time at the quantum level and is based on the concept that there is a physical limit to observation which is the Planck length. It shows how at the smallest level, the structure of space time is a fractal…So sit back and enjoy the beauty of naturally occurring mathematics in pure geometric form connecting E=MC2 (energy) to art. All are HAND DRAWN using only a pencil, ruler and compass.

(via proofmathisbeautiful: / staceythinx:)

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