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Mathmajik

Thinking Too Much
Sep 12 '14
dumb-science-jokes:

trigonometry-is-my-bitch:

"Glassified" Ruler by MIT Media Lab Automatically Measures Angles, Volume, and Shape Properties.
[source]


Will that ever be a practical device

dumb-science-jokes:

trigonometry-is-my-bitch:

"Glassified" Ruler by MIT Media Lab Automatically Measures Angles, Volume, and Shape Properties.

[source]

Will that ever be a practical device

Aug 26 '14

(Source: bassistdrix)

Aug 26 '14

ryanandmath:

Imagine you wanted to measure the coastline of Great Britain. You might remember from calculus that straight lines can make a pretty good approximation of curves, so you decide that you’re going to estimate the length of the coast using straight lines of the length of 100km (not a very good estimate, but it’s a start). You finish, and you come up with a total costal length of 2800km. And you’re pretty happy. Now, you have a friend who also for some reason wants to measure the length of the coast of Great Britain. And she goes out and measures, but this time using straight lines of the length 50km and comes up with a total costal length of 3400km. Hold up! How can she have gotten such a dramatically different number?

It turns out that due to the fractal-like nature of the coast of Great Britain, the smaller the measurement that is used, the larger the coastline length will be become. Empirically, if we started to make the measurements smaller and smaller, the coastal length will increase without limit. This is a problem! And this problem is known as the coastline paradox.

By how fractals are defined, straight lines actually do not provide as much information about them as they do with other “nicer” curves. What is interesting though is that while the length of the curve may be impossible to measure, the area it encloses does converge to some value, as demonstrated by the Sierpinski curve, pictured above. For this reason, while it is a difficult reason to talk about how long the coastline of a country may be, it is still possible to get a good estimate of the total land mass that the country occupies. This phenomena was studied in detail by Benoit Mandelbrot in his paper “How Long is the Coast of Britain" and motivated many of connections between nature and fractals in his later work.

Aug 26 '14

geometrymatters:

The geometry of DNA: a structural revision

This proposed structure for DNA is wholly founded upon mathematical
principles. Although the geometrical modification to the base pairings is
relatively minor, the resulting double helix manifests a clarity altogether
distinct from that offered by Crick and Watson and it would appear to
shed light upon a number of areas of continuing uncertainty.

• Geometric equations predict the dimensions of DNA’s structure. Not
only does the pentagonal geometry predict the helical dimensions but
it would also demonstrate ‘principle causation’.
• The pentagonal geometry provides the dynamics required to build a
consistent, stable and uniform helical structure and also establishes
why there should be consistently ten bases contained within a single
turn of the helix. Incidentally, when converted to the molecular
dimension I would certainly predict degrees of variation, certainly
between 9.5 and 10.5 bases per turn, but perhaps even more.
• Both the hollow centre and side-by-side structural formation ensure
instant access at any point within the helix. This would permit the
DNA (even circular) to open and close during its replication functions
without entangling itself.
• The modification to the base pairing would appear to be able to exist
in either the enol or keto formations.
• While the sugar-phosphate backbones will undoubtedly prove integral
to the stability of the helical structure, it is the geometry of the basepair
molecules themselves

© Mark E. Curtis

(Source: curtisdna.com)

Aug 22 '14
fuckyeahcalculus:

Can’t look away…

fuckyeahcalculus:

Can’t look away…

Aug 22 '14
lightprocesses:

Hyperbolic paraboloid

lightprocesses:

Hyperbolic paraboloid

Aug 20 '14
mathematica:

Howdy, everyone! Tonight we’re rolling out Mathemedia, a compendium of math-related media — and we’re looking for your submissions!
Send us your favorite books, movies, songs, art, essays, and articles (etc.) that prominently feature mathematics or mathematicians — and we’ll add them to Mathemedia. But first! a few ground rules:
We prefer that material be accessible to anyone with a love of mathematics and access to Wikipedia — assuming, at most, a typical high school education in math.
In general, we’re looking for stories about mathematics — told either from the inside or outside. Fiction is great; non-fiction is good too, as long as it’s designed for more than a mathematically-trained audience. There are several great lists out there on the interwebs of seminal papers and great textbooks and phenomenal websites — this is not one of them.
If you have the time, please include a short description of the book, movie, etc. in question. This will help anyone browsing the list figure out what’s most interesting and appealing to them. If the material is legally available on the Internet (e.g., an article or music video), feel free to include a link!
Thanks for helping spread our love of math! We’ll start putting up what you submit later tonight.
This list was inspired by math-is-beautiful's fabulous list of math-related blogs :)

mathematica:

Howdy, everyone! Tonight we’re rolling out Mathemedia, a compendium of math-related media — and we’re looking for your submissions!

Send us your favorite books, movies, songs, art, essays, and articles (etc.) that prominently feature mathematics or mathematicians — and we’ll add them to Mathemedia. But first! a few ground rules:

  1. We prefer that material be accessible to anyone with a love of mathematics and access to Wikipedia — assuming, at most, a typical high school education in math.
  2. In general, we’re looking for stories about mathematics — told either from the inside or outside. Fiction is great; non-fiction is good too, as long as it’s designed for more than a mathematically-trained audience. There are several great lists out there on the interwebs of seminal papers and great textbooks and phenomenal websites — this is not one of them.
  3. If you have the time, please include a short description of the book, movie, etc. in question. This will help anyone browsing the list figure out what’s most interesting and appealing to them. If the material is legally available on the Internet (e.g., an article or music video), feel free to include a link!

Thanks for helping spread our love of math! We’ll start putting up what you submit later tonight.

This list was inspired by math-is-beautiful's fabulous list of math-related blogs :)

Aug 20 '14
thedotisblack:

Like it? Follow thedotisblack

thedotisblack:

Like it? Follow thedotisblack

Aug 18 '14

mathhombre:

rhombitrihexagonal dual.

This sketch has a section of an Archimedean tiling (rhombitrihexagonal) and its dual.

The dual is found by taking the center of each polygon, then connecting those as vertices if the polygons they’re in are adjacent (share an edge).

When you move the slider or hit the play button, the sketch will shift between the original and the dual.

It’s called the dual, because if you do that again, you get back to the original (or a variation of the original.) You can use this technique to find the structure of tessellations.

If you download it, this sketch has tools to make your own. One tool finds the center of a polygon (barycenter), and the other family of tools is for making the animated dilations.

Inspired by bmk sketches like at [url]http://geogebrart.weebly.com/blog/duality-2[/url]

Sketch at GeoGebraTube.

Aug 18 '14

So why is Euler’s Identity so Beautiful?

In answer to obywan’s question:

The physicist Richard Feynman called the formula it is derived from “one of the most remarkable, almost astounding, formulas in all of mathematics”.

I’m sure you’ve seen all the symbols in the identity before, but isn’t it weird how they are not connected in any evident, but combine to give such a normal result?

The next part is an excerpt from Surein Aziz’s article the Beauty in Mathematics: 

'So, why does this happen? You might think that it is down to some really complex idea — how do we even take a number to the power of i? Well, actually, it isn’t too difficult to see how Euler’s identity comes about - that is one thing that makes the identity so wonderful! But first you have to see Euler’s formula, which leads to his beautiful identity, in full generality:

Doesn’t look quite as nice and neat now, does it? But don’t be put off. To understand how this formula comes about, we need something called Taylor series. These are just a way of expressing functions such as sin (x) or cos (x) as infinite sums. They were discovered by the mathematician Brook Taylor (who was also part of the committee which adjudicated the argument between Isaac Newton and Gottfried Leibniz about who first invented the calculus).

The Taylor series for the function e^ x is:

where n! denotes the product

You can verify this Taylor series using a calculator: choose a number $x$ and see what value the calculator gives you for $e^ x$. Now use the calculator to work out the value of the sum

for as many terms as you like, that is for a number $n$ as high as you like. You will find that the result very nearly equals the result you got for $e^ x$ and the more terms you add to the sum, the closer the two results become. At some point the two results will be the same on your calculator, as their difference becomes too small for the calculator to detect. In “reality”, the two results are equal when you have added an infinite number of terms to your sum.

The Taylor series for the other two functions appearing in Euler’s formula are:

Again you can check this using your calculator, bearing in mind that the angle x is measured in radians, rather than degrees.

Now let’s multiply the variable x in the Taylor series for e^ x by the number i. We get:

But certain powers of i can be simplified – for example, i^2 = -1 by definition, and so i^3 = -i and i^4 = +1, and so on. So we can simplify the above to:

We can gather the terms involving i together to give:

Now notice that these two series are the same as the series for sin (x) and cos (x) from earlier, so we can substitute these in to get:

which is Euler’s formula! 

All we have to do now is substitute x = pi. Since sin (pi ) = 0 and cos (pi ) = -1 we get:

so

So you see, after a sequence of fairly complex mathematics we arrive back where we started — at the (seemingly) simple numbers 1 and 0. That is what I think is so beautiful about this identity: it links very strange numbers with very ordinary and fundamental ones. Seeing why it works feels a bit like treading a little-known path through the mathematical jungle to reach a secret destination somewhere in the thick undergrowth.’

And what about Euler? (from storyofmathematics.com)

Today, Euler is considered one of the greatest mathematicians of all time. His interests covered almost all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, as well as optics, astronomy, cartography, mechanics, weights and measures and even the theory of music.

Much of the notation used by mathematicians today - including e, i, f(x), ∑, and the use of a, b and c as constants and x, y and z as unknowns - was either created, popularised or standardised by Euler. His efforts to standardise these and other symbols (including π and the trigonometric functions) helped to internationalise mathematics and to encourage collaboration on problems.

He even managed to combine several of these together in an amazing feat of mathematical alchemy to produce one of the most beautiful of all mathematical equations, eiπ = -1, sometimes known as Euler’s Identity. This equation combines arithmetic, calculus, trigonometry and complex analysis into what has been called “the most remarkable formula in mathematics”, “uncanny and sublime” and “filled with cosmic beauty”, among other descriptions. Another such discovery, often known simply as Euler’s Formula, is eix = cosx + isinx. In fact, in a recent poll of mathematicians, three of the top five most beautiful formulae of all time were Euler’s. He seemed to have an instinctive ability to demonstrate the deep relationships between trigonometry, exponentials and complex numbers.

@obywan

Aug 17 '14
matthen:

Creating the Sierpinski triangle fractal with rotating triangles. [more] [code] [inspiration]

matthen:

Creating the Sierpinski triangle fractal with rotating triangles. [more] [code] [inspiration]

Aug 16 '14

kqedscience:

Meet the First Woman to Win Math’s Most Prestigious Prize

As an 8-year-old, Maryam Mirzakhani used to tell herself stories about the exploits of a remarkable girl. Every night at bedtime, her heroine would become mayor, travel the world or fulfill some other grand destiny.

Today, Mirzakhani — a 37-year-old mathematics professor at Stanford University — still writes elaborate stories in her mind. The high ambitions haven’t changed, but the protagonists have: They are hyperbolic surfaces, moduli spaces and dynamical systems. In a way, she said, mathematics research feels like writing a novel. “There are different characters, and you are getting to know them better,” she said. “Things evolve, and then you look back at a character, and it’s completely different from your first impression.”

Learn more about Maryam Mirzakhani at wired.

Aug 16 '14

hppy-rbt asked:

Hi! Been following your blog for some time, and enjoyed it much. My interest in math was awaken when i tried parametric design. Rhino and Grasshopper is my tool of choice. Quite often I have to solve problems with the help of geometry, because my knowldege of math is, lets say, limited XD But its not always possible or effective to go that way. So, my question is: how to divide an ellipse into equal parts and with equal distances between division points? :) Sorry for grammar, I'm from Russia. )

Здравствуйте! Рад, что вы наслаждаетесь мой блог!

It is pretty tricky to split an ellipse equally, here are some instructions for a draftsman’s method that may be what you are after?

http://stackoverflow.com/questions/20197974/how-to-divide-an-ellipse-to-equal-segments

Alternatively you could find the ellipse perimeter, then use line integration….here are a couple of instruction sets that may help.

http://www.mathsisfun.com/geometry/ellipse-perimeter.html

http://www.mathwords.com/a/arc_length_of_a_curve.htm

Aug 16 '14

geometrymatters:

The geometry of diatoms, desmids and other algae.

http://www.micromagus.net/microscopes/pondlife_plants01.html

Map