ENGI

science:

Usually, when art and science, or science and religion, intersect, they are seen as being in opposition. Art is free-flowing where science is rigorous; religion is faith-based where science needs evidence. But sometimes, the three actually intersect in ways that, at least to my eye, actually heighten the beauty of all of them. One such example is medieval Muslim ornamentation.

Imagine you have a fixed set of tile shapes, but you can have as many of each as you want. Can you tile them in such a way that you fill an infinite plane, with no gaps? If you can, you’ve got yourself a tiling. If you can shift the pattern around in some way, say, one unit to the left, so that the end result is the same as you started with, you’ve got a periodic tiling. But if any shift at all in the pattern creates a unique pattern, the tiling is said to be non-periodic. And if you’ve got a set of tile shapes that can only form non-periodic tilings, no matter what pattern you make with them, the set of tiles is said to be aperiodic. Until the mid-20th century, mathematicians doubted that there could be aperiodic tilings. But in the 1970s, Roger Penrose discovered a set of very simple tiles that—if you apply a couple of restrictions to how they can be arranged (restrictions that can be made superfluous if you give the tiles some bumps)—are aperiodic, i.e., no matter how you arrange these tiles, and no matter how large a plane you tile, you will never find a periodic pattern. They’re called Penrose tiles.

This was new knowledge. No one knew about this until Western mathematics started exploring this in the mid-20th century. Or so we thought.

Because of Islam’s restrictions on religious iconography, such as depicting living beings, Islamic artists have found ways to make the most of abstract patterns and shapes. You see it in Arabic calligraphy, and you see it in the magnificent shapes on the walls of mosques and religious schools. In 2007, physicists Peter Lu and Paul Steinhardt discovered that the patterns on the walls of medieval Islamic buildings very closely resemble Penrose tilings. The crucial invention of girih tiles, basic shapes used to build more complex patterns, allowed Islamic architects to decorate their walls with non-periodic tilings. And in the Darb-e Imam shrine in Ishafan, Iran, built around 1450 (above), the tiles almost perfectly form a pattern that can be generalized as a Penrose tiling. If you deconstruct the pattern on the Darb-e Imam shrine into Penrose tiles, you’ll find that only 11 out of 3700 are mismatched, and the mismatch is so small that it’s “removable with a local rearrangement of a few tiles without affecting the rest of the pattern”. (more)

nikkigraziano:

 so imagine everything ever and then imagine a bunch of math that says “this is what everything ever is made up of, some crazy combination of only these things and nothing else.” those things make up a standard basis. crazy right? math!

nikkigraziano:

 in a fit of frustration earlier at the library, i realized for the n-millionth time that math textbooks suck. they’re just awful. they’re ugly. they use fonts like impact. they’re bad at explaining things. looking stuff up is either cheap and mindless, or a 10+ minute hunt through over-explanations. for the most part, mathematicians are bad at communicating with non-mathematicians. but publishers also make these things very easy to hate. it’s probably a big reason why people hate math, too. it’s not that the math is boring and makes no sense, it’s that most (no, not all) of the people explaining it to you are boring and make no sense.

i am going to write these textbooks one day if it kills me. but until i get a Ph.D., all i’ve got is a BFA. so redesigning the books i own and hate to look at is about all i can do for now.

thedailywhat:

Secret Golden Ratio of the Day: Twitter’s Creative Director Doug Bowman says: To anyone curious about #NewTwitter proportions, know that we didn’t leave those ratios to chance.”

Paging Tweeting Dan Brown.

[laughingsquid.]

(Source: thedailywhat)

unknownskywalker:

Quantum Time Machine Solves Grandfather Paradox

A new kind of time travel based on quantum teleportation gets around the paradoxes that have plagued other time machines. Of all the weird consequences of quantum mechanics, one of the strangest is the notion of postselection: the ability to trigger a computation that automatically disregards certain results.

Here’s an example: suppose you have a long expression in which there are a frighteningly large number of variables. The question you want answering is which combination of variables makes the expression logically true. And the conventional way to solve it is by brute force: try every combination of variable until you find one that works.

Postselection makes the solution easy to find by simply allowing the variables to take any value at random and then postselect on the condition that the answer must be true. This is controversial because it leads to all kinds of fantastical predictions about the power of quantum computers. Nobody is quite sure if these kinds of computations are possible or how to achieve them but quantum mechanics seems to allow them.

MIT researchers say that if you build a time machine combinig postselection with another strange quantum behaviour such as teleportation, this uses the phenomenon of entanglement to reproduce in one point in space a quantum state that previously existed at another point in space.

The idea is to use postselection to make this process happen in reverse, ensuring that only a certain type of state can be teleported. This immediately places a limit on the state the original particle must have been in before it was teleported. In effect, the state of this particle has travelled back in time.

What’s amazing about this time machine is that it is not plagued by the usual paradoxes of time travel, such as the grandfather paradox, in which a particle travels back in time and some how prevents itself from existing in the first place. This time machine gets around this because of the probabilistic nature of quantum mechanics: anything that this time machine allows can also happen with finite probability anyway, thanks to these probabilistic laws.

Another interesting feature of this machine is that it does not require any of the distortions of spacetime that traditional time machines rely on. In these, the fabric of spacetime has to be ruthlessly twisted in a way that allows the time travel to occur. These conditions may exist in the universe’s extreme environments such as inside black holes but probably not anywhere else.

Postselection can only occur if quantum mechanics is nonlinear, something that seems possible in theory but has never been observed in practice. All the evidence so far is that quantum mechanics is linear. In fact some theorists propose that the seemingly impossible things that postselection allows is a kind of proof that quantum mechanics must be linear.

However, if nonlinear behaviour is allowed, time travel will be possible wherever it takes place. It is possible for particles (and, in principle, people) to tunnel from the future to the past.

Source: Technology Review | The paper of this research is available via arXiv.org

Behind every student, there are dozens of teachers that are a little piece of that student. One of the teachers that stands out in my past is Mark Mikasa of Gabrielino High School, to whom I owe my enthusiasm for learning math to. I first met Mark in my sophomore year of high school, where he taught me and 30-odd other students Algebra 2.

We didn’t get along very well at first. I would always sleep in his class while he lectured, which he responded to by spraying me with a water bottle used for the very purpose of waking up students. I started wearing hoodies to class so that he couldn’t wake me up anymore, even with repeated water sprays. It was a teacher versus student escalation that seemed to be heading in a very bad direction…

But I started warming up to him as the semester continued. Mark had a Herculean sense of humor that seemed to be able to make anyone smile. Not just little smiles; they were always a person’s biggest and brightest smiles. This made for a very entertaining (and effective) way for him to teach mathematics. Whenever we learned a new concept, Mark would begin an example by saying “Let’s jump right in!” in his most enthusiastic voice.

Mark’s enthusiasm for math didn’t rub off on me at first. It took me the rest of high school and my first two years of community college to finally become the excited math student that I am today. Even so, I don’t fall in love with every math subject that I encounter; I’m currently taking differential equations and not enjoying it as much as I would like to. But when I get to a tough problem or a difficult concept, I hear Mark’s voice saying to me, “Let’s jump right in!”, and I can approach the material with a smile.

So thank you Mr. Mikasa! You’ve truly left me and hundreds of other students with a greater appreciation of what it means to be a student!

Photo: Mark and his son in his classroom (the same one I was taught in!) at GHS.