Fearful symmetry: Roger Penrose’s tiling

The Penrose tiling’s almost-perfect “forbidden symmetry”
Roger Penrose makes his own rules. He is one of the world’s most distinguished mathematical physicists and most inventive thinkers. Penrose’s work on the theory of general relativity in the 1960s led to the discovery that the gravity of collapsing stars can produce black-hole “singularities” in space-time. This, in turn, set Stephen Hawking on his course to rewrite black-hole physics. The research established Penrose’s name in science, but his thought continued to range much further. In The Emperor’s New Mind (1989) he proposed that the human mind can handle problems that are “non-computable,” which is to say that any computer trying to solve them by executing a set of logical rules (as all computers do) would chunter away forever without reaching a conclusion. This property of the mind, Penrose argued, might stem from the brain’s use of a quantum-mechanical principle, perhaps involving quantum gravity. In collaboration with anaesthetist Stuart Hameroff, he suggested in Shadows of the Mind (1994) what that principle might be, involving quantum behaviour in protein filaments called microtubules in neurons. Neuroscientists scoffed, glazed over, or muttered “Oh, physicists…”
So when I remarked that he is known for ideas that most others couldn’t even imagine, let alone dare voice, introducing a talk by Penrose yesterday, I didn’t expect that I would hear new ones that evening. Penrose was speaking about the discovery for which he is perhaps best known among the public: the so-called Penrose tiling, a pair of rhombus-shaped tiles that can be used to tile a flat surface ad infinitum without the pattern ever repeating itself. It turns out that this pattern is peppered with objects that have five- or ten-fold symmetry; like a pentagon, they superimpose on themselves when rotated a fifth of a full turn. That is very strange, because fivefold symmetry is known to be rigorously forbidden for any two-dimensional packing of shapes. (Try it with ordinary pentagons and you quickly find that you get lots of gaps). The Penrose tiling doesn’t have this “forbidden symmetry” in a perfect form, but it almost does.
These tilings – there are other shapes that have an equivalent result – are strikingly beautiful, with a mixture of regularity and disorder that is somehow pleasing to the eye. This is doubtless why, as Penrose explained, many architects have made use of them. But they also have a deeper significance. After Penrose described the tiling in the 1970s, the crystallographer Alan Mackay – one of the unsung polymathic savants of British science – showed in 1981 that if you imagine putting atoms at the corners of the tiles and bouncing X-rays off them, you can get a pattern of reflections that looks like that of a perfect crystal with the forbidden five- and tenfold symmetries. Four years later, such a material was found in the real world by the Israeli materials scientist Daniel Shechtman and his coworkers. This was dubbed a quasicrystal, and the discovery won Shechtman the 2011 Nobel Prize in Chemistry. Penrose tilings can explain how quasicrystals attain their “impossible” structure.
Roger Penrose on his tiling in the foyer of the
Mitchell Institute for Fundamental Physics and
Astronomy at Texas A&M University
In his talk Penrose explained the richness of these tilings, manipulating transparencies like a prestidigitator in ways that elicited several gasps of delight as new patterns suddenly came into view. But it was in the Q&A session that we got a glimpse of Penrose’s wildly lateral thinking. Assembling a tiling is a very delicate business, because if you add a tile in the wrong place or orientation, somewhere further down the line the pattern fouls up. But how could atoms in a quasicrystal know that they have to come together in a certain way here to avoid a problem right over there? Maybe, Penrose said, they make use of entanglement, the bizarre quantum-mechanical property that foxed Einstein, in which two particles can affect one another instantaneously over any distance. Crikey.
In Penrose’s mind it all links up: quasicrystals, non-computable problems, the universe… You can use these tiles, he said, to represent the rules of how things interact in a hypothetical universe in which everything is non-computable: the rules are well defined, but you can never use them to predict what is going to happen until it actually happens. But my favourite anecdote is of Penrose inspecting a new tiling being laid out on the concourse of some university. Looking it over, he felt uneasy. Eventually he saw why: the builders, seeing an empty space at the edge of the tiling, had stuck another tile there that didn’t respect the proper rules for their assembly. No one else would have noticed, but Penrose saw that what it meant was that “the tiling would go wrong somewhere in the middle of the lawn.” Not that it was ever going to reach that far – but it was a flaw in that hypothetical continuation, that imaginary universe, and for a mathematician that wouldn’t do.
From an article of Mr Philip Ball you can find here http://www.prospectmagazine.co.uk/ball/fearful-symmetry-roger-penroses-tiling

Simple Ways to Sleep a Lot Better at Night

If you sleep like a baby – meaning you wake up crying every two hours – forget the Ambien and warm milk. Take steps to eliminate the stress and anxiety that keeps you awake.
Try a few of these:

1. Step back from one thing you really care about… but have no ability to impact.

For some people it’s politics. For others it’s family. For others it’s global warming. You care — and you desperately want others to care.
Fine. Do what you can: Vote. Lend a listening ear. Recycle and reduce your carbon footprint. Do what you can do. Be your own change… but don’t try to make everyone else change.
They won’t – unless they decide to on their own.

2. Stay out of other people’s business.

Help. Offer guidance. Encourage. Motivate.
But don’t gossip. Don’t get mixed up in politics. It always ends badly. Never put yourself in a position where you’re worried that Phil will tell Allen you said something snarky about Stu and… (yeah, it’s a “Hangover” reference.)

3. Set up automatic warning systems.

The larger your scope of responsibility – professional or personal – the more you have to worry about. Your list of concerns is endless. You’re always on edge, especially at night. So you check your email. You text and call to make sure everything is OK.
The fear of the unknown drives you crazy.
Instead of worrying about what you don’t know, make sure you do know. Decide what you need to know when and set up systems to support you. Let your employees know what constitutes an emergency — and, just as importantly, what doesn’t. Create automated systems that notify you of problems.
A friend runs a 1,200-employee manufacturing plant. He has a separate phone for emergencies: Employees call that phone or send emails to emergency@. He turns off his regular phone at night and sleeps soundly, because he knows if something happens, he’ll know. He won’t have to check.
Determine what you need to know and create systems to ensure you will know. Then you won’t have to waste time and energy worrying about the unknown.

4. Be grateful for criticism.

When you get feedback, at least someone cares enough to want you to improve: Your product, your service, your work, your life…. You only need to worry when no one cares enough to criticize you.
Criticism creates an opportunity. Embrace that opportunity.

5. Write it all down.

David Allen, the author of Getting Things Done, told me this:
Most people try to use their psyche as their systemic process, which means issues gain importance based on your emotions. I’ve never met anyone who said they didn’t feel a little better if they sat down and made a list. Nothing changes when you write things down except how you engage with your issues: You can be objective and also be creative and intuitive.
Your head is for having ideas, not holding ideas, and it’s certainly not for filing things away. Without exception you will feel better if you get stuff out of your head.
Try it. Write down your challenges. List your problems or concerns.
I bet you’ll start to feel better right away. You’ll realize things aren’t as bad as you think. You’ll also start to figure out ways to make things better — because now you won’t worry passively. You’ll actively solve your problems.

6. Lay off the conspiracy theories.

No one is out to get you. Even if people are, they’re really not the problem – most of us do a better job sabotaging ourselves than someone else ever could. Besides, you can’t control what other people might do.
But you can control what you will do.

7. Reduce the number of judgment calls.

The more prepared you are to handle a situation, the easier it is to be objective — and to avoid stressing out later over whether or not you made the wrong call.
Create price lists that take into account unusual requests. Set up guidelines for responding to customer complaints. Create employee policies for objective areas like attendance, quality, and performance. Decide what you will and will not allow your kids to do before they start asking.
Think about situations you struggle with and decide what you will do before those situations get stressful or confrontational. Then you can make better decisions and greatly reduce your level of stress… and regret.

8. Create a cutoff time…

Yeah, I know, you consider yourself a 24/7 go-getter. But that’s impossible. Decide what time you’ll stop working each day, no matter what.
And if stopping makes you feel guilty?

9. …Then create a plan for tomorrow.

Write down what you need to do first thing tomorrow. You’ll rest easier knowing you have a plan to take care of what you didn’t get done today.
10. Spend a few minutes every day getting better at something else.
It doesn’t matter what you pick. Just make sure it’s not business: A musical instrument. A foreign language. A hobby. Whatever it is, spend a little time on it. Get a little better.
Step outside your daily grind and do something for yourself.
In the process, you’ll gain a little perspective. Perspective soothes the soul.

11. Count your blessings.

Take a second before you turn out the light. In that moment, quit worrying about what you don’t have. Quit worrying about what others have that you don’t.
Think about what you do have.
Thought so. You have a lot to be thankful for.
Feels pretty good, doesn’t it? Feeling better about yourself is the best sleeping pill of all.