Archive for November 8, 2016

Romanesco and the Golden Spiral

Posted in mathematics, The Universe and Stuff with tags , , , on November 8, 2016 by telescoper

Some time ago I mentioned that I received one of these in my weekly veggie box..


Actually, that reminds me that a new box is due tomorrow morning…

Anyway, the vegetable in the picture is called Romanesco. I’ve always thought of it as a cauliflower but I’ve more recently learned that it’s more closely related to broccoli. It doesn’t really matter because both broccoli and cauliflower are forms of brassica, which term also covers things like cabbages, kale and spinach. All are very high in vitamins and are also very tasty if cooked appropriately. Incidentally, the leaves of broccoli and cauliflower are perfectly edible (as are those of Romanesco) like those of cabbage, it’s just that we’re more used to eating the flower (or at least the bud).

It turns out that this week’s Physics World has a short piece on Romanesco, which points out that a “head” of Romanesco has a form of self-similarity, in that each floret is a smaller version of the whole bud and also displays structures that are smaller versions of itself. That fractal behaviour is immediately obvious if you take a close look. Here’s a blow-up so you can see more clearly:

However, one thing that I hadn’t noticed before is that there is another remarkable aspect to the pattern of florets, in that they form an almost perfect golden spiral. This is a form of logarithmic spiral that grows every quarter-turn by a factor of the golden ratio:

\phi = \frac{1+\sqrt{5}}{2}.

Logarithmic, or at least approximately logarithmic, spirals occur naturally in a number of settings. Examples include spiral galaxies, various forms of shell, such as that of the nautilus and in the phenomenon of phyllotaxis in plant growth (of which Romanesco is a special case). It would seem that the reason for the occurrence of logarithmic spirals  in living creatures is that such a shape allows them to grow without any change in shape.

Not really relevant to anything much, I know, but I thought you might be interested…

P.S. One thing the Physics World piece fails to mention is that, regardless of its geometrical properties, Romanesco is really delicious!

American Psycho

Posted in Politics with tags , , , on November 8, 2016 by telescoper


Well today’s the date of the election of the next President of the United States of America. Will it be Hillary Clinton? Or will it be an unstable racist misogynist, a pathological liar, and a man who has boasted of a string of sexual assaults? It remains to be seen. The polls are alarming close. Hillary Clinton is ahead by just 3 or 4 percent nationally but only a handful of states really matter and some of those are too close to call. I’ve been following Nate Silver’s 538 election forecast for a while now. It seems to me his methodology more accurately estimates the uncertainty in the opinion polls. After narrowing considerably when the FBI decided to throw a spanner into the works last week, the probability of a Clinton win is now over a little over 70%. Uncomfortable, but the odds have been below 2-1 very recently.

At the weekend I decided that I would follow my usual betting practice and place a wager on the outcome that I don’t want to happen. Hunting around, the best odds I could find were 18-5 against Donald Trump. I put a monkey on, so will walk away with £2300 if Trump wins. I plan to use the proceeds to begin work on the construction of a fallout shelter in my garden. If an unstable psychopath like Donald Trump gets his hands on the American nuclear codes I don’t hold out much hope for the future of civilization.

I followed the same strategy on Referendum Day as I felt it in my bones that Vote Leave was going to win. I ended up depressed but compensated to the tune of £1000. I’m afraid to say I feel the same way now about the likelihood of a Trump victory. Not very scientific, I know, but there you go.

I have never paid much attention to American politics in the past. It is as incomprehensible to me as British politics must be to them. Gore Vidal summed it up for me:

There is only one party in the United States, the Property Party … and it has two right wings: Republican and Democrat. Republicans are a bit stupider, more rigid, more doctrinaire in their laissez-faire capitalism than the Democrats, who are cuter, prettier, a bit more corrupt — until recently … and more willing than the Republicans to make small adjustments when the poor, the black, the anti-imperialists get out of hand. But, essentially, there is no difference between the two parties.

Things have changed this time. Although both parties still represent the moneyed classes more than anyone else, but this time the Republican contender has overtly fascist tendencies. No wonder Nigel Farage admires him so much. Americans are free to vote for whomever they wish, of course. I don’t have a say, as I’m a foreigner. All I can say is that you should be very careful what you wish for.

Although I find it deeply depressing that this race is even close, I won’t lose any sleep over the election night. I don’t have a television, and I’ll do what I did on the day of the EU referendum. Drink some wine, listen to music and then go to sleep. There’s no point in worrying about things that are out of your hands. And if Trump does win, at least it shortens the list of countries I will have to consider emigrating to if and when the UK does leave the European Union..