Fractal poetry, and other links

This post contains what purports to be a fractal poem. It’s not a bad poem in its own right, but the link to fractal geometry was too subjective for my taste. However, it got me thinking about what else a “fractal poem” might mean, and I was up till two that morning bringing my idea into fruition. I shared the poem I came up with in the comments, but a fuller explanation appears below.

I based my poem on a simple L-system. An L-system contains a set of rules, applied iteratively, for replacing one symbol with a sequence of symbols. For example, suppose we agree to replace “A” with “ABBA” and “B”, with “BA”. Then, starting with “A”, the first iteration gives “ABBA”, the second iteration gives “ABBABABAABBA”, the third “ABBABABAABBABAABBABAABBAABBABABAABBA” and so on. The connection to fractal geometry is that if we interpret the symbols graphically (e.g. “A” for “go forward” and “B” for “turn left”), we get a squiggly line whose squiggliness depends upon the number of iterations.

I used an L-system where “A” becomes “ABBA”, “B” becomes “BCCB”, and so on. (Using numbers rather than letters, this is: “n → n, n+1, n+1, n”.) After two iterations, we have “ABBABCCBBCCBABBA”, which is the structure I used for my poem, interpreting each letter as representing a line and requiring all lines assigned the same letter to rhyme. In other words, it had to be a 16-line poem in which lines 1, 4, 13 & 16 rhyme, lines 2, 3, 5, 8, 9, 12, 14 & 15 rhyme, and lines 6, 7, 10 & 11 rhyme.

Here is the result. It has, I think, an interesting aesthetic quality when read aloud.

This doggerel does not intend
To satisfy the reader’s would
For art that is remotely good;
It will not serve to meet that end,
So don’t imagine that it could.
But in its rhyming structure you
Might find, if you are able to,
A pattern to be understood
That’s relevant to trees of wood
And clouds of water vapour, too –
The applications are not few –
For it possesses fractalhood.
Look closely, and you’ll comprehend
The secret pattern, bad or good,
Which, if this text were longer, could
By iterative means extend.

After a third iteration, the structure would be a challenging ABBA BCCB BCCB ABBA BCCB CDDC CDDC BCCB BCCB CDDC CDDC BCCB ABBA BCCB BCCB ABBA. Four iterations would give you an epic poem of 256 lines. You’re welcome to give that a go, or maybe you’d prefer to write your own variation on a shorter poem like mine.

Here are some more links that I found over the Christmas holidays:

  • The development of a foetus, animated.
  • Strong Language is a new linguistics blog about swearing. Mostly. Along the way it covers a variety of topics and is worth a look.
  • A well-presented and informative video on placenames ending in -stan.
  • A curious difference between the Andromeda Galaxy and our own.
  • All of the best arguments against vaccination together on one page. (No, it’s not blank, but you’ve got the right idea.)

As for the holidays themselves, I don’t feel like writing a report, but rest assured I had an excellent time. Here are two photographs that capture some special moments.


The photo on the left shows my niece and her parents (my sister on the left, pregnant with her second child) at the Christmas table as it is being prepared. Of note are the origami mangers, complete with jelly baby and paper straw, alternating with paper trees. The brown paper bags are what we used instead of crackers.

On the right is a framed photo set showing miscellaneous moments in Elke’s life so far. This was Rebecca’s Christmas present to me, and it is now hanging above the light switch in my bedroom.

Links: early December 2014

Here are some links that I think are worth sharing. It’s a short list, but you’re all busy getting ready for Christmas so that’s appropriate.

Links: November 2014

I decided to skip the Late October installment, and this Early November installment is late, but here it is now.

  • A curious coincidence concerning the planets.
  • Everything you wanted to know about goldfish.
  • Chimpanzees plan ahead when it comes to breakfast. (This gave me an idea for a card game, but it needs a lot of work.)
  • How humans, birds and grasshoppers breathe.
  • Einstein and the nature of reality: a brief history.
  • A very nice 3D video from Mars.

In personal news, I recently bought my calendar for 2015. The theme I’ve chosen is a collection of Rob Gonsalves paintings — illusionist paintings in the tradition of M. C. Escher. [Update: larger collection here].

I’ll be going to see James Randi on December 1st. (I decided against the “Meet and Greet” ticket, though; it’s about a hundred dollars extra which is more than I can justify.) In preparation, I’ve rewatched this video from 2010.

Photos from family weekend

I spent Friday, Saturday and Sunday with my extended family, for reasons involving two birthday parties and a musical. I won’t go into details here, but we can talk about it in the comments if you like.

Here is a photo of my mother and my niece, Elke, who is now a confident walker, developing her vocabulary, and will be 18 months old later this month.


From later in the weekend: three generations of women on the marimba.


Left: My young relatives at play. Right: A community room in the town to which Dad has contributed dinosaurs.


I got some lovely photos of Elke with Leah, the middle child in my cousin Robert’s family. Here they are on the swing.



And here they are on the golf course.




And that’s all. It really was lovely to catch up with everyone.

Elgar’s intriguing scribble

I recently read an article on Nautilus about the Dorabella Cipher scribbled by Edward Elgar in 1897, before he became a famous composer. The cypher is made up of geometrically regular glyphs, and was given by Elgar to his young friend Dora on what seems to have been a whim. Some people claim to have decrypted it, but these solutions cannot be verified and there is no reason to expect that definitive decryption is possible.

My own thoughts and illustrations follow. It’s not that I have much to say, but a blog post is a good medium for integrating text and pictures. Also, all images embedded in this blog post are public domain, so if you want to play with the cipher yourself you’re welcome to make use of them.

First, the original cypher:

elgar original

As you can see, each glyph consists of one, two or three cusps, facing in any of eight directions. Four of those directions (left, right, up and down) I will call upright and the others (the diagonals) I will call slanted.

The geometric regularity is part of the intrigue, but when you start examining the cypher, you soon notice that Elgar’s handwriting is not exactly regular. It isn’t always clear which direction a particular glyph is supposed to be facing. For example, take the fifth glyph on line two, or the third glyph from the end of line three — in isolation these look mostly upright, but if you interpret them that way then other glyphs, ever so slightly more tilted, become ambiguous. No matter where you draw the line there are always edge cases, and I think the only solution is to document how you have interpreted the glyphs so that if other people want to change anything they can look it up and see precisely what to change.

Below is the cypher again, with the glyphs I’ve interpreted as upright in black, and the glyphs I’ve interpreted as slanted in red.

elgar interpretations

When I decided to play with the cipher, I began by replacing each glyph with a perfectly regular design, on the grounds that this might make any patterns become more obvious to the eye. I then compiled these designs into the following animation.


The outer circle represents glyphs with three cusps, the middle circle those with two cusps, and the inner circle those with one. The eight sectors represent the different directions the glyphs can face: the mapping is arbitrary, but I took the direction faced by the open side of the cusps and mapped it to the first sector clockwise from the corresponding line.

Here are the 87 frames in sequence (left to right, top to bottom, as you would read).

elgar all

The table below shows the number of times each element occurs. You should be able to translate these back to the original glyphs.

elgar tally

Of course it’s possible that Elgar’s scribble isn’t a substitution cipher at all. It could represent a piece of music, or dance steps, or the location of buried treasure. Music seems unlikely, though, as nothing in the tally suggests a pattern indicative of a bias toward dominant notes in the scale, or other regularities you’d expect.

The directions most often faced by the open sides of the glyphs — using the standard map convention — are: northwest (corresponding to the column where the numbers are 8, 11, 4), southeast, south, and then a tie for fourth place between north and east. Curiously, if we give precedence to north, then the most and least common orientations alternate in pairs (probably a coincidence but worth noting nonetheless). Below is the sequence of frames again, with the four most common orientations shaded maroon.

elgar top4

The first pattern I noticed after watching the animation you saw earlier is that a particular glyph often has either the same orientation as its immediate predecessor or the exact opposite orientation. This occurs 42 times (take note, Douglas Adams fans) out of a possible 86, whereas by chance it would occur a quarter of the time. But taking into account the distribution of the four most common orientations — which I had not tallied at the time — increases the chance prediction from 21.5 to 34 occurrences (I think), so the discrepency is greatly reduced. The 42 occurrences are highlighted below.

elgar transitions

Examining the above further shows that 23 of the 42 cases (more than half) involve the northwest and southeast directions. Of these, the most common are a northwest followed by a southeast (9 occurences), or a southeast followed by a northwest (8 occurrences). A northwest follows a northwest 5 times, and there is only one pair of consecutive southeasts.

That’s basically all I have. I just wanted to mess around with the cipher for a while, to see what I could make of it. I didn’t expect to make a historical breakthrough; I expected to have fun. I hope this has been fun for you too, and perhaps you are inspired to explore further.

Links: Early October 2014




  • I occasionally wonder if certain idiotic laws have been changed since I last heard about them. And then I find out they haven’t.


  • Last month I bought some origami-related material (instructional video & paper) as a gift for a younger relative. Noteworthy items I found in my research include these paper reviews and this video tutorial of a model that proved easy enough for me to make myself (compare with this even simpler one). I see potential for decorating this year’s Christmas dinner table.

Links: Late September 2014

A short assortment of links.

Links: Late August 2014

Here is a collection of links, several of which I bookmarked before my holiday in Sydney.


  • I include this article on quantum computers only because I once wrote an article of the sort it condemns as hyped and misinformed. Sorry. (I wrote the article as an undergraduate, and you can find it via this old blog post.)
  • Good and simple video showing how gravity works in general relativity. It represents my own level of understanding.
  • You have fingers for essentially the same reason that cheetahs have spots.
  • I found this nice tool for calculating apparent angular diameters. From a satellite orbiting 225 km above the earth (not quite high enough to be stable), the planet would have an apparent angular diameter of 150 degrees — the same as analogue clock hands at 7:00 or 5:00. From 2,640 km (comparable with the 2000 km outer bound of low earth orbit), it would be 90 degrees across. From 18,260 km (roughly halfway to geosynchronous orbit), it would take up 30 degrees. What else can you discover?


  • This mathematical question involving squares in conjunction with curves is largely solved but interesting to contemplate (and a good excuse to doodle).
  • Not delightful at all, but marking the end of something that is: I recently learned that the collaborative short story writing site is closing its doors (but there’ll be an archive).


Winning Cosmos

This is the story of a competition I recently won.

In an episode of the Skeptic Zone podcast published while I was in Sydney, Richard Saunders announced a competition: to take a photograph on the theme Billions and Billions for a chance of winning a DVD set of the recent Cosmos remake.

Being on holiday I wasn’t planning to enter, but I did spend some time thinking about what sort of photograph one might take (it also made a good conversation topic). I decided that hundreds and thousands should feature somehow, and my best idea — given the time, resources and skills to pull it off — was as follows. (1) Bake a cake in a small, hemisphere-shaped bowl, and ice it to look like (half of) a giant hundred-and-thousand. (2) Choose a background for the photograph that represents the void of space  — perhaps a dark cloth laid over some surface — and sprinkle hundreds and thousands all over it. (3) Place the cake amidst the hundreds and thousands; and on top of the cake, place a lego figure with a telescope.

I don’t have the resources to create this, but obviously there are people out there who could pull it off, and probably do something even better that I hadn’t thought of. So I didn’t think I had any hope of winning the competition, and was just hypothetically contemplating what I would do.

Then I went along to the August 7th Skeptics in the Pub (as described in my Sydney report), and chatted to some people from the Skeptic Zone podcast. The photograph competition came up in conversation with Jo Alabaster, who strongly encouraged me to enter, saying that there had been very few entries, and that even a diagram of my idea would be worth sending in.

My original idea might have been at the edge of possibility given enough time and borrowing of resources, but with a deadline just two weeks after the original announcement (more like one week by the time I got home from Sydney), it was completely impossible. Still, by now I knew that a simpler photograph was in with a chance, and the Cosmos DVD set was a pretty alluring prize. Then — as I was contemplating what resources I might find on an upcoming grocery shop — I hit on an idea that was easily within my grasp, and a multi-layered interpretation of the challenge. All I needed to buy was one bag of icing sugar.

At some point I looked up the other entries on the Skeptic Zone facebook page, and indeed there weren’t many. This surprises me: the much-talked-about Cosmos series is surely an attractive prize, and not something many Australians would have seen already (people who subscribe to non-free-to-air TV are a small minority); I saw one episode on Youtube before it was taken down.

You can find my entry here, and I’ve also replicated it below. Here is the photograph:


And here is the explanation I sent with it. (A note on the calculation: if you google the size of an icing sugar particle, you’ll find figures between 10 and 100 micrometres. I used 100 cubic micrometres for my estimate … wait, that’s wrong, isn’t it? A cube 10 to 100 micrometres across is actually 1,000 to 1,000,000 cubic micrometres … call it 100,000 cubic micrometres … and a billion particles that size would take up a tenth of a litre … whoops, now I’m embarrassed.)

It’s a pair of equations, one horizontal, the other vertical, like a crossword. Physical objects stand in for quantities

The first equation reads: “100s & 1000s cubed is less than icing sugar”.

If hundreds and thousands (the famous confectionary) are called hundreds and thousands, then icing sugar could very reasonably be called billions and billions. In fact, I calculate that just one millilitre of icing sugar contains more than ten billion grains!

The second equation reads: “earth multiplied by icing sugar is less than universe”.

This ties the photograph to a cosmological theme, asserting that the universe contains the equivalent of billions and billions of earths.

By the time the deadline arrived I was expecting to win (although I liked the elegance of Jon Frary’s entry), and throughout the next day, tension was high. I checked the relevant links over and over, and as time passed I thought maybe I hadn’t won after all.

Then — about 31 minutes into episode 304 of the Skeptic Zone — the official announcement … I won!

I look forward to the DVDs. I’ve read enough reviews to know that the series is flawed — just like everything else in the real world — so I am not expecting perfection. I do, however, expect it to be very good, and that its strengths will outweigh its flaws by a considerable margin. Perhaps I will tell you what I thought.

Incidentally, long time readers will know this isn’t the first time I’ve won a science-related online competition. Last time I wrote a limerick.

What we did in Sydney

Together with my parents, I spent the first week of August in Sydney. It was the first holiday in which I used my smartphone to navigate. Here is a summary of what we did.

Some background: I was born in Sydney, and not coincidentally my parents have a number of friends over there. I’ve been back a couple of times, most recently in 1989 when I attended a Doctor Who convention and we also had a family holiday.

As for this trip, we arrived early in the afternoon of Saturday August 2nd. The main event for our first evening was a classical music program at the Opera House — the first time I’ve ever been inside.

On Sunday we visited the Justice and Police Museum (notable exhibits: maces & other home-made weapons; phrenology model; whips for adult vs juvenile offenders), and then met our friend Ian and his current guide dog at the Botanic Gardens. After lunch, Dad and I visited the Jewish Museum (notable exhibits: talmud; ceremonial objects; art installations; immigration history; maps of ghettos, concentration camps, etc) and one other destination which turned out not to be worth our while, before returning by bus to meet Mum and Ian at the hotel.

On Monday we took the train north to visit friends of my parents, and went for a coastal walk in Bouddi National Park.

View from train Bouddi

Bouddi Bouddi

On Tuesday, a place where we’d planned to have breakfast turned out not to exist anymore, so instead we had breakfast at the place now occupying the same location. Breakfast was followed by a visit to the small but welcoming Kerrie Lowe Ceramics Gallery on our way to the Sydney University museums. (These were smaller than I expected, but the Macleay Museum includes a display on the history of photography, an orang-utan skeleton, and a deformed horse skull connected to a bunyip anecdote; while the Nicholson Museum includes quite a sizeable section on the Etruscans.) After that we visited the Glass Artists’ Gallery (which has some very nice stuff but would benefit from a more spatious venue) and browsed a second-hand bookshop. We spent the evening with more friends of my parents.

On Wednesday we visited the outstanding Australian National Maritime Museum (notable exhibits: whale photographs and artifacts; information on little-known historical alliances; a beer-can boat; a panel on scurvy; some excellent paintings; and a dinosaur on a submarine … that last being a model that belonged to one of the crew), the Chinese Garden of Friendship, and the Sydney Aquarium. In the evening we went to the Capitol Theatre (a lovely venue, btw) to see the Lion King musical (parts of which are amazing).

View from Maritime Museum

Chinese Garden Chinese Garden

On Thursday, we each visited central Sydney museums independently, which in my case meant the Australian Museum (where my favourite exhibits were the crocodiles and the ankylosaur model), the Hyde Park Barracks (which, between the dead rats and the detailed reports on flogging, is a truly delightful place), and the Art Gallery of NSW (from which, here are links to selected works: [****]). In the early afternoon, my parents and I met one of the people I follow on Twitter for coffee.

In the evening I went by myself to the monthly Skeptics in the Pub (going to the wrong address at first, but finding the right one with the help of my Internet and GPS enabled phone) where I enjoyed conversing with Richard Saunders and Jo Alabaster from the Skeptic Zone podcast, as well as other people connected with the Australian Skeptics (including Ian Bryce and Tim Mendham). I gave Richard one of my customised coffee mugs.

On Friday we took the ferry to Nutcote (home of famous author May Gibbs, which I loved) and then took the bus to Taronga Zoo. Animals at Taronga which I don’t think I’ve seen at a zoo before include Komodo dragon and condor. After finishing there, we returned by ferry and got ready for our evening flight home.

Nutcote  Nutcote

Nutcote  Nutcote

Taronga - Komodo Dragon  Taronga - Condors

Taronga - lizard Taronga - lizard 3D

The last two photos above show a lizard of the genus hydrosaurus, the second time in 3D along with a bonus tortoise that I hadn’t noticed at the time. I had previously misidentified this as a tuatara, because it was adjacent to a seemingly unoccupied tuatara enclosure and I really wanted it to be one. (The enclosure shown actually belongs to the tortoise, so I’m not sure what the lizard was doing there.)

I bought several souvineers and gifts while I was in Sydney, mostly from the Maritime Museum and Nutcote. Overall I had an excellent time, which I hope comes across despite the brevity of my report.