Links: Early May 2014

A few links to inform and entertain you:



Religious reminiscence: Crucifixion

Filed away in my desk drawer and certain locations on my computer, I still have various items from the days when Christian faith was a part of my identity. These include:

  • Some evangelistic web pages I wrote;
  • Religious discussions from online forums;
  • A notebook for jotting down theological speculations;
  • Various other items.

It’s been a few years since I’ve mentioned my religious past on this blog, but see my 2009 post about the afterlife and my 2011 post about prayer. As I aim to blog on a variety of subjects, now seems a good time to dig into those archives in search of an idea for a new post.

Since Easter wasn’t long ago, why not look at what my thoughts were, when I was a Christian, on the death and resurrection of Christ?

There is more diversity than often admitted, within thoughtful Christian communities, about what the crucifixion of Christ has to do with the reconciliation of sinners and God. Christians agree that the crucifixion was necessary, but there are different explanations for why (the New Testament’s account is largely metaphorical). Unfortunately, certain interpretations are more culturally dominant than others and the fact the discussion exists tends to be obscured. And of course, many people are content to accept the necessity on faith and leave the theologising to others.

The most familiar explanation is that it is an aspect of God’s nature that he literally cannot allow sin to go without its punishment, and that in the death of Christ he reconciles that necessity with the demands of his love by taking the punishment upon himself. We have all been exposed to that idea — not least from the allegories of C. S. Lewis — and there are many who insist on it as a defining article of Christian faith. But in all my years as a Christian, that viewpoint never made any sense to me, and it was never a part of my theology.

A less widespread explanation — but one with its share of advocates — is that the crucifixion was at heart the ultimate demonstration of God’s love. According to this idea, the normal expectation of humanity is to see God as judging and demanding, to be preoccupied with earning God’s acceptance and fearful of his rejection. And so Jesus comes to offer humanity a different view of God: a God who will forgive us even if we crucify him, a God whose love is (quoting Geoff Bullock) “beyond humankind’s capacity to earn it”.

That explanation also never felt like it could be the whole answer — largely because that view of God is clearly not as universal as it makes out — although I could envisage it as part of the answer. Anyway, those were the two competing explanations I found in books,* and, finding them inadequate, I speculated.

One line of thought I followed was that before a person can be healed of their sinful condition and fitted for Heaven, it is necessary to first develop a visceral appreciation of how serious sin is. That is to say, God can no more operate on the soul of someone who is insufficiently horrified by immorality, than a dentist can fix the tooth of someone who won’t open their mouth wide enough. And by bearing in mind the image of Jesus on the cross — the most perfect life ever lived tortured to death for the approval of the crowd — and learning to associate that image with all sin including one’s own seemingly trivial transgressions, it’s possible to put sin into its proper perspective and prepare the human heart to be transformed by God.

The main problem I acknowledged was that this doesn’t explain why some other illustration of the enormity of sin shouldn’t do just as well: why should the crucifixion and only the crucifixion be adequate to elicit the proper level of repulsion? I had no answer to that, but it seemed plausibly on the right track.

A further speculation which unites some of the above ideas is best presented as a metaphor, like the Scylla and Charybdis of Greek mythology. In this account there are two opposing obstacles on the spiritual journey, both of which must be negotiated if one is to make it to Heaven. The first is the danger of thinking our moral imperfections are no big deal — that we are good enough without God’s intervention — which in Christian thought is often looked upon as the ultimate folly. The second is the danger of being so overcome by a sense of our own inadequacy that we imagine God would never accept us, and are afraid to make the approach. Above this landscape — a narrow passage between two opposing and terminal errors — the Cross shines like a lighthouse, warning us at one and the same time not to underestimate sin, but also not to underestimate forgiveness.

That is as good an answer as I ever had. It works pretty well as a metaphor, but (just like every other explanation) not so well under clinical observation. In typical mythological style, the obvious questions — like why one universal lighthouse is better than several smaller ones — can be answered only by appealing to the seductive simplicity of the story. I realised that my answers were inadequate, even in conjunction, but they reinforced the impression that there was an answer to be found. At the end of the day, though, I accepted the idea that the crucifixion was somehow necessary for humans to be fitted for Heaven, even if I didn’t see why.

No religious belief can be understood in isolation, and everything I’ve said relates to topics that are outside the scope of this post (for example, this is not the time to discuss how I understood the identity between Jesus and God). But I think it’s worth saying that I always believed God shares all human suffering — like a divine mirror-touch synaesthete — and that I was absolutely sure no-one’s destiny hinges on ideas they may or may not encounter in this lifetime (as stated in my 2009 post).

We are all shaped by beliefs we no longer hold, and no doubt I wouldn’t be the same person today if I hadn’t in the past been a devoted believer. I am convinced of the value of occasionally discussing our former convictions not to defend nor refute them, but in the same spirit that one might share an old photograph. I hope this reminiscence has contained some thought for everyone to take away, Christians and atheists and all others as well.

* For further reading, see The Plain Man Looks at the Apostle’s Creed by William Barclay. (Also, Power of Your Love — Jesus: The Unexpected God by Geoff Bullock, although that book is irritatingly prone to speculation masquerading as fact.)

Page and sidebar renovation

Just a quick announcement here: I’ve written four new pages for this blog and made some adjustments to the sidebars.

The new pages are linked to from the left sidebar and also below. They replace the old pages, which were several years out of date.

If you’re reading this from a subscription feed of some kind, please visit the blog directly to see the updated sidebars.

If you’d like to offer feedback on the changes, please leave a comment below.

A cruel-coloured scathed crow

Audio pareidolia applied to song lyrics is a potent source of comedy. A single misheard line is often amusing enough, but the illusion is taken to another level when the imposed and original lyrics are in different languages. You may remember the Four Tuna video that went viral several years ago, in which a very well-known Latin song is given English captions that somewhat resemble the Latin phonetics. The result is simultaneously hilarious and fascinating.

I have listened countless times to the Youtube video of Karan Casey singing A Chomaraigh Aoibhinn O — an extraordinarily beautiful Irish song. I first linked to it in 2008, and it’s still a favourite. The lyrics, alongside an accurate English translation, are available here.

Recently I decided to give it the Four Tuna treatment, and my fake English lyrics are given below, underneath the original video. You can follow along and see how well my false lyrics fool your brain.


Moving hard down creek,
Nor the kid is still related;
I come back heaving gold.
Store the vine, though a cool work,
And your do will clearly fail all;
I come back heaving gold.
Laugh, ha-ha, yellow well,
Spoke a kind-hearted greybuck;
The gloam does swallow
The font of the layerer.
Oh, Grandma Creek,
It’s suet lower K, lol,
I come back heaving gold.

If stone grew barren
At a cruel-coloured scathed crow,
I come back heaving gold.
Through the last seven sewers,
Little heart’ll want a grainer;
I come back heaving gold.
Our fastest larrikin,
Here gone the pathed fields;
My wrath I show,
Let it show o’er the glazed fair.
Wrote a scar, penned and drew it,
And a new sort of spare art;
I come back heaving gold.

Nor veer sour or solar,
Shadowly we came, which
I come back heaving gold.
He knew we’d get fond
And yearn, let it stay here;
I come back heaving gold.
Our barber carried on,
Got a new sort of cradle;
If a heart got barred,
Fish can like a late shift.
It’s cost me a legion,
I flew a lot of leisure;
I come back heaving gold.

I vaguely entertain the notion that a skilled satirist could weave some story around these words — as another layer of pareidolia, that would be somewhat fitting. They’re utterly meaningless, of course, but it does no harm to caution readers in Ireland to watch out for the cruel-coloured scathed crow, just in case.

Does it work for you?

[I originally shared this on Google Plus --- which is sometimes useful for sharing things that are still taking shape in my mind, with minimal attention to presentation or how they might seem in retrospect --- but I've decided it merits a place on the blog.]

Links: April 2014

Links from the last month:


  • Widely-reported astronomical discoveries included an asteroid with rings and the second object (after Sedna) ever to be found in the inner Oort cloud. Links go to the best articles I’ve seen on each topic.
  • Also in astronomy, a model of Enceladus’s ocean derived from gravitational anomolies. This doesn’t tell us much we didn’t already know, but it’s an interesting confirmation.
  • A lovely tribute to a strange sea slug.
  • And in more creature strangeness, a sexually-reversed insect.
  • A relatively obscure 17th century alchemist and inventor.
  • British Pathé released an extensive archive of historical television clips. It’s well worth a browse, and if you find some favourites I encourage you to link to them in the comments. Meanwhile, each of the following asterisks links to a different video to get you started: [********]. In my opinion the most interesting ones are often those that reflect changing social attitudes.


  • For when you have plenty of time: the original Cosmos series on Youtube.
  • Christie Wilcox shares a dream. (Have you ever dreamed about a new species of animal? One of mine last year featured a migrating desert snail monotreme, but even in the dream I didn’t get to touch it; just saw the news report.)
  • Cityscape timelapse with a difference.


Buying my second mobile phone

I typically publish links at this time of month, and the following personal update was drafted as a footnote below just such a collection. But on reflection I’ve decided to let the links accumulate for another two weeks before publishing them, so the wouldbe footnote is now the entire post.

So … recently I bought myself a new mobile phone, replacing the one I bought in 2008 and reported on this blog at the time. After almost six years, the ringtones in the old one stopped working in January (broken speaker I guess), and I’ve been procrastinating getting a replacement ever since.

The new phone is an android (Samsung GT-S7500T) and cost me about $150. I’ve installed the following apps: LocSMS, Battery Widget, Brightness ControlRocket Music Player, Date in Status Bar, File Shortcut, MuPDF [list updated July 2014].

For SMS notifications I’ve chosen the Pizzicato tone that came with the phone, and for incoming calls I’ve imported the ringtone I made for the old phone. To create wallpaper for the new phone, I rotated and trimmed one of my fractal images, and you can see the result below. (Incidentally, the same source image is also used in the background of my profile on Twitter, Gravatar and other places.)


Of course there have been disappointments — ranging from conspicuously absent features to Google doing funny business without my consent — but I don’t like to use the blog as a place to rant. Besides, my feelings about a new phone are not interesting; what’s important is how I’ll feel after I’ve had a chance to get used to it. If it lasts another five or six years I will, on balance, be happy.

Navigating the night sky

Which stars do you recognise when you look up at the night sky, and how do you find them?

I’ve been planning to re-introduce shorter, lighter blog posts on random topics, and one topic that seems fit for that purpose is to describe how I personally find my way around the sky at night. This is not meant as advice on how other people should navigate the sky; it’s just a description of how I do it, and you are welcome to reciprocate in the comments. It is also only a summary and not a guide, so please look things up if you need more information. Bear in mind that I’m in the southern hemisphere.

When the sky is not yet dark and only a few stars are visible, I like to identify those stars as early as I can. This requires being familiar with the shapes formed by only the brightest stars, without relying on dimmer ones for context.

I start by looking for the triangle formed by Sirius, Rigel, and Canopus, which is back-to-back with the triangle formed by Sirius, Rigel and Procyon. All of these stars are among the brightest in the sky, and Sirius is the brightest of all, so once you’ve identified Sirius you know that anything brighter is not a star. The angles between these stars can look quite different depending on whether they’re overhead or near the horizon, but I can usually identify them quickly. (From Sirius, the line to Canopus is longer than the line to Procyon, so I find it helpful to remember that this is the opposite of what you’d expect given that pro- invokes words like prolonged.)

Once I’ve identified Rigel I can easily find Betelgeuse, and thus predict where the belt of Orion will appear even before it is visible (good party trick). If the sky is dark with little light pollution, I also look nearby for the Hyades (a.k.a. the horns of Taurus) and identify Aldebaran.

My usual next step is to follow the zigzag line from Procyon to Sirius to Canopus and then take one more zag to Achernar. Achernar is the only bright star anywhere near its vicinity, so you don’t need much precision to find it. The line of best fit through these four bright stars I have nicknamed the false ecliptic, because if you didn’t know your way around the night sky you might assume they were planets.

We’ll come back to Achernar in a moment, but let’s now turn our attention to the stars of the Southern Cross (featured on the flags of many Southern Hemisphere countries) and the adjacent Pointers. These are easy to find, and not hard to identify individually. The Pointers are Alpha and Beta Centauri (also known as Rigel Kentaurus and Hadar, but I prefer the former names) and the stars of the Southern Cross are Alpha, Beta, Gamma, Delta and Epsilon Crucis. Telling them apart is easy when you know that the sequence from brightest to dimmest proceeds clockwise, and that they are named in that order.

The classic way to find the South Celestial Pole is to draw a line through the long axis of the Southern Cross and another line passing between the Pointers at 90 degrees, and to find the point where those two imaginary lines meet. An alternative method — slightly more accurate but less often practical depending on what clouds and the horizon obsure — is to identify Beta Centauri (the lesser Pointer) and Achernar (which I promised we would return to), and find the point halfway between the two. It’s no bad thing to have two different methods for finding the Pole, because then you can verify one against the other and also use this as an alternative way to find Achernar.

In total, I can point to and name 14 stars in the night sky — Sirius, Rigel, Canopus, Procyon, Betelgeuse, Aldebaran, Achernar, plus the seven stars of the Pointers and Southern Cross. I’ve never made the effort to memorise any more. It’s perhaps worth noting that I live in a fairly light-polluted city, but spent most of my childhood on a farm with many more stars visible.

That’s about as much as I can say from a personal perspective; most other details are things you can look up. If you have a perspective you’d like to share, it is now your turn.

Links: Late March 2014

Here are some links that I’ve seen recently and recommend. It’s an intentionally short list; I’ve shared several more on Twitter.

The big news in physics this week is the detection of polarisation patterns in the cosmic microwave background radiation that match what is expected if inflation theory is true. However, I’m not able to recommend a single best article on this topic, so I’ll leave it out.


  • The origami microscope is amazing. The TED talk will probably leave you with unanswered questions, like “Huh?”
  • LONG article involving autism and cartoons. Worth the effort.
  • The role of sponges in oxygenating the early oceans. Article could be better written, but it’s an intriguing idea.
  • You probably already know why Americans are weird (and, at the risk of spoiling the joke, so are Australians). But read this anyway; it’s a well-written overview.


  • I tried out the demo version of Osmos — a game in which you play the part of a blob absorbing (or being absorbed by) other blobs. I like it a lot, but with one deal-breaking reservation: the game saves information files in the user’s Documents folder. I feel strongly that the Documents folder should be reserved for the user’s own files, and that any software that stores files there automatically (without letting the user select an alternative location) is intruding on their personal space. (Note: At the time of writing, the most recent Windows version is 1.6.0.)
  • Hilarious clip involving a dog at a restaurant.

Elke at 10 months

The last time I posted photographs of my sister’s daughter, Elke Adele Smith, she was six months old. She’s now ten months, and I have new photographs.

Here is my favourite, rolling about by the River Torrens in Adelaide:


More from the same place. Mum (her grandmother) also appears, in her role as chair and climbing frame. You can see Elke is trying to learn crawling, but her legs haven’t figured it out yet.






We passed through the Festival Theatre, where I made this video …

(I used Youtube’s frankly awful background music selection tool — which I’d rant about at length were it not for the fact that doing so would be incongruous in a blog post about a cute baby — but for a 30 second video it can be made to work)

… and took these photos:


Here we are walking by the Torrens and later reading a book:


Finally, more playing around. This time in the playground by the caravan park:






I see no need to add much commentary; the pictures speak for themselves. She’ll soon be a whole year old, which is pretty amazing.

Operational algebra

I vaguely remember wondering — as a child of seven or eight — how new mathematics became established. I pictured a long line of people in front of a guy with a desk (or maybe a tent), waiting to register their latest idea — such as a new mathematical operation.

I had one such idea myself. I knew by this age that multiplication consisted of repeated addition, so I thought why not define something else in terms of repeated multiplication? Thus I invented integer exponentiation. (It’s an obvious idea: I blame the Childcraft books for arousing my interest.)

When I was older and learned that exponentiation actually existed, I was troubled by a curiosity of language. When people talk about 104, they say “ten to the power of four” — and yet 102, 103, 104 are collectively referred to as “powers of ten”. In one context, the number after the “of” is the exponent, and in the other context, it’s the base. Why is this? Isn’t mathematics supposed to be consistent?

If I could answer my childhood self, I’d point out that the same ambiguity crops up in everyday language. We say that x-ray vision is one of the powers of Superman, but we also say that Superman has the power of x-ray vision. When we use terms like “power of” we are translating mathematical notation into English, which has no use for consistency.

I not surprisingly came to speculate about an operation defined in terms of repeated exponentiation, another operation defined in terms of that, and so on ad infinitum. Some of you will find this familiar.

Let’s adopt the symbol ➀ for addition, ➁ for multiplication and ➂ for exponentiation, and let’s have ➊ for subtraction, ➋ for division and ➌ for the calculation of roots (although exponentiation actually has two inverse operations — the other being the calculation of logarithms — so we’ll leave that well alone).

Now we can specify operations using algebraic variables — and for that reason I referred to this system as operational algebra. I’ll use the symbol ✪ for “operation”, and write 2 [✪3] 5 to mean the same as 2 ➂ 5 which means the same as 25. For inverse operations I’ll insert a minus sign so that 2 [-✪2] 5 means the same as 2 ➋ 5 which means the same as 2/5. My original notation was handwritten, obviously, but this is a reasonable compromise. I won’t pretend it’s perfect.

For positive integers, higher-order operations can be defined as:

a [✪n+1] b = a [✪n] a [✪n] … [✪n] a {b terms}

But because exponentiation is non-associative, this presents us with a choice. If we bracket the expression from right to left — as in 2(2^(2^(2^…))) — then we are in the same realm as Knuth arrow notation (which I didn’t know about at the time). But if we bracket it from left to right — as in (((22)2)2) — then because (ab)c=abc, it follows under this definition that ➃ can be extended to non-integer operands using the formula a ➃ b = a(a^(b-1)), and then a ➄ b can be defined for non-integer a. I always used the latter (left-associative) definition.

As for higher operations in this sequence, I was convinced at the time that they could all be extended to non-integer operands, but my reasoning was faulty.

The fun wasn’t all at the high end, though. Early on I recorded the trivial fact that there is no zeroth operation. That is, no operation can be consistently defined such that a + b = a ✪ a ✪ … ✪ a {b terms}.

I don’t propose to share everything I did with this system. For one thing, a lot of it was wrong, and for another, I threw my notes away years ago. But here is something worth finishing on. Recall that an inverse operation can be defined such that:

(a [-✪n] b) [✪n] b = a

Now observe:

  • 2 ➊ 2 = 2 – 2 = 0
  • 2 ➋ 2 = 2 / 2 = 1
  • 2 ➌ 2 = sqrt(2) = 1.41421…
  • 2 ➍ 2 = [solution of nn = 2] = 1.55961…
  • 2 ➎ 2 = [solution of (n^(n^(n-1)) = 2] = 1.6498…

Based on this, it seems intuitively likely that if we could extend all higher operations to non-integer operands, then as n approaches infinity, we’d find that 2 [-✪n] 2 approaches 2, giving us a glimpse into the infinite operation. This intrigued me.

I abandoned my speculations in operational algebra around the time I left school and started university. I am vaguely aware that everything I explored (except the wrong bits) exists in established mathematics, if often subsumed under much more abstract generalisations. This makes no difference to the fact that a teenager had fun messing about.

As a final word: please don’t take from this blog post the idea that I’m good at maths. I lack the rigour to be good at it; I am merely playful.