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I am rewriting this post in early March because I am in the process of rewriting the linked PDF. The new version is more self-contained and much more comprehensive. This introduction will serve mostly as a space where comments can be posted.

Brief introduction: I came across (https://cameroncounts.wordpress.com/2012/06/21/fibonacci-numbers-2/) a discussion of the fact that any positive integer can be expressed as a sum of non-consecutive Fibonacci numbers (e.g. 50 = 3 + 13 + 34), and this inspired my mathematical muse.

A PDF of my observations can be found at https://app.box.com/s/tnrxizgqosvr2cjnf7hf2sk89h9j47jt
This is a warning about mermaids. Stay away from mermaids; those things are dangerous.

OK, for those of you who want to know more, here are the details.

One thing a lot of people get wrong about mermaids is that they tend to think of them as half human and half fish. In fact, no part of a mermaid is human and no part of a mermaid is fish, in the same way that no part of a stick insect is stick. Evolutionarily speaking, mermaids are closer to sea slugs than to any kind of vertebrate. You can think of them as a highly differentiated form of sea slug with a unique camouflage.

Another thing to know about mermaids is that they're like bees and ants in that they have different body forms for different social roles. The ones you hear about, the ones that look like a woman with a fish's tail, let's call them the foragers. They gather food and bring it back to the nest. Now, they have a kind of magic whereby anyone who is touching the mermaid can survive underwater. You don't need to breathe, you won't be harmed by the pressure, etc. Which is how they catch their food. First they'll lure you close, and then as soon as you have a firm grip on the mermaid, the mermaid will plunge. And then you're trapped. If you let go, you drown. If you don't let go, you live a little longer, but your fate won't please you.

What happens next is that the mermaid hands you over to the queen. Now, the queen mermaid does not look like a woman with a fish's tail. More like a huge squishy slimy blob with tentacles that sits on the ocean floor and does nothing much except lay eggs. And you're going to be attached to one of those tentacles. It begins with your genitals being physically fused to her body, somewhere not already occupied by the remains of a previous meal. Eventually your whole body will be melted into her as well, but it takes a while so you'll have time to be aware of the process. And in your final moments, immobile and covered in putrid, sticky slime, you might reflect on the fact that after she's absorbed all the nutrients your flesh contains, she will turn them into eggs and make more mermaids to lure more unwary victims to the same fate.

As I was saying: stay away from mermaids; those things are dangerous.
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Last updated 10 January 2016.

Mark Rosenfelder's China Construction Kit is one of two books currently on my to-read list. I've not yet read it in an organised, linear fashion, but I have opened it to random pages.

One thing that has intrigued me is the discussion of the classic 8th Century Chinese poem Lu Chai (~ Deer Park) by Wang Wei on pages 232-3. Rosenfelder provides the following translation of his own, as well as two other English versions for comparison. He also mentions the book 19 Ways of Looking at Wang Wei, which discusses multiple attempts made by translators over the years.

An empty mountain. No one is seen
Yet we hear the sound of voices.
Evening light penetrates the deep forest
It shines again on the green moss.

This made me want to create a version. I don't know any Chinese and did not do any prior research; I just wanted to try and distill what I intuitively felt were the most essential features of the translations provided into an aesthetically pleasing form.

(If any purists are reading this, I say only that there is a place for all kinds of adaptations, suitable for different purposes, and there is only a problem if one kind tries to pass itself off as another.)

Here's what I came up with after a day (v2). Hear me read this version aloud at https://app.box.com/s/swuxsin9k4tizkbu0l8cdcfs9z7nb6ya and scroll down for commentary and variations.

An empty mountain. No human presence seen.
Yet voices are carried on the air.
The light of dusk between the forest branches
Strikes green moss that breaks its shaded depths.

An earlier draft (v1) ended somewhat differently: "The light of dusk between the trees is shining / And strikes green moss where forest depths begin." At the time I imagined the sunlight penetrating the edge of the forest, but on reflection I decided it must refer to a beam (or multiple beams) of light fully surrounded by shadow, like the circle of yang surrounded by yin on the yin/yang symbol.

The following image captures the landscape and atmosphere that I imagine Wang was talking about. https://s-media-cache-ak0.pinimg.com/736x/3c/0b/6e/3c0b6ec1f5aa0013cc547cd623b8cfb1.jpg

The reference to the forest being deep seemed essential to me, but I couldn't fit it in a pleasing way on line three, so I moved it to line four. It also pleased me aesthetically to finish by evoking the image of dark forest punctuated by patches of illuminated moss.

I completely ignored the reference to periodicity (the "again" in the last line), as the significance wasn't apparent to me. Every translation adds some things and subtracts others, as you will see if you look it up online. But I've since realised this aspect can be reinstated by changing "between the" to "again through" in the third line, giving us the following (v3).

An empty mountain. No human presence seen.
Yet voices are carried on the air.
The light of dusk again through forest branches
Strikes green moss that breaks its shaded depths.

Researching the poem after completing my attempt (v2), I learned, among other things, that the Chinese does not explicitely mention it being evening. But apparently one phrase can refer to the evening idiomatically, and most but not all sources interpret as such it this context. Anyway, at the time I assumed the evening setting was significant because of being the time of transition between day and night — another yin/yang reference, but it seems to be more complicated.

Had I researched the poem BEFORE composing a version, I would not have written the same version. (I would not have written any version at all, being quite overwhelmed by the multitude of interpretations.) But this was an aesthetic exercise, not a scholarly one, so accuracy was quite beside the point.

My version makes no pretence to be anything it is not (such as "accurate" or "informed"). But I hope that in its unassuming whimsy, it pleases you.

Update: I've come up with a more literal but less evocative translation (v4):

Empty mountain. No-one seen
But voices echo in the breeze.
Setting sun through forest trees
Shines again on mossy green.

Some comments on this new version:

● I am now slightly better informed — roughly as informed as someone might be after finishing a school project or browsing ten pages of Google results. Not as informed as someone might be after having a school project marked.
● There's no question that v3 fits Western tastes better than v4. I tested both versions on Dad, who liked v3 and found it evocative but thought v4 was "very cold". I'd add that v4 sounds a bit like something out of a children's book.
● On the other hand, v4 is much closer to Wang and might be useable in a "just the facts" publication such as a textbook. I'm comfortable calling v4 a "translation" whereas I prefer to call v3 a "version".
● So, v3 is a better poem but v4 is a better translation. There's no escaping the fact that different versions have different purposes and a poet has to decide on their priorities. The poetic shortcomings of v4 might be largely due to my limitations and the limitations of English, but there are also cultural differences in what people expect poetry to be.
● If you like, you can ask which version comes closer to having the same effect on a modern Western audience that Wang's original would have in eighth century China, and whether any English versions by other translators come closer than either. But that would mean knowing what effect Wang was aiming for (e.g. whether it's to evoke a sensory image or something else), and since he isn't alive to correct you, you are largely free to conclude what you like.
● Commentary on Wang often finds Buddhist themes in the poem, the main points being (a) mention of emptiness, (b) indirect perception of reality, (c) symbolic significance of sunset and its direction, and (d) enlightenment of the green moss. I think this is probably correct, but it is also a nature poem, and Wang surely intended the two readings (natural and spiritual) to be in harmony and not in competition. (A few versions by other poets make the spiritual reading dominant.)
● My tentative suggestion is that the natural reading is a better guide for understanding what Wang put into the poem, whereas the spiritual reading is a better guide for understanding what he left out. I wrote v4 on the assumption that I'm right about this.
● Commentary also points out that ancient Chinese poetry was rich in parallelism, and I think this too is better reflected in v4. In the first two lines, the parallelism is sight vs sound, and in the last two lines, it's something like cause vs effect, or origin vs destination (of sunlight).

This is a PDF copy of the book reviewing the multiple translations.

http://www.textetc.com/workshop/wt-wang-wei-1.html

This is another article with interesting perspectives, although it rather bafflingly refers to "word-for-word pinyin translations" and to Pinyin differing in meaning from the original characters, which makes no sense as Pinyin is a romanisation scheme and has nothing to do with translation.

Rosenfelder (in a tweet responding to this post) is critical of this source, primarily on the grounds that it overanalyses the characters. I don't disagree, but with all its eccentricities I expect it still brings something to the table. And if all analyses are flawed, the cure is to read more of them so that no one source dominates our thinking.

https://zompist.wordpress.com/2016/01/03/wang-wei/

Written by Rosenfelder in response to this post. A quick overview of the poem word by word and line by line.

A Youtube video from a course on Chinese language. Another quick overview that also includes the entire poem read aloud. Clearly tries to keep things simple, possibly to a fault (as opposed to other sources that make things too complicated).

A Google Books result critical of Weinberger and Paz. Always good to have a second opinion. (See also Footnote #2 in the textetc article.) Page 125 is unfortunately not available in the preview; I'd be interested to know what it says.
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I was reflecting on "interesting number" lists such as http://www2.stetson.edu/~efriedma/numbers.html and http://www.archimedes-lab.org/numbers/Num13_23.html and thinking of doing my own.

My plan was to find one quirky mathematical fact about every integer from 1 to some arbitrary limit. I wanted to try and think of or discover the facts for myself, looking online for inspiration only after I'd had a fair crack at it.

I had various preferences for the kinds of facts I was interested in. They should be elegant and simple to state. They should be easy to verify with pencil and paper, rather than requiring a sophisticated proof. They should not depend on parochial things such as writing the numbers in base ten. They should not seem to make a number stand in the shadow of another. They should not cite a number as the second or third to have some property, but they might cite it as the first to have two properties simultaneously.

On the other hand, I was not looking for the most profound fact about a number, and often chose quirky ones that might have been picked randomly from a choice of multitudes. The very fact of their insignificance appeals to my aesthetic tastes.

If different people independently constructed their own lists it would be interesting to compare them, both to see the variety and distribution of facts arrived at and to see how different tastes influence the choices made. The one by Erich Friedman is less quirky and references a richer variety of mathematical concepts, but sometimes cites a number that isn't the first to have some property (e.g. 28, 29) or uses a fact that only works in decimal (e.g. 18).

In the end I decided to stop at 20. My complete list should be thought of as a first draft, as I would certainly expect to make changes before, for example, printing them on a T-shirt. You might suggest changes and alternatives yourself.

1 — Every circle has this many centres.

2 — Every line segment has this many ends.

3 — The least possible number of sides of any polygon.

4 — The least number of sides of any polygon in which each side is parallel to another.

5 — The least number of sides of any polygon in which all angles are obtuse.

6 — The greatest number of sides of any regular polygon that can be tiled to fill the plane.

7 — The first prime number that's not in the Fibonacci sequence.

8 — In a square grid, each cell is adjacent to this many others, including diagonals.

9 — The only square number that's greater than two raised to the power of its root.

10 — The only triangular number that's equal to the sum of all smaller triangular numbers.

11 — The first number that's a both square plus two and a cube plus three.

12 — The least number of equal sectors in a circle such that the sectors fit evenly into both a right angle and the angle between tangents of an identical adjacent circle.

13 — Exactly half of the integers from zero to this number inclusive are in the Fibonacci sequence.

14 — The greatest pyramidal number that's less than half of the next pyramidal number.

15 — The greatest triangular number that's greater than the square root of the corresponding factorial.

16 — The area and perimeter of the only square for which those values are equal.

17 — The first number expressable as aᵇ+bᵃ where a and b are distinct integers greater than one. Also expressable as aᵇ-bᵃ with the same value restriction.

18 — The total number of connections in an triangular grid graph in which the whole graph is triangular and exactly half the connections are on the boundary.

19 — The first sum of two consecutive pyramidal numbers which, represented as an octahedron, contains an internal vertex.

20 — The total number of connections in a pentagram.
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April Fools' Day was just a few days before Easter this year, and I'd thought that maybe I could cover both in the one blog post. But that isn't going to work — it would be too much of a break in the narrative flow. So instead I'll write about Easter on the blog and April Fools' here.

On Twitter I became a girl for a day, having discovered a ritual that, performed just before going to bed, causes a person to change gender overnight. As is normally the case on April Fools', the purpose wasn't to trick people (I mean, really!) but to entertain them.

I'd developed the ritual two or three weeks previously, coming up with something that to my mind has the right sort of atmosphere. The verbose version is as follows:

Kneel naked on your bed with a shot glass containing a little opal nera (http://opalnera.com/) on the mattress in front of you. Dip a finger in the glass and rub the drink onto your opposite nipple for three finger revolutions, then do the same with the inverse finger and nipple. With both hands, lift the glass to your mouth and drink the rest of the liquid. DROP the glass onto the mattress (note: my shot glasses are very bottom-heavy and resistant to tipping over) and finally clap your hands (only after which may you look to see how the glass landed).

The tweetable summary (which I could have expanded on if anyone had questions) was as follows:

If you kneel naked, rub opal nera on nipples 3 revs each, pick up glass 2 hands, sip, drop it on bed & clap, you'll change gender overnight.

I tweeted this at about 8:15am and quickly followed up with:

*WARNING* Aforetweeted ritual actually works. Do *NOT* try just for a laugh or to debunk it. That's what I did and now I'm a girl!!!!!

I also changed my username to Veronica C. Morgan (the choice of which we can talk about if you like), replaced my profile picture with one created on http://avachara.com/portrait/ and modified in an image editor (https://outerhoard.files.wordpress.com/2015/04/veronica-c-morgan-profile.png), and replaced my profile description with the following:

Abruptly transformed on 31/3/2015 from 180cm guy who didn't believe in magic into 165cm woman who does! Details in tweets. Scary amount of adjusting to do.

At 11:45am I posted another follow-up tweet, in the hope of being noticed by people who had missed my earlier ones. (The bit about coming across the ritual in my reading matter is part of the fiction; in reality, I made it up.)

Ritual cited incidentally in something I read. Tried in a spirit of ha-ha-bet-my-masculinity-this-doesn't-work. Must now accept consequence.

Traditionally, April Fools' Day ends at midday, though it's a worthy topic of conversation how to interpet that rule in the Internet age, when we routinely play jokes across timezones. I put my profile back to normal at 8:15pm, intentionally twelve hours after my original tweet.

Incidentally, I had not originally intended to replace my profile picture (on the grounds that if you really changed gender unexpectedly overnight, getting your camera out is not the first thing you would do), but when I discussed the idea with on Skype she advocated that I should.

Was it a success? Not really, because no-one replied. I think it could have been a lot of fun if some interaction had come of it, but you know what it's like when you put effort into something and no-one cares. I hope that people were quietly amused.
A BRIEF MEDITATION ON DICE

Suppose you want to memorise the precise arrangement of dots on a six-sided dice. What are the best mnemonics to use?

Everyone knows that opposite faces add up to seven. That's a good start, but there are still lots of possible arrangements within that constraint.

You might observe that the line through the 3 is almost a continuation of the line through half of the 6. That would not be the case if the 3 were located where, for example, the 2 actually is.

To say it more precisely, if you flattened out the cube, the line connecting all the dots of the 3 would be at an obtuse angle to a line connecting half the dots of the 6. Or else you can imagine rotating the 3 by an acute angle so that it becomes a continuation of half the 6.

You might also observe that one of the vertices has no dots next to it. The only place where that can possibly occur is between the 1, 2 and 3, and it does.

Those three rules -- that opposite faces add to seven, that the 3 is almost a continuation of half the 6, and that there exists a vertex with no dots around it -- leave only two possible arrangement of dots, one of which is the mirror image of the other.

Both arrangements are found. Wikipedia says most Western dice are right-handed, which means the 1, 2 and 3 proceed anti-clockwise around their common vertex, but left-handed dice are not particularly rare. Mine happen to be left-handed and not otherwise distinguished.

Incidentally, every possible set of three faces with a common vertex has something in common:

1, 2, 3 -- These are the lowest three numbers on the dice.
1, 2, 4 -- These are the powers of two.
1, 3, 5 -- These are the odd numbers.
1, 4, 5 -- These are the faces with a fully symmetric arrangement of dots.
2, 3, 6 -- These are the faces that are not fully symmetric.
2, 4, 6 -- These are the even numbers.
3, 5, 6 -- These are the numbers that are not powers of two.
4, 5, 6 -- These are the highest three numbers on the dice.
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Near the end of November, Irish artist @eolai-- assisted by @brandalisms-- spent 24 hours painting as many scenes as possible from photos that people sent in via the hashtag #100paintings1day . There was a Youtube livestream.

It was the second time such an event was held. Last year Eolaí painted 43 paintings out of more than 180 submitted. This year he only painted 29, but spent a lot more time on each. (I think he decided that rushing a painting is not something he enjoys all that much.)

The gallery is http://bicyclistic.com/index.php/one-day-of-painting-2014/ and includes a link to last year's gallery as well.

This year I decided to submit a photo of my own, and so scoured my archive in the days preceding the event.

I might have chosen a photo of my mother overlooking the place where a friend and amateur artist's ashes were scattered (see it here: http://outerhoard.files.wordpress.com/2010/06/pamsplace1.jpg), because that would have been a fitting tribute. But I didn't get an enthusiastic response to the idea from her surviving family, so instead I went with the other photo on my shortlist -- a quirky rather than sentimental choice.

The photo I sent in was https://twitter.com/GoldHoarder/status/538447666892402688. I chose it because I figured a dinosaur might be something Eolaí had never tried to paint before, and thought he might enjoy that. Also, I felt the scene was simple enough that it might work in broad strokes, but not so simple that it would be predictable.

My photo wasn't among the 29 lucky winners, but I knew from the start that it probably wouldn't be. The whole point of the exercise was for Eolaí to have as much fun as possible, and the rest of us were mere facilitators to that end. That's how it should be, and that is how it was.

I tuned in to the livefeed during the seventh painting, and retired for the night when the count was somewhere in the low twenties. In between I kept an eye on the feed -- sometimes continuously, and sometimes at fifteen minute intervals or so.

Watching the event reminded me of how, some time ago, I started doing a virtual tour of Ireland in Google Street View, looking for the most scenic route down the west coast and recording it. That's been on hold for about a year now, but I do mean to get back to it someday. (My worst nightmare is that Google withdraws support for Classic Maps before I get around to it!)
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This short reminiscence was prompted by a recent article by Darren Naish on the history of dinosaur art: http://blogs.scientificamerican.com/tetrapod-zoology/2014/09/01/changing-life-appearance-of-dinosaurs/

One of the first dinosaur books I ever owned -- and certainly the first that wasn't written specifically for children -- was the 1977 "The World of Dinosaurs" by Michael Tweedie (see http://www.amazon.com/gp/customer-media/product-gallery/0688032222/ ). This book is as old as I am, as 1977 is also the year I was born.

But at some point I got rid of it ... and with hindsight, I wish I hadn't.

My very fuzzy recollection is that in the late 1980s -- when changes in the depiction of dinosaurs hit popular culture in a big way -- I decided that any old books on dinosaurs had to be outdated and worthless. I didn't realise at the time that in specialist circles -- as opposed to popular culture -- the revolution had taken place two decades earlier, so the depictions in Tweedie probably weren't all that antiquated at all (or at least were transitional). Not yet a teenager, I wasn't mature enough to assess them for myself, and simply assumed they must be hopelessly out of date given how persistently more recent publications were going on about "new-look dinosaurs".

So yeah, throwing out Tweedie's World of Dinosaurs is one of my many regrets in life. I'd honestly be quite interested to know where it actually fits into the story of our changing assumptions about dinosaurs, because that would constitute a kind of closure. But without access to more than the front cover illustration, there's little I can say.
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I was curious about the angular diameter of the earth as measured from different heights above the surface, and inversely, about the heights at which the angular diameter of the earth would have particular values.

A search led me to the following useful tool: http://www.neoprogrammics.com/spheres/apparent_angular_diameter.php

At about 2,640 km above the earth's surface (comparable to the outer bound of low earth orbit), the planet would have an angular diameter of 90 degrees. At 18,260 km (roughly halfway to geosynchronous orbit), the angular diameter would be 30 degrees.

Closer to home, the angular diameter from 55 km above the earth's surface (roughly the top of the stratosphere) is about 165 degrees, so its curvature would be well noticeable by then. From 225 km the angle is about 150 degrees, the same angle as between analogue clock hands at 5:00 or 7:00.

Apply this information as you will.
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It was in 2008 that I linked to https://www.youtube.com/watch?v=sJWRkCfq1K0 from my blog, and the video has stayed close to my heart ever since. Honestly, I rank it among the most beautiful songs I've ever heard.

Recently, the idea occured to attempt a pareidolic English transcription (in the tradition of Four Tuna, etc). My best effort appears below. If you can suggest improvements, please do -- either to improve the pareidolic resemblance or to improve the semantic sense without sacrificing the resemblance.

(You can google the real lyrics.)

=====

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,