Thought Association

I was jogging on the beach, trying to think of something else because the last couple of days had been rather upsetting. I settled on thinking about an essay I am trying to write about The End of The World. Very soon I found that I had the REM song It’s the end of the world as we know it running through my head on repeat.

After a while I noticed somebody running along next to the concrete promenade, where the sand is softest because it is furthest from the water and almost never gets wet from the sea. The sand was pretty soft where I was, about halfway between the promenade and the water. But maybe it was softer over near that other guy. In any case, we’d had heaps of rain recently, so if water makes sand pack together harder, presumably where I was would be just as water-hardened as next to the promenade.

But then maybe seawater has a different effect. Perhaps it makes the sand stick together better than rainwater does. If so then the sand next to the promenade really would be softer, unless the sea ever gets up to there.

That led to me wondering about whether, in the wildest sorts of weather, the sea ever came all the way up to the concrete wall below the promenade (about fifty metres from the high tide mark).

Thinking of stormy weather made me think of the scene in the movie The French Lieutenant’s Woman where the female lead stands at the end of a long jetty in a storm, only a metre or two above the rough sea – a precarious position, deeply evocative.

That led me to wonder whether it is sexist to refer to the character as somebody’s ‘woman’, thereby seeming to suggest ownership. That led to my thinking about the reverse phrase ‘somebody’s man’, which led me to think of the Tammy Wynette song Stand by your man.

And without any conscious decision to do so, there I was, jogging along the beach, mentally humming Stand by your man instead of It’s the End of the World as we know it.

Thought association.

Andrew Kirk

Bondi Junction, April 2017

Featured Image is from the 1981 movie The French Lieutenant’s Woman, showing the jetty called ‘The Cobb’ at Lyme Regis UK.

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The End of the World

One day the sun will grow so large that it will first dessicate, then bake, then engulf and vaporise, the Earth and everything on it. No life will survive that. Perhaps some people will have escaped to habitable places in other solar systems, but it’s hard to imagine it would be many, given the enormous energy that is likely to be involved in any interstellar travel. I expect ordinary people will be unable to escape.

Even escapees will be wiped out eventually, as the universe, many billions of years from now, slides inexorably into heat death. No life will survive that.

So there it is: the end of the world is a matter of when, not if. We are powerless to prevent it.

That background makes it a bit confusing to work out what moral obligation we have to take actions that prevent a near-term end of the world, and to avoid actions that would hasten it.

If we are talking about preventing the end of the world in our lifetime, it can be easier to resolve, because that would affect people that are alive now, and most people recognise that they have at least some obligation of care to other people that cohabit the world with them.

But that obligation is less widely accepted when it comes to future generations, and the farther away those generations are, the fewer people tend to feel an obligation towards them. Politicians sometimes talk about intergenerational equity and caring for the future of our children, maybe even our grandchildren. But it’s a rare politician that argues for a policy on the basis of its effect on our great-great-great-great-great-great-great-great-children.

At some stage, life on Earth will come to an end, and it seems likely that that end, unless it occurs in the blink of an eye – which it is hard to imagine happening – will be accompanied by tremendous suffering. If that is inevitable then how can we work out whether it matters whether it occurs sooner or later?

We cannot solve this by reason alone. As David Hume so acutely observed “Tis not contrary to reason to prefer the destruction of the whole world to the scratching of my finger.” Of course he wasn’t saying he did prefer the destruction of the world. He was saying that one must look to one’s emotions to find an answer.

Looking to my own emotions, I confess that I am more alarmed at the prospect of the world ending in a catastrophe in 100 years than in 100 million years, despite the fact that I will not be here to see either..

I use the word ‘catastrophe’ rather than ‘cataclysm’ because I think the end would be lingering and painful. We should be so lucky as to be extinguished in the blink of an eye. Philosophers who have nothing better to do with their time make up thought experiments involving a button you could press to instantaneously end the world, and ask under what circumstances you would press it. But there are no such buttons, nor ever likely to be, so we need to contend with the end being a drawn-out, painful process. I suspect widespread famine would be a major part of it. That would lead to outbreaks of uncontrolled violence as people compete for the dwindling resources of food and water. Disease would spread to accompany the famine – perhaps providing a more merciful end for some. We see this sort of catastrophe already in some parts of the Earth, and we will see them more often as climate change becomes more severe.

Is a ‘soft landing’ possible? What if, realising that the world will become uninhabitable within 200 years, we were to decide that we were morally obliged to not have children, in order not to inflict on those new people the pain of experiencing the world’s slow death? What would a world with no new children be like? Most of us, including me, feel that it would be very sad. I know of two novels that explore this: ‘The Children of Men‘ by PD James, and ‘I who have not known men‘ by Jacqueline Harpmann. In the first, for some unknown reason, humans cease to be able to conceive. The novel is set about twenty-six years after the last baby was born. In the second novel, a group of female prisoners escape from their underground dungeon to find the Earth deserted. They wander for many years in vain search of other survivors and after a while start to die of old age, with no replacement.

Both novels are confronting, bleak and sad. The James also has a thriller element to it (which I won’t spoil for you), but the basic premise is still bleak.

It would be very hard for us now to decide ‘No more babies’. Imagine us all gradually dying one by one, deprived of that feeling of continuity – the circle of life – that one gets from seeing younger generations. But what if society had the time to work up to that over several generations? What if, realising that all life would cease within ten generations, society worked to change its culture in order to equip people to feel more positive about non-procreation and less reliant on younger generations. It would be a very difficult psychological shift to accomplish. It would have to counteract the powerful impulse embedded in our psyche by evolution – to perpetuate the species. But who knows what techniques of psychological manipulation humans may have managed to invent in a thousand or more years’ time? Maybe they could condition future humans to find fulfilment in bringing their species in for a soft landing – for instance in working as a childless carer for old people until one becomes too old to work. Things could be set up so that the last remaining people have all the food, water, clothes, medicine, shelter, power and entertainment they need to survive solo (we would also need to train people to be comfortable with isolation, which we current humans are definitely not). They might also be provided with pills to provide a painless end to life once they near the point where they will no longer be able to feed themselves. That is not how it happens in ‘The Children of Men‘. But that book is set in 2021, not 3021, and with no notice for society to prepare for the landing (for some reason fertility just suddenly ceases in 1994).

If a soft landing were possible then, while an end of the world may be inevitable, its accompaniment by great suffering would not be. It would then become easy to argue for doing what we can to delay the end of the world. It is simply to prevent a great suffering.

What if it’s not possible, so that the great suffering is simply a matter of ‘when’ rather than ‘if’? What if the amount of suffering accompanying the end of the world will be roughly the same regardless of whether it occurs in 200 years or 200 million years? Are we morally obliged to do what we can to defer it beyond 200 years? I pick 200 years by the way because that should be long enough to be fairly certain that nobody currently alive will be around to experience a world’s end in 200 years.

It seems to me that the main difference between the two end dates is all the currently-unconceived humans that would experience life in the intervening 199,999,800 years. Is it a good thing or a bad thing that such lives should come to pass? There is very little moral guidance on this. Even religions have little to say about this, with only a very few religions (albeit big powerful ones) forbidding contraception.

A group that has a decisive opinion that is the direct opposite of the anti-contraceptionists is the anti-natalists, led by the prominent South African philosopher David Benatar. Benatar argues that, since all life contains some suffering, it is immoral to create any new life. He does not accept that suffering may be offset by pleasure at other times in a life. Even a few moments of mild pain in an otherwise long, happy life makes the creation of that life a moral mistake, in Benatar’s book. Less extreme anti-natalists argue that procreating is OK if we think the new life will have more pleasure than suffering but that, since we can’t be sure, we are obliged to not procreate. A more folksy version of this is the comment uttered at many a late-night D&M discussion, that ‘this is no world to bring an innocent child into‘.

Not many people are anti-natalists. Most people, despite the exaggerated doom and gloom on the news – terrorist this and serial-killer that – (of course no mention of the real dangers like climate change, malaria, poverty, road carnage and plutocratic hijack of our democracies) see life as a generally pleasant experience and look positively on conferring it on new humans. But that tends to be a very personal feeling, in which the moral dimension cannot be disentangled from the powerful personal urge to procreate.

For those of us who are neither anti-natalists nor anti-contraceptionists, the question of those lives in the intervening 199,000,800 years remains a mystery to be explored. Is it important that they come to pass? Is it good that they do so?

Lest you decide I sound like a homicidal maniac and ring Homeland Security to have me ‘dealt with’, let me state here that I feel that it is better to do what we can to delay the end of the world. That’s a major factor in why I think action on climate change is the most important issue facing humanity today. But I won’t go into the reasons why in this essay, because this topic will be discussed at my upcoming philosophy club meeting and I want to avoid spoilers. In any case, I’m more interested in what other people think about this.

The dilemma posed by this essay was first raised by Oliver Kirk.

Andrew Kirk

Bondi Junction, April 2017

Questions

  1. What, if any, obligations do we have to unborn generations? Do they include an obligation to ensure their existence?
  2. Does the nature or strength of the obligation change with the remoteness of the future generation?
  3. If we accept that the end of humanity will occur, and will be accompanied by great suffering, are we obliged to do what we can to delay it for as many centuries or millennia as possible (taking as agreed that we are obliged to delay it beyond the lifespan of anybody currently alive)?
  4. If we do feel obliged to delay, does that imply an obligation to maximise the population of the Earth, subject to being able to maintain adequate living standards?
  5. How do I feel about the fact that a time will come when there is no more life? Does it strip life of meaning? Or does it enhance meaning? Or neither?
  6. How would I feel about a world in which human reproduction became impossible?
  7. Do I feel differently about the world ending in 200 years from how I feel about it ending in 200 million years?
  8. What implications do our opinions on the above have on our feelings of what stance we should take on current future-oriented issues like climate change, balancing government budgets, infrastructure building, asteroid mapping, solar flare prediction?

References

PD James: ‘The Children of Men’

Jacqueline Harpmann: ‘I Who Have Never Known Men’ (‘Moi qui n’ai pas connu les hommes’)

Peter Singer: “Practical Ethics’. Discussion of obligations to future generations on p108-118 of Third Edition (2011, Cambridge University Press).


More fascinating facts about stars

I’ve had a few thoughts and discussions since I wrote this article about drawing stars. I thought it was worth sharing.

Stars within stars

The first observation is that stars, drawn in the way I described, contain other stars, nested within one another like a set of Russian dolls. Recall that we use the term ‘n-k star‘ to indicate a star with n points such that, if we draw a circle through all the points and consider the boundary of that circle as split up into n curve segments bounded by the points, then the straight line from one point to another traverses k of those curve segments. Like this, for a 7-3 star:

star 7-3 with arrows

Yuriy made an interesting observation about the stars in my last article, that the sum of the angles of all the points of an n-k star is 180 × (n-2k). In the course of thinking of ways to prove that formula true, I came upon the realisation that a n-k star contains a n-(k-1)star, which contains a n-(k-2) star and so on down to the n-1 star, which is a regular, n-sided polygon.

Here’s a picture that shows this for an 11-5 star.

11-5 with labels

The red, outer 11-5 star contains a green 11-4 star, which contains a red 11-3 star, which contains a green 11-2 star, which contains a red 11-1 star (polygon). The points of each inner star are the innermost vertices of the star that immediately contains it. Since we will be referring to those vertices again later, let’s make up a name for them. We’ll call such a vertex a ‘tniop‘, since it is in a sense the opposite of a point. The above diagram shows a point and a tniop. We’ll call the stars inside a star ‘sub-stars‘.

We saw in my last essay that, when a star cannot be drawn without taking the pencil off the paper, it is made of a number of ‘component stars’, that are rotated copies of one another. Here is a picture of a 16-6 star, which uses different colours to highlight the two component stars. We have two 8-3 stars, one light blue and one red.16-6 [1,0,0] star

And here is a picture that uses colour variation to show the sub-stars of each of the two components.

16-6 [1,1,0] star

The 8-3 light blue star contains an 8-2 star which is made up of two components that are 4-1 stars (also known as ‘squares’) coloured blue and pink. Similarly, the 8-3 red star contains an 8-2 star which is made up of two components that are 4-1 stars (also known as ‘squares’) coloured yellow and green.

The 8-2 stars each contain an 8-1 star (octagon) as the intersection of two squares -pink and dark blue for one octagon and yellow and green for the other.

Finally, those two octagons between them, bound a hexadecagon (16-sided polygon or 16-1 star). So altogether, in the one picture, we have:

  • one 16-6 star (red and light blue)
  • one 16-5 star (also red and light blue) inside that
  • one 16-4 star (pink, green, dark blue and yellow) inside that
  • one 16-3 star (also pink, green, dark blue and yellow) inside that
  • one 16-2 star (also pink, green, dark blue and yellow) inside that
  • one 16-1 star (also pink, green, dark blue and yellow) inside that
  • two 8-3 stars (one red, one blue) making up the 16-6 star
  • two 8-2 stars (one pink and dark blue, one yellow and green), one inside each of the 8-3 stars
  • two 8-1 stars (octagon: one pink and dark blue, one yellow and green), one inside each of the 8-2 stars
  • four 4-1 stars (squares: coloured pink, dark blue, yellow and green) which, in pairs, make up the 8-2 stars.

That’s sixteen stars altogether. What a lot of stars in one drawing! Can you see them all?

Here’s a different colouring that makes it easy to see all five 16-point stars:

16-6 [0,1,1] star

Although the stars get smaller as k reduces, they do not shrink away to nearly nothing. In fact they get closer together as they go inwards, as if they are asymptotically approaching a circle of some fixed, minimum size.

To investigate this, I drew a 101-50 star:

101-50 (cc_0,cs_1,st_0,bn_0,bg_FALSE) star

You’ve probably noticed by now that I’m no longer drawing these by hand. My drawing is much too wobbly to capture the intricacies of stars-within-stars. So I wrote a computer program to draw them for me. I’ll try to remember to attach it at the end of the article, so that those of you who like mucking about with computers can muck about with it.

Anyway, that 101-50 star pretty well killed my hypothesis that the inner stars can’t get very small. In this one they almost disappear out of sight. I like the swirly patterns. I haven’t yet worked out whether they are really features of this very spiky, very complex, star, or whether they are just artefacts of the crudeness introduced by the computer’s need to pixillate.

Here’s a zoomed-in image of the interior of that star. Cool, eh?

zoom of big star

This is a low-resolution image. I have saved a moderately high-resolution image of this star here. Zooming in and out is fun. It seems almost fractal as more patterns emerge from the inside when we zoom in. Also, the stars give the illusion that they are rotating as we zoom. To get the best effect you need to download the file (a .png image file) and then open it up, so that zooming is not limited by your internet connection’s speed.

Ratio of Outer to Inner radius

Let me briefly pick up on that idea above about whether there is some minimum inner radius for these stars. For each n, the spikiest n-k star is where k is the largest integer less than n/2, and this contains another k-1 stars, nested one within the other, down to the innermost, which is a n-sided regular polygon. We can work out the ratios of each star to the one immediately inside it, and use that to work out the ratio of the outermost, n-k, star, to the innermost, n-1, star. The attached computer program contains trigonometric formulas to do that. Here are the ratios of the radii of the innermost to the outermost star for each n from 1 to 109:radius ratios table

We observe that the ratios generally go down as n increases, but the decline is not steady. It bumps up and down. I have highlighted the prime numbers with asterisks. Notice how the ratio for those is always lower than for the numbers around them. The two drivers of the ratio seem to be:

  1. the size of n: the ratio generally declines as n increases; and
  2. the number of different factors n has. Note how 16 (divisible by 2, 4, 8) has a higher ratio than 15 (divisible by 3, 5) and 56 (divisible by 2, 4, 7, 8, 14) has a higher ratio than 55 (divisible by 5, 11).

This would be interesting to look into further, and to see if there is some neat, sweet, compact formula for the ratio that highlights the relationship to size and number of factors (if there really is one). But I have to stop thinking about that now or I’ll never post this essay.

The most general form of symmetrical stars

We can make an awful lot of stars using the above approach. For an integer n the number of different n-pointed stars is the biggest integer less than n/2.

But in fact there is an infinite number of different n-pointed stars, without having to loosen our standards by allowing asymmetry. After a bit of thought, I realised that the most general form of n-pointed star can be specified by a single number, which is the ratio of its inner radius to its outer radius. The outer radius is the radius of the circle on which the points sit. The inner radius is the radius of the circle on which all the tniops sit. Given n and that ratio – call it θ, we can draw a star as follows:

  1. Draw two concentric circles with ratio of the inner to the outer radius being θ.
  2. Mark n equally-spaced dots around the outer circle and draw faint lines connecting each of these to the centre. These will be the points of our star.
  3. Mark a dot on the inner circle halfway between each of the faint, radial lines drawn in the previous step. These dots will the the tniops of our star.
  4. Working consistently in one direction around the circle, draw a zig-zag line fro point to tniop to point to tniop and so on, always connecting to the nearest dot.

We will describe such a star as a n/θ star. We use a slash rather than a dash in order not to mix it up with the former type of star. Here is a sequence of six-point stars, with the ratio of the inner to outer radius going from 0.2 up to 0.8:

row of 6-0.2 stars

It is nice that this gives us more options for stars – infinitely many different kinds of star for each n in fact. But they are not as much fun to draw as the n-k stars, and it is harder to make them come out right without geometric instruments – which rules them out as an effective doodling pastime.

Note how, unlike with the n-k stars, the line leading away from a point does not intersect any other point. Nor do we get any inner stars for free. The price of gaining more variety is a loss of structure. The inner structure of a n-k star provides a great richness by enforcing all sorts of relationships between the vertices.

Only very specific values of the radius ratio θ give us n-k stars. I worked out the formula for the ratio by the way, with a bit of trigonometry. The ratio of the tniop radius to the point radius for a n-k star is:

2 sin(π(k-1)/n) sin(πk/n) sin(π(1-(2k-1)/n)) / (sin(2π/2)+sin(π(1-(2k-1)/n)) )

– sin(π((2k-1)/n – ½) )

For a n/θ star, if there are no integers n and k that give values of that formula equal to θ, the star will not have the array of inner stars that a n-k star has.

Computer Program to draw pretty stars

This whole diversion started as an exercise in drawing stars by hand. But there’s a limit to how intricate those stars can get without getting too messy with smeared ink or graphite. For those that like to look at pretty, intricate geometrical pictures, here’s my computer program in R that can draw stars of any of the types discussed.

Sums of angles

For those that like mathematical proofs, there are outlines of proofs here of Yuriy’s observation about the sum of the internal angles at points of a n-k star.

Andrew Kirk

Bondi Junction, April 2017