The clumsy secret agents who used Zwiefacher dance patterns as an encryption key. One broke an ankle trying to decrypt a message.

I saw that recently and I was very intrigued, but I couldn’t find any recordings. If anybody comes across a recording of that, I’d love to know!

     I don’t know if it’s been performed yet, but the full score can be seen here:

     http://imslp.org/wiki/Vuvuzela_Concerto_(Matthews%2C_John-Luke_Mark)

Alright, let me take a crack at this! I’ve taken a college class on evolution and written a book about fictional time travel. ( https://www.elidupree.com/stories/time-travelers-and-how-to-kill-them-a-practical-guide )

That was a fun read!

Unfortunately, I don’t know any model of time travel where situations like Carol can occur naturally (i.e. without a human writer deciding what happened, which makes discussion of probability meaningless). Not even the “consistent coilers” model that I made to rationalize consistent time travel in general. Sorry! 😛 (I will make up some math later in this post, though.)

I think I was picturing a model where the universe is internally consistent, but external forces can add consistent temporal loops. I.e., probability is a useful tool in the universe, but there are some events with undefined probability because the cause was (is? will be?) not in the universe. So there’s no way to determine the in-universe probability of Carol existing, but given that she does exist, it’s possible to reason about her genetics. I haven’t thought about this model very rigorously though, so maybe it falls apart.

All genes have only two alleles, which are equally frequent. There are no mutations

I don’t think it’s safe to assume that alleles are equally frequent. Carol is effectively the only member of her species, so by assuming the alleles are equally frequent, you either have to get 50% homozygous, or contradict the assumption.

Normal inbreeding calculation just lumps every thing other than homozygosity due to common ancestors in the same probability bucket as heterozygosity, which I’m pretty sure avoids needing to make any assumptions about number or frequency of alleles, or about mutations. So something that would be homozygous from descent, but mutates to be heterozygous is still counted as homozygous. And something that would be homozygous due to an allele just happening to be in unrelated ancestors is counted as heterozygous.

But wait! Inbreeding often causes offspring to be inviable. And another thing we know about Carol is that Carol is viable!

That’s a very good point, and one that it seems I conveniently missed in my original analysis. 🙂

I think “coefficient of inbreeding” is a simplification that ignores natural selection, so it would still be 80% regardless.

Yup, that’s pretty much correct for the non-time-travel case. Coefficient of inbreeding is used to guess at how likely it is that a (sometimes hypothetical) offspring will be viable/healthy/etc., so it’s not useful to include the offspring’s viability as a prior observation in the calculation. With a time loop, I’m not sure whether or not it makes sense to include it as a prior. Side note: the word “prior” amuses me in the context of time travel.

Anyway, I hope my comments here have been helpful (or at least interesting)!

Yes, thank you for the great counter-analysis! It’s good to see that I’m not the only person interested in tackling the important problems in time travel inbreeding calculation. 😉

Unfortunately, I don’t know any model of time travel where situations like Carol can occur naturally (i.e. without a human writer deciding what happened, which makes discussion of probability meaningless). Not even the “consistent coilers” model that I made to rationalize consistent time travel in general. Sorry! 😛 (I will make up some math later in this post, though.)

Here’s the problem: normally, the idea of probability assumes that causality is acyclic. So the final result is just what happens when you roll each random event in order. With time loops, you can’t do that. So you need a *timeless* notion of the probability that each universe-history (including loops) would occur – you need to determine those probabilities “before time starts”, so to speak. That’s not impossible, but what probabilities would you use? Here are some easy options that don’t actually work:

– “The probability of a history is proportional to the product of the probabilities of all the individual events in it”. This doesn’t even produce a world like our own. The vast majority of worlds are ones where, for each random event, the most likely result happens (almost) every time, and even if you observed some unlikely things happening, you could still expect that you would never see unlikely things again. Same thing for time travel – worlds where time loops exist are much less likely, so even if you observed a time loop, you could expect that no more time loops would happen.
– “All *possible* histories are equally likely (weighted by exact uniqueness)”. This one MIGHT be reasonable, depending on the underlying physics. At least, it might be able to produce a reasonable world where time travel DOESN’T exist. But if loops are possible, then from any starting time, the number of futures that proceed according to regular physics is vastly smaller than the number of possible time travelers that could return to that moment (because time travelers can be ANYTHING, not just things that would arise from straightforward physics). So the world would be a chaos of time loops, everywhere, forever. Subject to any physical limits on the amount of time travel, I guess. If you could somehow make the AMOUNT of time travel correct, stable loops that come from nowhere still have a larger solution space than time travel arising from straightforward human activities, so time loops would basically be a spontaneous physics-event rather than a human thing.
– My “consistent coilers” model, where loops are created by iterative re-running of straightforward physics. This creates a reasonable world (I hope), but never produces Carol because the Carol situation can’t occur on any of the finite iterations.

Now, we COULD make some simplifying assumptions about the exact situation, and get some probabilities. Here goes:
– Carol is the only thing that exists. Nothing else can interfere with the time loop.
– Oversimplifications of genetics are the only physics. Don’t worry about the physics of any other part of Carol.
– All genes have only two alleles, which are equally frequent. There are no mutations.
– All loci in each gamete have independent probabilities of which parent allele goes to the offspring. (Unlimited crossovers with no errors!)
– Given that fixed set of probabilities, what arrangements of alleles have what probabilities of choosing the same arrangement of alleles?
In this model, your math isn’t quite right, especially where you got the counterintuitive result that adding additional independent genes makes the existing genes more likely to be homozygous. Here, let me simplify the math by writing out all of the (independent, equally likely) possible cases for each gene:

AA->AA
AA->AA
AA->AA
AA->AA
AB->AA
AB->AB
AB->BA
AB->BB
BA->AA
BA->AB
BA->BA
BA->BB
BB->BB
BB->BB
BB->BB
BB->BB

After you observe that the output was the same as the input, the remaining possibilities are 8 cases of homozygous and 2 cases of heterozygous, so the coefficient of inbreeding is always 80%, regardless of the number of genes.

So, based on this math, we would assume that Carol will be 80% homozygous. But wait! Inbreeding often causes offspring to be inviable. And another thing we know about Carol is that Carol is viable! In the most extreme case, if we assume that every gene uses strict dominance and the homozygous recessive variant is inviable ( 😛 ), then we can also remove all of the BB cases, so Carol is now 2/3 homozygous dominant, 1/3 heterozygous.

If you need multiple, interacting homozygous recessive genes to get inviability, the math is harder, so I won’t do it. 😛 And anyway, I don’t think this changes the “coefficient of inbreeding” – I think “coefficient of inbreeding” is a simplification that ignores natural selection, so it would still be 80% regardless.

Adding a bit more “realism”, the 80% would probably be the chance that Carol had 2 copies of the same *entire chromosome* (assuming no crossover). And then even if the chromosomes were different, each gene would also have a 50% chance to be homozygous, so you’d end up with 90% homozygous. You could probably add various other forms of “realism”, too, but pretty soon it would break the simplifying assumptions, so that seems like a fool’s errand unless we can come up with a more versatile model for looping time travel.

Anyway, I hope my comments here have been helpful (or at least interesting)!

– Eli

Aha. We did see the spectacular waterfall, but no one mentioned any myths of dancing there as a surprise, and in any case, our train did not stop, and there were no dancers to be seen!

We also did not experience the entire ride from Flam to Myrdal, as the ship had arranged for the ride to end at a hotel where waffles, whipped cream and jam were served. We would have preferred the entire train ride, and wished we had just booked it separately.

However, cruise stops, the train is reserved entirely, except for the first trip, so those who tried to book on the spot were disappointed.

Here’s a description (see the section Flam Railway: Going up!) and a video of the surprise on the Flåm railway. I thought you saw the same thing when you went?

Yours is the first blog I truly enjoyed. Glad you detailed so much of your trip and your reactions. Because we had been on the Flam railway recently, it was neat to have it described from your perspective.

So, what was the surprise?

Bashful3 (aka Granny)