Sunday, January 13 (65 minutes)
Saturday, January 12 (6400m progression run)
I decided it was time to stop dicking around and go run at least some sort of workout. My goal was to hit a disciplined 6:00/5:50/5:40/5:30. Of course I did not. The first lap of the second mile was too fast, and after that the plan was about as effective at enforcing itself as the U.N.
As soon as I put a watch on, I go delusional and begin to think I'm not running a workout, but operating a ratchet. The pace cannot get slower. Even if you began by sprinting the first turn to impress the hot chick in short shorts, the pace cannot get slower. If you ever get slower in a workout you're training to get slower in a race.
I'd like to change the focus, and perhaps tell myself that running with discipline in a workout is training to run with discipline in a race. But that's a big cognitive leap when your manly honor is on the line.
The splits were 5:55, 5:37, 5:24, 5:19. Not too impressive, considering that I was working pretty hard by the end, but it's a decent starting point. And then afterwards I was walking past the start line for the hundred, and the sky was unusually black, the moon a small, sharply-focused crescent, the lanes seemed so immaculately clean and straight next to the green of the infield, and a bat was swooping around, and all this for some reason seemed absolutely perfect to me. I remember thinking that it was preposterous for anyone to have problems when the world could look like that, and that I couldn't understand how anyone could ever have a body and not want to run, gulping down this perfect air and circling the transparent darkness. That's an odd sort of scene to find inspiring, but there you have it.
Thursday, January 10 (70 minutes)
It's a problem you might have heard before, but I'll recap it anyway, since the makers of a BBC documentary about China seem to be confused.
In a certain country, every adult has a spouse, and every couple starts off having one child. If the child is a boy, they stop having children. If it is a girl, they have a second child, and if that one is a girl, a third, etc until they get a boy. What will the male/female ratio be in this country (assuming men and women live equal lifespans)? Also, what is its population growth per generation (assuming life expectancy does not change over time)?
At first brush, it might seem there will be more boys than girls, because every family has a boy, half the families don't have any girls. However, there will be some families with many girls and only one boy. These two effects cancel each other exactly, so that the male/female ratio is 50/50 (this also assumes male and female births are equally likely).
Simple proof: each child born has a 50/50 shot at either sex. Regardless of the couple's history, it's still a 50/50 shot for their next child, so there's no way around having a 50/50 split for the entire population. So the high boy/girl ratio in China (6:5) is due to illegal abortion of girls. Also for a 6:5 ratio of births, you'd need to abort one of every five girls. That's about two million black market abortions of girls every year.
Second part of the problem (population growth):
A family has a 1/2 chance of having just one child (first child boy), a 1/4 chance of having two children (first girl and second boy), a 1/8 chance of having three children, etc. To get the expected number of children per couple, we multiply the # children by the probability of getting that many, and sum.
I know two ways to do it. First, evaluate the sum explicitly. A good trick is to break the sum into a grand sum of easier sums, like this:
1/2 + 2/4 + 3/8 + 4/16 + 5/32 +...
= (1/2 + 1/4 + 1/8 + 1/16 +....) + (1/4 + 1/8 + 1/16 +...) + (1/8 + 1/16 + ...) + ...
These you probably recognize. You can solve them by multiplying by two, and seeing that you get the same sum back again, plus twice the first element. For example:
S = 1/2 + 1/4 + 1/8 + ...
2S = 1 + 1/2 + 1/4 + ... = 1 + S
S = 1
Using this result on the sum-of-sums, the grand sum simplifies to
1 + 1/2 + 1/4 + 1/8 +....
Which we now recognize to be 2. So in this scenario the population of the country remains stable. (In China, even without abortion, they officially don't go beyond 2 children, even if they're both girls. Also, not everyone marries and raises a family. Finally, you can also reduce the population significantly my marrying at an older age rather than having less children. That particular trick is a one-time shot though, and in the very long run population will start to rise again.)
There's a quicker way to get the answer than the sum-of-series. We already know there are just as many boys as girls. Further, every family has exactly one boy, so on average they must have one girl as well. That makes two.
Finally, the reason I'm watching documentaries about China and posting silly math problems related to their demographic regulations is that I'm pretty much decided to apply for the job in Beijing this summer. Anyone familiar with how to get a visa to visit China?
Also, I got an email today from Carlsbad describing the bib pick up instructions and free hotel accommodations arranged by the race. They never took my name off the list after I withdrew. But damn, if I'd known there was a free hotel room, course tour, special gear-stashing station, and free massages included in the deal I'd never have broken the arm to begin with.
Wednesday, January 9 (70 minutes)
I was relieved to wake up this morning (Thursday) and see that my foot was not covered in blood. Not my own, anyway. I feel like that's the sort of thing we all ought to check for once in a while. Wednesday afternoon I began running on the North Field, but was brought to a halt by a sharp but fairly inconsequential pain on the middle of the bottom of my left foot. When I looked at it it was actually bleeding. This was a relief.
The day before, I had been running barefoot on the south field, and cut the run short due to pain in the same spot. At the time I wasn't sure if it was a real injury or if I just stepped on some of the multifarious random crap that is strewn liberally across that field.
An injury is much worse, because it indicates something pernicious and possibly recurring, whereas stepping on something is a fluke. And at another level, an injury I psychologically perceive to be something wrong with me, and it casts a pall of weakness on my character. A fluke accident makes me a martyr, and allows me to complain indignantly to anyone foolish enough to stick around long enough to listen.
Tuesday, January 8 (40 minutes)
Stopped early, see above.
Monday, January 7 (30 minutes)
Stopped early. I was going out for a good long one, but my head felt weak and faint and I came back inside and slept all afternoon.
1/7/08 - 1/13/08 Once a Runner, Twice the Nipple Chafing (325 minutes, one tempo run)
12/31/07 - 1/6/08 Year of the Retarded Gay Panda Bears With Their Heads on Fire (320 minutes)
1/5/07 (60 minutes)
Things have felt different, lately. I feel unfit, but this seems to elicit alternating resignation and anger at my plight rather than motivation to improve. The training environment is far from ideal here in Maryland, but my opinion is that as an intelligent adult I ought to be able to cope with such minor inconveniences as the sun going down and the ground being too cold to admit bare feet.
My parents gave me a breadmaking machine for Christmas (at my suggestion), and today it broke. I realized after fooling with it for a while that I was getting severely distressed by simple, unimportant machine. So after half an hour I stopped screwing with the thing, kneaded the dough by hand and baked it in the oven, instead. It was the obvious solution. I knew early on that I wouldn't be able to make the thing go. But I persisted, and as I did I wasn't thinking about making the breadmaker work, but about the incompetence of the manufacturer, the idiocy of the automated phone system (it WAS poorly-designed, even in retrospect, but still) and in general all sorts of negative things that were irrelevant to the facts of the situation. I adopted the correct solution slowly and grudgingly rather than readily and impartially, as I would normally expect from myself.
I pride myself on an equanimity which has been tenuous and fleeting since the time, two or three months ago now, when my training started to deteriorate. I think this is evidence that my physical and mental states are more closely intertwined than I like to imagine.
I hold an irrational, egotistical belief that my "mental self", the "me-ness" of who I am, is an independent, invincible entity whose main purpose is the systematic mastery of whichever tasks either present themselves or are selected. The mastery of the physical body is one example. This is how I view the source of competitive drive.
This outlook, I am beginning to realize, is sophomoric and ultimately untenable. It is philosophically questionable to suppose there exists some sort of "external Mark" existing as an ineffable emergent phenomenon, floating around in nether-space and too strong to be overcome by such trivialities as brain chemistry. Why I have been genetically programmed to believe in such ghosts (as most humans seem to do) is a mystery that I may speculate upon at another time. But for the moment, it's mostly just an illusion I wish to disavow myself of.
I don't mean that I want to stop believing in sentient existence, free will, and the human soul. I mean that I want to stop believing that mine, in particular, are so damn privileged.
1/4/07 (90 minutes)
Good true long run, the first in a while. Felt great after two easy days. Check out the YouTube of last year's conference championships.
1/2/07 (30 minutes)
Pontificating on philosophy:
part 1
part 2
My tibilias anteriors were starting to get sore from all the road running (I ran after dark again and had to use the road), so I came back in early. But honestly I was happy to do so, because it was truly cold for the first time since I came home (a few degrees below freezing, with strong winds), and California has made into a sissy in that regard. I think freezing weather is fine, but with wind it's nasty, and my hands were getting numb.
Enough bitching. I'm done writing for now.
1/1/07 (70 minutes)
Managed to get out while there's still some daylight, and ran laps around the back yard. The only problem is that these are only one minute long, which means I'm turning almost constantly, and I can feel that wearing on my ankles and knees somewhat while I run. My legs feel fine now, though.
I decided to name this the Year of the Retarded Gay Panda Bears With Their Heads on Fire because it's a new year, but calling it the year of the rooster or dog or Quetzalcoatl or whatever name they already have for it did not seem apt. The Olympics are coming to Beijing, and here are the official mascots:
Incidentally, there's a chance I will be going to Beijing this summer as well. The summer camp I've worked at the past two years is starting an overseas program with a campus in Beijing, where they'll offer cosmology. Technically, I don't know cosmology (not on a college course level), but I my boss doesn't know that I don't know it. Also, they have a picture of me on their website.
I'm allowing them to use my gorgeous visage for free advertising, so they clearly owe me one. How could they not send me wherever I want to go? And let's see - your options are to do the exact same thing you've done the past two summers, living on the Stanford campus for six weeks, or to go to freaking Beijing, travel expenses paid, right before the start of the Olympics. Not a hard choice.
Also from their site, this guy
lived in our dorm last year. He's Kenyan, but he's fat and slow and never runs except when there's peanut butter at the end of it. One night he saw some non-affiliated kids sneaking around the dorm and beat the shit out of them all, simultaneously, until the cops came and handcuffed the camp director in his pajamas. True story.
12/31/07 (70 minutes)
ran around my backyard breaking wind in two senses. when i finally ran out of gas (in one sense), i felt abandoned and started making farting noises with my mouth, but they came out all wrong and so i gave up and just listened to the empty silence of winter.
later my parents went to bed at 10:30PM so i stayed up alone and watched the ball fall in times square, at which point i spontaneously ejaculated. true story.
12/24/07 - 12/30/07 Backlogged Week
12/24
No run. Ran in the rain yesterday and opened up a sore on the back of my achilles tendon.
12/25 - 12/26 No run. Same reason.
12/27
75 minutes on a hotel treadmill. Visiting my sister's family, including new nephew, in North Carolina. Dad's photos
He spelled Bryon's name wrong many times, but later fixed it.
12/28
55 minutes on the roads. Had to waddle it in at the end due to severe gastric distress.
12/29
70 minutes.
12/30
no run. legs sore from being on the roads so much, and being out of shape.
Learned that Google keeps track of your search history. Actually, I thought I might have heard this somewhere, but it didn't sink in. Here are the sites I visit most frequently, all time:
Top sites
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| 5. | |
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Oh, running.caltech.edu, why did you have to go? And why is our current web-running-community foundering, too?
12/17/07 - 12/23/07 sqWeek (30 minutes)
Saturday, 12/22/07 (60 minutes)
Ran at night, on the roads. My legs seemed to hold up fine. Also, it was one less hour that I had to share the house with that killer robot masquerading as a floor-cleaning device.
Friday, 12/21/07 (30 minutes)
I didn't sleep last night because I was trying to get on a more normal schedule, and that was the only way I could think to do it. So I didn't run much.
Thursday, 12/20/07 (no run)
I always have a hard time adjusting my running when I move to a new place. My schedule goes according to some sort of routine, even if the outside observer would disagree. It's very hard to run after dark here, because the my backyard is too uneven and I'm avoiding the roads as much as possible. That means if I haven't started running by 4pm I won't get a full run in.
Wednesday, 12/19/07 (60 minutes)
Ran around the back yard.
I'm trying to figure out why my picture on the header got chopped down. I did not do it - it was all the internets operating of their own accord. Believe me that I would not purposefully crop my own photo down to my crotch.
Tuesday, 12/18/07 (no run)
First: my sister popped one out today. As in, my parents are now my nephew's grandparents. It's name is Bryon Charleston Reed. I had to answer the phone call from North Carolina because my mom was driving. I relayed a quick series of questions from my mom over to whoever I was talking to on the other end, until my mom could be satisfied enough to look back up at the road again. We only sideswiped one traffic cone coming out of the airport parking lot.
Second: My other sister will be coming home on Friday. She now claims to look like this:
I believe her.
Third: I spent the entire day traveling and did not have time to run. While in the airplane I attempted to test my equation for how far you can see around the Earth at a given height when the pilot informed us we were at 37,000 feet. The difficulty was that there was so much atmosphere between me and the horizon 150km away that I could not see a distinct line separating them. I did see a faraway mountain, and estimating rather arbitrarily that it was 5000 higher than the surrounding ground, and holding my thumb up at the end of my arm (to the annoyance of the chick sitting next to me, but it's okay, because she wasn't hot, even though she was trying to be), I figured that 150km was probably about right.
Fourth: My dad hired a robot to clean the floors in the house for him. It lives downstairs, where they're "conveniently" keeping me, as well. If this is my last blog post, let me just say this: BIOTA RULES! YOU CAN TAKE OUR LIVES, BUT YOU CAN NEVER TAKE OUR ABILITY TO OCCASIONALLY BEAT YOU AT CHESS ON A GOOD DAY, BUT NOT CHECKERS, EVER AGAIN!
Monday, 12/17/07 (30 minutes)
I got a call from my Dad this afternoon. My parents don't call me to chat.
"Mark. Where are you?"
"Umm, in my room?"
"Your mother is looking for you at the airport."
"But I'm not at the airport."
"I know. If you were at the airport, she would have found you already."
Very true.
At first I thought they had the date of my flight wrong (I was planning on leaving tomorrow morning), but then I remembered mom is never wrong about anything in a schedule book. I missed my flight and made my mom drive out to the airport, not a quick trip from where we live. Not the most auspicious start to a winter break.
Ran half an hour, because I had to change my afternoon plans to accommodate packing, etc. before tutoring in the evening. I'm heading out to the airport now. There'll be no barefoot running or gym available at home, but no sprinkler heads or chained fences, either.
I'm looking forward to a quiet three weeks.
Further Intellectual Masturbation
J.R. once told me about a friend who wrote a computer program that found the shape of a hanging spring. The shape of a hanging chain is a "catenary" [i.e. y=cosh(x)], a fact I was aware of but didn't know how to prove. The shape of the hanging spring (specifically, a spring with zero rest length but finite mass, and whose density is therefore a delta function) turns out to be a parabola.
The program apparently discovered this by starting with a string of many springs of zero mass and zero rest length connecting point particles with mass. The point particles start off stretching in a straight line between two supports of the same height, horizontally separated by some fixed distance. So it's like the spring hanging from the towers that support a suspension bridge. The difference is there's no bridge, and the spring is stretchy whereas the cables on the bridge basically are not.
The program then calculated the forces from gravity and the springs, took a small step forward in time using F=ma, recalculated the forces, took another step, etc. until the thing came to rest. There must have been a small damping coefficient in the program as well to keep the spring from oscillating forever.
I wanted to solve this problem analytically, but at the time I didn't know how. Later, I saw the solution to the catenary problem in a discussion of the calculus of variations. In this approach, you write a function for the energy of the chain/spring, then use a variational principle to minimize the energy.
However, I today I read about how to treat a similar problem, where you have a massless, nonstretchy chain supporting a massive, flat road (i.e. suspension bridge). Modifying the book's solution a bit, I found that you can solve all three problems (massive chain, massive spring, massless chain with bridge) using only single-variable calculus. Here is how:
Assume the spring has some shape given by the function height = y = f(x), where x is the distance from the middle of the two supports.
Our plan will be to find a differential equation for y using the following assumptions:
- The system is in equilibrium
- The tension in the spring/chain at a point must be along the direction of the tangent to that point.
- The only external force on the spring is gravity, which is in the y-direction.
Not only must the entire spring be in equilibrium, but every differential element ds of the spring must be in equilibrium. So consider the forces acting on an element of the spring ds, as shown below:
The condition of equilibrium in the x-direction requires
T1x - T2x = 0
T1x = T2x = T0.
In the y-direction, equilibrium requires
-T1y + T2y - Fg = 0
rewriting Fg as px(x)*g*dx, where px(x) is the mass density per unit distance in the x-direction at the point x (NOT necessarily the same as the mass density of the spring, ps), this becomes
-T1y + T2y = px(x)*g*dx
Now we invoke the fact that the tension is along the spring.
Ty/Tx = y'
but we found earlier that Tx = T0, so
Ty = T0*y'
so going back to the eqn for equilibrium in the y-direction:
T0*[y'(x2) - y'(x1)] = px(x)*g*dx
for sufficiently small segment ds, this becomes
T0*y''(x)*dx = px(x)*g*dx
y''(x) = px(x)*(g/T0)
Now we have the desired differential equation in y. So far, the solution to all three problems is the same. Now they diverge, because the mass density is different in the three cases.
Massless Chain Supporting a Road:
Here, the mass density is just the mass of the road per unit in the x-direction
px = p0
y''(x) = p0*g/T0 = C
where C is some constant. Integrating, we get a quadratic - the massless chain supporting a road hangs in the shape of a parabola.
This could potentially be a useful result, because people are interesting in making parabolas, because a parabola is the ideal shape for telescopes, radio antennas, etc.
Massive chain:
The mass density of the chain is constant:
ps = p0
From the geometry of the situation, we can get px
ds2 = dx2 + dy2
ds/dx = (1 + y'2)1/2*dx
px*dx = ps*ds
px = (1 + y'(x)2)1/2*p0
substituting into the D.E.
y'' = (1 + y'2)1/2 * C
where I mashed the constant together
make a substitution z=y', z'=y''
dz/dx = (1 + z2)1/2*C
dz/(1 + z2)1/2 = C*dx
Look this integral up and you get sinh-1(z) = C*x
z = sinh(C*x) = y'
y = cosh(c*x), which is the shape of a catenary
Massive Spring
For the massive spring, we have to take a look at Hooke's Law:
T = k*x, where x is the displacement from equilibrium.
Note that if we stretch a string to a length x, its density becomes m/x, where m is the length of the spring.
ps(x) = 1/T(x)
where T is the tension and I suppressed a constant.
T = (Tx2 + Ty2)1/2
T = [T02 + (T0y')2]1/2
using earlier results for Tx and Ty
T = T0*(1 + y'2)1/2
ps = 1/T = 1/(1 + y'2)1/2
px = (1 + y'2)1/2*ps, as in the previous case
px = [(1 + y'2)1/2]/[(1 + y'2)1/2]
px = 1
again suppressing constants.
So this case is the same as the original problem with the road hanging from a chain, and the hanging spring of zero rest length takes the shape of a parabola.
Additionally, by modifying the function px, you could find the differential equation for the shape of the bridge under an arbitrary load, such as combining all three methods to model a suspension bridge supporting a road where the cables have some finite mass and also some Young's modulus, and finite rest length. You could add traffic going across, acceleration of the road when lifting it up to allow a boat to go underneath, etc. Note that a point mass traveling across the bridge hanging from a chain would cause the density function to become a delta function at that point. So y'' is a delta function, y' has a discrete jump, and the result is that a heavy truck driving across such a bridge causes a kink in the cable directly above the truck.
12/10/07 - 12/16/07 Fuck You, Bones!
Who cares if my bones are broken? Fuck that shit. I'm running anyway.
Monday (1:53)
One minute and fifty-three seconds of running, before I was kicked off the North Field. The South Field was also closed, and I didn't have time to go in search of new grass. Still, this represented a significant improve on my recent running mileage.
Tuesday (60 minutes)
Ran in Lacy, where all the moms explained to their little kid why the man was running. I felt great considering I'm out of shape.
Wednesday (EPIC)
Beer Mile 5:56!!!!!
Thursday (65 minutes)
Easy run on the Eichenlaub Grassy Special, which I haven't used much recently. Ran past Garrett on the way back. I realized that Ian runs my route more frequently than I do, and I run Ian's Arroyo Tempo Loop more frequently than he does. Ah, the absurdities of life.
Friday ( 65 minutes)
Repeat of Thurs.
Saturday (60 minutes)
SFTC w/ Kangway, who told me how atrophied I am. Thanks dude.
Sunday (no run)
Math II
Kangway wins Math I. He receives a score of 95%, because he got the answer right, but his answer was really long and included references to things like the "zero product property" and i don't know what he's talking about. the only interesting part is to note that by rewriting 32x as (3x)2, we can turn it into a quadratic equation in 3x. Also, a neater way to express log324 is log3233 = 1 + 3log32.
Also, Ian gets honorable mention for correctly solving the problem, but he didn't follow the full instructions, and so receives a score of 80%. His method worked, but it wouldn't work for the same problem with some different values plugged in.
Here is my next question:
I was trying to prove that d/dx(ax) = ln(a)*ax
limh->0 (ax+h - ax)/h =
limh->0 ax(ah - 1)/h
and so now all i have to do is fix the constant
limh->0(ah - 1)/h
which is of the form 0/0, and i want to show that it is equal to ln(a)
normally you would use L'Hospital's rule and take a derivative with respect to h of the numerator and denominator, but that's illegal here since the form of the derivative is exactly what we're trying to prove.
the next thing i thought to do write ah as (1-(1-a))h and use the binomial theorem to expand this, ignoring all powers of h greater than 1.
the resulting series is not too complicated, but it is an alternating series whose terms individually diverge, and i don't know how to evaluate the sum.
so my question is: who has another trick to prove that
limh->0(ah - 1)/h = ln(a)?
of course i could look this up in a calculus book/the internet, but that's not as fun. also, i don't understand textbooks because they have no souls.
finally, i'm aware that my task can be accomplished by taking the definition of e to be the number "a" s.t.
limh->0(ah - 1)/h = 1,
but then my problem is to show that this definition of e is equivalent to the common definition
e=limn->inf(1 + 1/n)n
Math
I was tutoring a pre-calculus student today, and she had a homework problem I didn't know how to do. I played it smooth, saying, "Well, if you're not sure how to do it, let's check through the book for an example." So here's the problem. Go:
solve for x by algebra
32x + 3x+1 - 4 = 0
Cells Are Big
I didn't realize before what the scale of a cell is, but I was reading some from a bio textbook today. They say a typical eukaryotic cell would be 10-100 microns, so take 30 microns on a side. Then if a sugar molecule were as big as a person, the cell would be as big as Los Angeles (50km), except that it's 3D, so it's more like a thousand copies of Los Angeles stacked on top each other.
That allows for a hell of a lot of complexity. It gives some appreciation for just how grand the scale of life is. In particular, that of Haile Gebrselassie.
The Round, Pythagorean Earth
My good news today is that the orthopedist agreed to let me go completely free - no cast or even a splint. I have a sling that I should wear most of the time, but in general my orders are to take it easy, be careful with my arm, but move it around a little bit now and then. I'll see him again after a week. I shouldn't regain full range of motion in that time, but I'm already up to 75%. I can't run or exercise yet, but I can type.
Some time ago I wrote in a piece for English class that if you go to the beach and hold a postcard up against the horizon, you can see the curvature of the Earth. I didn't really fact-check this beforehand, though. Now that I think about it some more, I know that not only is it true, but that the ancient Greeks should not have had a very hard time figuring out that the Earth is round, and even could have measured its radius by making simple observations from the shore or from a boat.
The boat's lookout sits in the crow's nest - high above the main deck. This was probably practiced before people knew that the Earth is round. (Although as I Google it now, it turns out that many ancient Greeks did believe the Earth was round, perhaps beginning with Pythagoras, who unfortunately held that belief for retarded reasons. Later Aristotle talked about gravity pulling the Earth into a sphere in and some dude name Eratosthenes actually measured it using the lengths two shadows measured at the same time but at places far away from each other.) Why should the lookout seek a high vantage point? Because he can see farther, of course. This wouldn't be the case with a flat Earth.
On a flat Earth with no atmosphere, you would be able to see all the way to the edge, no matter how high up you are to start. The same is true for the crow's nest. You would have a slightly better vantage point from the crow's nest because objects in the sea far away would subtend a larger angle, but your total distance of visibility would be the same.
On the other hand, on a curved Earth the horizon is in fact further away if you are higher up. The man in the crow's nest can see the mermaid, but the man on the deck cannot.
The first time I sat down to work this out, I calculated the distance to the horizon for a 6-foot tall man standing at the beach, and got something close to an even 5000m. I thought that had to be wrong, since I had seen horizons much further away - when I was climbing mountains.
Only later did I realize that of course you can see further when climbing mountains - you're in a great tall crow's nest.
I originally calculated the distance to the horizon using analytic geometry, but it's an ugly way to do the problem. There's a much cleaner way, as shown here:
So, by the fact that you can see further from a crow's nest, people should have known the Earth is not flat. If you hypothesize a sphere for the Earth's shape, you could measure the radius quite easily by letting a ship sail out to sea, then measuring how far away it is and how high up you have to be to see it. This gives you an estimate of R for every time you measure the distance to the ship. The quality of the estimate will depend on how well you can measure distances out to sea (triangulation from the shore should work well), how accurately you can measure how high you are (probably easy), how spherical the Earth is (spherical enough), whether the ocean is truly flat (over long distances it is), and whether light is refracted on its way from the buoy back to you (variable and hard to control for). But I'd say you should be able to get a pretty good estimate of the radius of the Earth this way, and it can be done with measurements all from one place.
So finally, can you see the curvature of the Earth with your naked eye? Certainly if you go high enough up you can. From far out in space you can see the entire Earth at once. If you want the Earth to curve 1 degree, then you'd need to view about 100km of horizon under your postcard. So the horizon would need to be on the order of 200km away, because if the postcard subtends more than about a 30 degree angle, you can't see both sides of it at once. That means you'd have to be 3000m up. So you probably won't be able to see the curvature of the earth with the naked eye by holding a postcard up at the beach, but you could easily do it from a mountain overlooking the sea.