It's been a while since we had one of these. But gosh, people have
sent me quite a lot of really interesting mail lately.
I related my childhood disappointment at the limited number of cool
coordinate systems. Norman Yarvin directed me to prolate
spheroidal coordinates which are
themselves a three-dimensional version of elliptic
coordinates which are a system of
exactly the sort I escribed in the article, this time parametrized
by a family of ellipses and a family of hyperbolas, all of which
share the same two foci; this article links in turn to parabolic
coordinates in which the two families
are curves are up-facing and down-facing parabolas that all share a
focus. (Hmm, this seems like a special case of the ellipses, where
one focus goes to infinity.)
Walt Mankowski also referred me to the Smith
chart, shown at right, which is definitely
relevant. It is a sort of nomogram, and parametrizes certain points
by their position on circles from two families
Electrical engineers use this for some sort of electrical engineer
calculation. They use the letter !!j!! instead of !!i!! for the
imaginary unit because they had already used !!i!! to stand for
electrical current, which is totally reasonable because “electrical
current” does after all start with the letter !!i!!. (In French!
The French word is courant. Now do you understand? Stop asking
questions!)
Regarding what part of the body Skaði was looking at when the Norse
text says fótr, which is probably something like the foot,
Alexander Gurney and Brent Yorgey reminded me that Biblical Hebrew
often uses the foot as a euphemism for the genitals. One example
that comes immediately to mind is important in the book of Ruth:
And when Boaz had eaten and drunk, and his heart was merry, he
went to lie down at the end of the heap of corn: and she came softly,
and uncovered his feet, and laid her down. (Ruth 3:7)
M. Gurney suggested Isaiah 6:2. (“Above him were seraphim, each with
six wings: With two wings they covered their faces, with two they
covered their feet, and with two they were flying.”) I think
Ezekiel 16:25 is also of this type.
I mentioned to Brent that I don't think Skaði was looking at the
Æsir's genitals, because it wouldn't fit the tone of the story.
Alexander Gurney sent me a lot of other interesting material. I had
translated the Old Icelandic hreðjar as “scrotum”, following
Zoëga. But M. Gurney pointed out that the modern Icelandic for
“radish” is hreðka. Coincidence? Or was hreðjar a euphemism
even then? Zoëga doesn't mention it, but he doesn't say what word
was used for “radish”, so I don't know.
He also pointed me to Parts of the body in older Germanic and
Scandinavian
by Torild Washington Arnoldson. As in English, there are many words
for the scrotum and testicles;
some related to bags, some to balls, etc. Arnoldson does mention
hreðjar in the section about words that are bag-derived but
doesn't say why. Still if Arnoldson is right it is not about
radishes.
I should add that the Skáldskaparmál itself has a section about
parts of the body listing suitable words and phrases for use by
skálds:
Hönd, fótr.
… Á fæti heitir lær, kné,
kálfi, bein, leggr, rist, jarki, il, tá. …
(… The parts
of the legs are called thigh, knee, calf, lower leg, upper leg,
instep, arch, sole, toe … [ Brodeur ])
I think Brodeur's phrase “of the legs” here is an interpolation.
Then he glosses lær as “thigh”, kné as “knee”, kálfi as
“calf”, and so on. This passage is what I was thinking of when I said
Many of the words seem to match, which is sometimes helpful but
also can be misleading, because many don't.
I could disappear down this rabbit hole for a long time.
Regarding mental estimation of the number of primes less than 1,000,
which the Prime Number Theorem says is approximately
!!\frac{1000}{\ln 1000}!!, several people pointed out that if I had
memorized !!\ln 10\approx 2.3!! then I would have had that there are
around !!\frac{1000}{3·2.3}!! primes under 1,000.
Now it happens that I do have memorized !!\ln 10\approx 2.3!! and
although I didn't happen come up with it while driving that day, I
did come up with it a couple of days later in the parking lot of a
Wawa where I stopped to get coffee before my piano lesson. The next
step, if you are in a parking lot, is to approximate the division as
!!\frac{1000}{6.9} \approx \frac{1000}7 = 142.857\ldots!! (because
you have !!\frac17=0.\overline{142857}!! memorized, don't you?) and
that gives you an estimate of around 145 primes.
Which, perhaps surprisingly, is worse than what I did the first
time around; it is 14% too low instead of 8% too high. (The right
answer is 168 and my original estimate was 182.)
The explanation is that for small !!n!!, the approximation
!!\pi(N)\sim\frac{N}{\ln N}!! is not actually very good, and I think
the interpolation I did, using actual low-value counts, takes better
account of the low-value error.
I read this, but seemed to skip over the part where he explains
why this changed suddenly, when the behavior was documented?
What changed to make the perl become capable whereas previously it
lacked the low port capability?
So far, we don't know! Frew told me recently that he thinks the
TMPDIR-losing has been going on for months and that whatever
precipitated my problem is something else.
In
my article on the Greek clock,
I guessed a method for calculating the (approximate) maximum length
of the day from the latitude: $$ A = 360 \text{ min}\cdot(1-\cos L).$$
Sean Santos of UCAR points out that this is inaccurate close to the
poles. For places like Philadelphia (40° latitude) it is pretty
close, but it fails completely for locations north of the Arctic
Circle. M. Santos advises instead:
$$ A = 360 \text{ min}\cdot \frac{2}{\pi}\cdot \sin^{-1}(\tan L\cdot
\tan\epsilon)$$
where ε is the axial tilt of the Earth,
approximately 23.4°. Observe that when !!L!! is above the Arctic
Circle (or below the Antarctic) we have
!!\tan L \cdot \tan \epsilon > 1!! (because
!!\frac1{\tan x} = \tan(90^\circ - x)!!)
so the arcsine is undefined, and we get no answer.
I looked into several books on Unix shell programs, including:
Linux Shell Scripting with Bash (Burtch)
Unix Shell Programming 3ed. (Kochan and Wood)
Mastering UNIX Shell Scripting (Michael)
All of these contained examples of flag variables in Bourne shell, and
none used the technique I described. (In fact, most books wanted to
switch to if [ ... ]; right away, or even to pretend that
that was the only possible syntax.) So it may be obvious, but it
doesn't seem to be widely used. I also looked into The Unix
Programming Environment, by Kernighan and Pike, which is the
book from which I learned shell programming, to see if it was there.
I couldn't find any examples of boolean variables at all! But there
were surprisingly few shell programs; they switched to awk rather
quickly.
But two readers sent me puzzled emails, to tell me that they had been
using the true/false technique for years are were
surprised that I found it surprising. Brooks Moses says that at his
company they have a huge build system in Bourne shell, and they are
trying to revise the boolean tests to the style I proposed.
And Tom
Limoncelli reports that code by Bill Cheswick and Hal Burch (Bell Labs
guys) often use this technique. Tom speculates that it's common among
the old farts from Bell Labs.
Also, Adrián Pérez writes
that he has known
about this for years.
It's tempting to write to Kernighan to ask about it, but so far I have
been able
to resist.
My first meta-addendum: In October's
addenda I summarized the results of a paper of Coquand,
Hancock, and Setzer about the inductive strength of various theories.
This summary was utterly wrong. Thanks to Charles Stewart and to
Peter Hancock for correcting me.
The topic was one I had hoped to get into anyway, so I may discuss it
at more length later on.
data Nat = Z | S Nat
data Ordinal = Zero
| Succ Ordinal
| Lim (Nat → Ordinal)
In particular, I asked
"What about Ω, the first uncountable ordinal?" Several readers
pointed out that the answer to this is quite obvious: Suppose
S is some countable sequence of (countable) ordinals. Then the
limit of the sequence is a countable union of countable sets, and so
is countable, and so is not Ω. Whoops! At least my intuition
was in the right direction.
Several people helpfully pointed out that the notion I was looking for
here is the "cofinality" of the ordinal, which I had not heard of
before. Cofinality is fairly simple. Consider some ordered set S. Say
that an element b is an "upper bound" for an element a
if a ≤ b. A subset of S is
cofinal if it contains an upper bound for every element of
S. The cofinality of S is the minimum
cardinality of its cofinal subsets, or, what is pretty much the
same thing, the minimum order type of its cofinal subsets.
So, for example, the cofinality of ω is ℵ_{0}, or, in the language
of order types, ω. But the cofinality of ω + 1 is only 1
(because the subset {ω} is cofinal), as is the cofinality of
any successor ordinal. My question, phrased in terms of cofinality,
is simply whether any ordinal has uncountable cofinality. As we saw,
Ω certainly does.
But some uncountable ordinals have countable cofinality. For example,
let ω_{n} be the smallest ordinal with
cardinality ℵ_{n} for each n. In
particular, ω_{0} = ω, and ω_{1} =
Ω. Then ω_{ω} is uncountable, but has
cofinality ω, since it contains a countable cofinal subset
{ω_{0}, ω_{1}, ω_{2}, ...}.
This is the kind of bullshit that set theorists use to occupy their
time.
A couple of readers brought up George Boolos, who is disturbed by
extremely large sets in something of the same way I am. Robin Houston
asked me to consider the ordinal number which is the least fixed point
of the ℵ operation, that is, the smallest ordinal number
κ such that |κ| = ℵ_{κ}. Another
way to define this is as the limit of the sequence 0, ℵ_{0}
ℵ_{ℵ0}, ... . M. Houston describes κ as
"large enough to be utterly mind-boggling, but not so huge as to defy
comprehension altogether". I agree with the "utterly mind-boggling"
part, anyway. And yet it has countable cofinality, as witnessed by
the limiting sequence I just gave.
M. Houston says that Boolos uses κ as an example of a set
that is so big that he cannot agree that it really exists. Set theory
says that it does exist, but somewhere at or before that point, Boolos
and set theory part ways. M. Houston says that a relevant essay,
"Must we believe in set theory?" appears in Logic, Logic, and
Logic. I'll have to check it out.
My own discomfort with uncountable sets is probably less nuanced, and
certainly less well thought through. This is why I presented it as a
fantasy, rather than as a claim or an argument. Just the sort of
thing for a future blog post, although I suspect that I don't have
anything to say about it that hasn't been said before, more than once.
Finally, a pseudonymous Reddit user brought up a paper of
Coquand, Hancock, and Setzer that discusses just which ordinals
are representable by the type defined above. The answer turns
out to be all the ordinals less than ω^{ω}. But
in Martin-Löf's type theory (about which more this month, I hope)
you can actually represent up to ε_{0}. The paper is
Ordinals
in Type Theory and is linked from here.
Thanks to Charles Stewart, Robin Houston, Luke Palmer, Simon Tatham,
Tim McKenzie, János Krámar, Vedran Čačić,
and Reddit user "apfelmus" for discussing this with me.
[ Meta-addendum 20081130: My summary of Coquand, Hancock, and Setzer's
results was utterly wrong. Thanks to Charles Stewart and Peter
Hancock (one of the authors) for pointing this out to me. ]
Regardinghomophones of
numeral words, several readers pointed out that in non-rhotic
dialects, "four" already has four homophones, including "faw" and
"faugh". To which I, as a smug rhotician, reply "feh".
One reader wondered what should be done about homophones of
"infinity", while another observed that a start has already been
made on "googol". These are just the sort of issues my proposed
Institute is needed to investigate.
One clever reader pointed out that "half" has the homophone "have". Except
that it's not really a homophone. Which is just right!
The story features a race of humanoid aliens with a "public" and a "private"
mind. The "public" mind is fairly stupid, and handles all interactions
with the real world; and the "private" mind is intelligent and psychic.
The private mind communicates psychically with the private minds of
other members of the race, but has only limited influence over the
public mind; this influence manifests as visions and messages from God.
This would not be so remarkable, since Jaynes' theories have been widely
taken up by some science fiction authors. For example, they appear in
Neal Stephenson's novel Snow Crash, and even more
prominently in his earlier novel The Big U, so much so
that I wondered when reading it how anyone could understand it without
having read Jaynes first. But Schmitz's story was published in 1956,
twenty years before the publication of The Origin of
Consciousness.
Also in connection with Jaynes: I characterized his theory as
"either a work of profound genius, or of profound crackpottery". I
should have mentioned that this
characterization was not lost on Jaynes himself. In his book, he
referred to his own theory as "preposterous".
I had said that "[The one-letter word 'i'] appears in my sample
in connection with Sukselaisen
I hallitus, whatever that is". Several people
explained that this "I" is actually a Roman numeral 1, denoting the ordinal number
"first", and that Sukselaisen
I hallitus is the first government headed by V. J. Sukselaisen.
I had almost guessed this—I saw
"Sukselaisen
I" in the source material and guessed that the "I" was an ordinal, and
supposed that "Sukselaisen
I" was analogous to "Henry VIII" in English. But when my attempts to look
up the putative King Sukselaisen
I met with failure, and I discovered that
"Sukselaisen
I" never appeared without the trailing "hallitus", I decided that
there must be more going on than I had supposed, as indeed there was.
Thanks to everyone who explained this.
Marko Heiskanen says that the (fictitious) word
yhdysvalmistämistammonit is "almost correct", at least up to the
nonsensical plural component "tammonit". The vowel harmony failure
can be explained away because compound words in Finnish do not respect the
vowel harmony rules anyway.
Several people objected to my program's generation of the word
"klee": Jussi Heinonen said "Finnish has quite few words that begin with
two consonants", and Jarkko Hietaniemi said "No word-initial "kl":s
possible in native Finnish words". I checked, and my sample Finnish
input contains "klassisesta", which Jarkko explained was a loanword,
I suppose from Russian.
Had I used a larger input sample, oddities like "klassisesta" would have
had less influence on the output.
I acquired my input sample by selecting random articles from
Finnish Wikipedia, but my random sampling was rather unlucky, since it
included articles about Mikhail Baryshnikov (not Finnish), Dmitry Medvevev
(not Finnish), and Los Angeles (also not Finnish). As a result, the
input contained too many strange un-Finnish letters, like B, D, š, and
G, and so therefore did the output. I could have been more careful in
selecting the input data, but I didn't want to take the time.
Medvedev was also the cause of that contentious "klassisesta", since,
according to Wikipedia, "Medvedev pitää klassisesta
rock-musiikista". The Medvedev presidency is not even a month old and
already he has this international incident to answer for. What
catastrophes could be in the future?
Another serious problem with my artificial Finnish is that the
words were too long; several people complained about this, and the
graph below shows the problem fairly clearly:
The x-axis is word length, and the y-axis is frequency,
on a logarithmic scale, so that if 1/100 of the words have 17 letters,
the graph will include the point (17, -2). The red line, "in.dat",
traces the frequencies for my 6 kilobyte input sample, and the blue
line, "pseudo.dat", the data for the 1000-character sample I published
in the article. ("Ävivät mena osakeyhti...") The green
line, "out.dat", is a similar trace for a 6 kb N=3 text I
generated later. The long right tail is clearly visible. My sincere
apologies to color-blind (and blind) readers.
I am not sure exactly what happened here, but I can guess. The Markov
process has a limited memory, 3 characters in this case, so in
particular is has essentially no idea how long the words are that it
is generating. This means that the word lengths that it generates
should appear in roughly an exponential distribution, with the probability of a
word of length N approximately equal to
!!\lambda e^{-\lambda N} !!,
where 1/λ is the mean word length.
But there is no particular reason why word lengths in Finnish (or any
other language) should be exponentially distributed. Indeed, one
would expect that the actual distribution would differ from
exponential in several ways. For example, extremely short words are
relatively uncommon compared with what the exponential distribution
predicts. (In the King James Bible, the most common word length is 3,
then 4, with 1 and 8 tied for a distant seventh place.) This will
tend to push the mean rightwards, and so it will skew the Markov
process' exponential distribution rightwards as well.
I can investigate the degree to which both real text and Markov
process output approximate a theoretical exponential distribution, but
not today. Perhaps later this month.
My thanks again to the many helpful Finnish speakers who wrote in on
these and other matters, including Marko Heiskanen, Shae Erisson,
Antti-Juhani Kaijanaho, Ari Loytynoja, Ilmari Vacklin, Jarkko
Hietaniemi, Jussi Heinonen, Nuutti-Iivari Meriläinen,
and any others I forgot to mention.
Karinthy Frigyes got married two times, the Spanish flu epidemic took
his first wife away. A son of his was born from his first marriage,
then his second wife brought a boy from his previous husband, and
a common child was born to them. The memory of this the reputed remark:
"Aranka, your child and my child beats our child."
A number of readers, including some honest-to-God Italians, wrote
in with explanations of Boccaccio's term
milliantanove, which was variously translated as
"squillions" and "a thousand hundreds".
The "milli-" part suggests a thousand, as I guessed. And "-anta" is
the suffix for multiples of ten, found in "quaranta" = "forty", akin
to the "-nty" that survives in the word "twenty". And "nove" is
"nine".
So if we wanted to essay a literal translation, we might
try "thousanty-nine". Cormac
Ó Cuilleanáin's choice of "squillions" looks quite apt.
for my $k (keys %h) {
if ($k eq $j) {
f($h{$k})
}
}
could be replaced with:
f($h{$j})
But this is only true if $j actually appears in %h.
An accurate translation is:
f($h{$j}) if exists $h{$j}
I was, of course, aware of this. I left out discussion of this
because I thought it would obscure my point to put it in, but I was
wrong; the opposite was true.
I think my original point stands regardless,
and I think that even programmers who are unaware of the existence of
exists should feel a sense of unease when presented with (or
after having written) the long version of the code.
We then discussed the use of nonstandard adjectives in biology. I
observed that there seemed to be a trend to eliminate them, as with
"jellyfish" becoming "jelly" and "starfish" becoming "sea star". He
pointed out that botanists use a hyphen to distinguish the standard
from the nonstandard: a "white fir" is a fir, but a "Douglas-fir" is
not a fir; an "Atlas cedar" is a cedar, but a "western redcedar" is
not a cedar.
Several people wrote to discuss the use of "partial" versus "total",
particularly when one or the other is implicit. Note that a total
order is a special case of a partial order, which is itself a special
case of an "order", but this usage is contrary to the way "partial"
and "total" are used for functions: just "function" means a total
function, not a partial function. And there are clear cases where
"partial" is a standard adjective: partial fractions are fractions,
partial derivatives are derivatives, and partial differential
equations are differential equations.
In my utterly useless review of Robert Graves' novel King
Jesus I said "But how many of you have read I,
Claudius and Suetonius? Hands? Anyone? Yeah, I didn't think
so." But then I got email from James Russell, who said he had indeed
read both, and that he knew just what I meant, and, as a
result, was going directly to the library to take out King
Jesus. And he read the article on Planet Haskell. Wow! I am
speechless with delight. Mr. Russell, I love you. From now on,
if anyone asks (as they sometimes do) who my target audience is, I
will say "It is James Russell."
A number of people wrote in with examples of "theorems" that were
believed proved, and later turned out to be false. I am preparing a
longer article about this for next month. Here are some teasers:
Cauchy
apparently "proved" that if a sum of continuous functions converges
pointwise, then the sum is also a continuous function, and this error
was widely believed for several years.
I just learned of a major
screwup by none other than Kurt Gödel concerning the decidability
of a certain class of sentences of first-order arithmetic which went
undetected for thirty years.
Robert Tarjan proved in the
1970s that the time complexity of a certain algorithm for the
union-find problem was slightly worse than linear. And several people
proved that this could not be improved upon. But Hantao Zhang has a paper
submitted to STOC
2008 which, if it survives peer review, shows that that the
analysis is wrong, and the algorithm is actually O(n).
[ Addendum 20160128: Zhang's claim was mistaken, and he retracted the paper. ]
Finally, several people, including John Von Neumann, proved that the
axioms of arithmetic are consistent. But it was shown later that no
such proof is possible.
A number of people wrote in with explanations of "more than twenty states"; I
will try to follow up soon.
Stan Yen points out that I missed an important aspect of the
convenience of instant mac & cheese: it has a long shelf life, so
it is possible to keep a couple of boxes on hand for when you want
them, and then when you do want macaroni and cheese you don't have to
go shopping for cheese and pasta. M. Yen has a good point. I
completely overlooked this, because my eating habits are such that I
nearly always have the ingredients for macaroni and cheese on hand.
M. Yen also points out that some of the attraction of Kraft
Macaroni and Cheese Dinner is its specific taste and texture. We all
have occasional longings for the comfort foods of childhood, and for
many people, me included, Kraft dinner is one of these. When you are
trying to recreate the memory of a beloved food from years past, the
quality of the ingredients is not the issue. I can sympathize with
this: I would continue to eat Kraft dinner if it tasted the way I
remember it having tasted twenty years ago. I still occasionally buy
horrible processed American cheese slices not because it's a good deal,
or because I like the cheese, but because I want to put it into
grilled cheese sandwiches to eat with Campbell's condensed tomato soup
on rainy days.
Regarding the invention of
the = sign, R. Koch sent me two papers by Florian Cajori, a
famous historian of mathematics. One paper, Note on our sign
of Equality, presented evidence that a certain Pompeo
Bolognetti independently invented the sign, perhaps even before Robert
Recorde did. The = sign appears in some notes that Bolognetti made,
possibly before 1557, when Recorde's book The Whetstone of
Witte was published, and certainly before 1568, when Bolognetti
died. Cajori suggests that Bolognetti used the sign because the dash
----- was being used for both equality and subtraction, so perhaps
Bolognetti chose to double the dash when he used it to denote
equality. Cajori says "We have here the extraordinary spectacle of
the same arbitrary sign having been chosen by independent workers
guided in their selection by different considerations."
The other paper M. Koch sent is Mathematical Signs of
Equality, and traces the many many symbols that have been used
for equality, and the gradual universal adoption of Recorde's sign.
Introduced in England in 1557, the Recorde sign was first widely
adopted in England. Cajori: "In the seventeenth century Recorde's
===== gained complete ascendancy in England." But at that time,
mathematicians in continental Europe were using a different sign , introduced by René Descartes.
Cajori believes that the universal adoption of Recorde's = sign in
Europe was due to its later use by Leibniz. Much of this material
reappears in Volume I of Cajori's book A History of Mathematical
Notations. Thank you, M. Koch.
The sun itself looks slightly redder ... this effect
is quite pronounced ... when
there are particles of soot in the air ... .
Neil Kandalgaonkar wrote to inform me that there is a web page
at the site of the National Oceanic and Atmospheric Administration
that appears at first to dispute this. I said that I could not
believe that the NOAA was actually disputing this. I was in San Diego
in October 2003, and when I went outside at lunchtime, the sun was
red. At the time, the whole
county was on fire, and anyone who wants to persuade me that these
two events were entirely unrelated will have an uphill battle.
Fortunately, M. Kandalgaonkar and I
determined that the NOAA web page not asserting any such silly thing
as that smoke does not make the sun look red. Rather, it is actually
asserting that smoke does not contribute to good sunsets. And
M. Kandalgaonkar then referred me to an amazing book,
M. G. Minnaert's Light and Color in the Outdoors.
This book explains every usual and unusual phenomenon of light and
color in the outdoors that you have ever observed, and dozens that you
may have observed but didn't notice until they were pointed out.
(Random example: "This explains why the smoke of a cigar or cigarette
is blue when blown immediately into the air, but becomes white if it
is been kept in the mouth first. The particles of smoke in the latter
case are covered by a coat of water and become much larger.") I may
report on this book in more detail in the future. In the meantime,
Minnaert says that the redness of the sun when seen through smoke is
due primarily to absorption of light, not to Rayleigh scattering:
The absorption of carbon increases rapidly from the red to the violet
of the spectrum; this characteristic is exemplified in the blood-red
color of the sun when seen through the smoke of a house on fire.
A couple of people have written to suggest that perhaps the one
science question any high school graduate ought to be able to answer
is "What is the scientific method?" Yes, I quite agree.
Nathan G. Senthil has also pointed out Richard P. Feynman's suggestion
on this topic. In one of the first few of his freshman physics
lectures, Feynman said that if nearly all scientific knowledge were to
be destroyed, and he were able to transmit only one piece of
scientific information to future generations, it would be that matter
is composed of atoms, because a tremendous amount of knowledge can be
inferred from this one fact. So we might turn this around and suggest
that every high school graduate should be able to give an account of
the atomic theory of matter.
I said that Pick's theorem
Implies that every lattice polygon has an area that is an integer
multiple of 1/2, "which I would not have thought was obvious." Dan
Schmidt pointed out that it is in fact obvious. As I pointed out in
the Pick's theorem article, every such polygon can be built up from
right triangles whose short sides are vertical
and horizontal; each such triangle is half of a rectangle, and
rectangles have integer areas. Oops.
Seth David Schoen brought to my attention the fascinating
phenomenon of tetrachromacy. It is believed that some (or all) humans
may have a color sensation apparatus that supports a four-dimensional
color space, rather than the three-dimensional space that it is
believed most humans have.
As is usual with color perception, the complete story is very
complicated and not entirely understood. In my brief research, I
discovered references to at least three different sorts of
tetrachromacy.
In addition to three types of cone cells, humans also have rod
cells in their retinas. The rod cells have a peak response to photons
of about 500 nm wavelength, which is quite different from the peak
responses of any of the cones. In the figure below, the dotted black
line is the response of the
rods; the colored lines are the responses of the three types of cones.
So it's at least conceivable that the brain could make use of rod cell
response to distinguish more colors than would be possible without
it. Impediments to this are that rods are poorly represented on the
fovea (the central part of the retina where the receptors are densest)
and they have a slow response. Also, because of the way the higher
neural layers are wired up, rod vision has poorer resolution than cone
vision.
I did not find any scientific papers that discussed rod tetrachromacy,
but I didn't look very hard.
The most common form of color blindness is deuteranomaly,
in which the pigment in the "green" cones is "redder" than it should
be. The result is that the subject has difficulty distinguishing
green and red. (Also common is protanomaly, which is just the
reverse: the "red" pigment is "greener" than it should be, with the
same result. What follows holds for protanomaly as well as
deuteranomaly.)
Genes for the red and green cone pigments are all carried on the X
chromosomes, never on the Y chromosomes. Men have only one X
chromosome, and so have only one gene each for the red and green
pigments. About 6-8% of all men carry an anomalous green pigment gene
on their X chromosome instead of a normal one and suffer from
deuteranomaly.
Each of these men inherited his X chromosome from his mother, who must
also therefore carry the anomalous gene on one of her two X
chromosomes. The other X chromosome of such a woman typically carries
the normal version of the gene. Since such a woman has genes for both
the normal and the "redder" version of the green pigment, she might
have both normal and anomalous cone cells. That is, she might have
the normal "green" cones and also the "redder" version of the "green"
cones. If so, she will have four different kinds of cones with four
different color responses: the usual "red", "green" and "blue" cones,
and the anomalous "green" cone, which we might call "yellow".
The big paper on this seems to be A study of women heterozygous
for colour deficiencies, by G. Jordan and J. D. Mollon,
appeared in Vision Research, Volume 33, Issue 11, July
1993, Pages 1495-1508. I haven't finished reading it yet. Here's my
summary of the abstract: They took 31 of women who were known to be
carriers of the anomalous gene and had them perform color-matching
tasks. Over a certain range of wavelengths, a tetrachromat who is
trying to mix light of wavelengths a and b to get as
close as possible to perceived color c should do it the same
way every time, whereas a trichromat would see many different mixtures as
equivalent. And Jordan and Mollon did in fact find a person who made
the same color match every time.
Another relevant paper with similar content is Richer color
experience in observers with multiple photopigment opsin genes,
by Kimberly A. Jameson, Susan M. Highnote, and Linda M. Wasserman,
appeared in Psychonomic Bulletin & Review 2001, 8 (2),
244-261. Happily, this
is available online for free.
My own description is highly condensed. Ryan's Sutherland's article Aliens
among us: Preliminary evidence of superhuman tetrachromats
is clear and readable, much more so than my explanation above.
Please do not be put off by the silly title; it is an excellent
article.
The website Processes in
Biological Vision claims that the human eye normally contains a
color receptor that responds to very short-wavelength violet and even
ultraviolet light, but that previous studies have missed this because
the lens tends to filter out such light and because indoor light
sources tend not to produce it. The site discusses the color
perception of persons who had their lenses removed. I have not yet
evaluated these claims, and the web site has a strong stink of
crackpotism, so beware.
In discussing Hero's
formula, I derived the formula
(2a^{2}b^{2} + 2a^{2}c^{2} +
2b^{2}c^{2} - a^{4} - b^{4}-
c^{4})/16 for the square of the area of a triangle with sides of lengths
a, b, and c, and then wondered how to get from
that mess to Hero's formula itself, which is nice and simple:
p(p-a)(p-b)(p-c),
where p is half the perimeter.
François Glineur wrote in to show me how easy it is.
First, my earlier calculations had given me the simpler expression
16A^{2} = 4a^{2}b^{2} -
(a^{2}+b^{2}-c^{2})^{2}, which, as he
says, is unfortunately not symmetric in a, b and
c. We know that it must be expressible in a symmetric form
somehow, because the triangle's area does not know or care which side
we have decided to designate as side a.
But the formula above is a difference of squares, so we can factor it
to obtain (2ab + a^{2} + b^{2} - c^{2})(2ab +
c^{2} - a^{2} - b^{2}), and then simplify the
a^{2} ±2ab + b^{2} parts to get
((a+b)^{2} - c^{2})(c^{2} -
(a-b)^{2}). But now each factor is itself a difference of
squares and can be factored, obtaining
(a+b+c)(a+b-c)(c+a-b)(c-a+b). From here to Hero's formula is
just a little step. As M. Glineur says, there are no lucky
guesses or complicated steps needed. Thank you, M. Glineur.
M. Glineur ended his note by saying:
In my opinion, an even "better" proof would not break the
symmetry between a, b and c at all, but I don't
have convincing one at hand.
Gareth McCaughan wrote to me with just such a proof; I hope to present it
sometime in the next few weeks. It is nicely symmetric, and its only
defect is that it depends on trigonometry.
Carl Witty pointed out that my equation of the risk of Russian
roulette with the risk of driving an automobile was an
oversimplification. For example, he said, someone playing Russian
roulette, even at extremely favorable odds, appears to be courting
suicide in a way that someone driving a car does not; a person with
strong ethical or religious beliefs against suicide might then reject
Russian roulette even if it is less risky than driving a car. I
hadn't appreciated this before; thank you, M. Witty.
I am reminded of the story of the philosopher Ramon Llull
(1235–1315). Llull was beatified, but not canonized, and my
recollection was that this was because of the circumstances of his
death: he had a habit of going to visit the infidels to preach loudly
and insistently about Christianity. Several narrow escapes did not
break him of this habit, and he was eventually he was torn apart by an
angry mob. Although it wasn't exactly suicide, it wasn't exactly not
suicide either, and the Church was too uncomfortable with it to let
him be canonized.
Then again, Wikipedia says he died "at home in Palma", so perhaps it's
all nonsense.
It also occurred to me that my cousin Alex Scheeline is a professor of
chemistry at UIUC, and my wife's mother's younger brother's daughter's
husband's older brother's wife's twin sister is Laurie J. Butler, a
professor of physical chemistry at the University of Chicago. Both of
these have surely used gas chromatographs, so they bring the total to
five.
a collection of 2 billion points is completely enclosed
by a circle. does there exist a straight line having
exactly 1 billion of these points on each side
This has the appearance of someone's homework problem that they
plugged into Google verbatim. What struck me about it on rereading is
that the thing about the circle is a tautology. The rest of the
problem does not refer to the circle, and every collection of
2 billion points is completely enclosed by a circle, so the
clause about the circle is entirely unnecessary. So what is it doing
there?
All of my speculations about this are uncharitable (and, of course,
speculative), so I will suppress them. I did the query myself, and
was not enlightened.
If this query came from a high school student, as I imagine it did,
then following question probably has at least as much educational
value:
Show that for any collection of 2 billion points, there is a
circle that completely encloses them.
It seems to me that to answer that question, you must get to the heart
of what it means for something to be a mathematical proof. At a
higher educational level, this theorem might well be dismissed as
"obvious", or passed over momentarily on the way to something more
interesting with the phrase "since X is a finite set, it is
bounded." But for a high school student, it is worth careful
consideration. I worry that the teacher who asked the question does
not know that finite sets are bounded. Oops, one of my uncharitable
speculations leaked out.
Regarding the
manufacture of spherical objects, I omitted several kinds of
spherical objects that are not manufactured in any of the ways I
discussed.
One is the gumball. It's turned out to be surprisingly difficult to
get definitive information about how gumballs are manufactured. My
present understanding is that the gum is first extruded in a sort of
hollow pipe shape, and then clipped off into balls with a pinching
device something like the Civil-War-era bullet mold pictured at right.
The gumballs are then sprayed with a hard, shiny coating, which tends
to even out any irregularities.
Glass marbles are made with several processes. One of the most
interesting involves a device invented by Martin Frederick
Christensen. (US Patent #802,495, "Machine For Making Spherical
Bodies Or Balls".) The device has two wheels, each with a deep groove
around the rim. The grooves have a semicircular cross-section. The
wheels rotate in opposite directions on parallel axes, and are aligned
so that the space between the two grooves is exactly circular.
The marble is initially a slug of hot glass cut from the end of a long
rod. The slug sits in the two grooves and is rolled into a spherical
shape by the rotating wheels. For more details, see the Akron Marbles web
site.
Fiberglas is spun from a big vat of melted glass; to promote melting,
the glass starts out in the form of marbles. ("Marbles" appears to be
the correct jargon term.) I have not been able to find out how they
make the marbles to begin with. I found patents for the manufacture of
Fiberglas from the marbles, but nothing about how the marbles
themselves are made. Presumably they are not made with an apparatus
as sophisticated as Christensen's, since it is not important that the
marbles be exactly spherical. Wikipedia hints at "rollers".
The thingies pictured to the right are another kind of
nearly-spherical object I forgot about when I wrote the original
article. They are pellets of taconite ore. Back in the 1950s, the
supply of high-quality iron ore started to run out. Taconite contains
about 30% iron, but the metal is in the form of tiny particles
dispersed throughout very hard inert rock. To extract the iron, you
first crush the taconite to powder, and then magnetically separate the
iron dust from the rock dust.
But now you have a problem. Iron dust is tremendously inconvenient to
handle. The slightest breeze spreads it all over the place. It
sticks to things, it blows away. It can't be dumped into the smelting
furnace, because it will blow right back out. And iron-refining
processes were not equipped for pure iron anyway; they were developed
for high-grade ore, which contains about 65% iron.
The solution is to take the iron powder and mix it with some water,
then roll it in a drum with wet clay. The iron powder and clay
accumulate into pellets about a half-inch in diameter, and the pellets
are dried. Pellets are easy to transport and to store. You can dump
them into an open rail car, and most of them will still be in the rail
car when it arrives at the refinery. (Some of them fall out. If you
visit freight rail tracks, you'll find the pellets. I first learned
about taconite because I found the pellets on the ground underneath
the Conrail freight tracks at 32nd and Chestnut Streets in
Philadelphia. Then I wondered for years what they were until one day
I happened to run across a picture of them in a book I was reading.)
When the pellets arrive at the smelter, you can dump them in.
The pellets have around 65% iron content, which is just what the smelter
was designed for.
With stock Blosxom, however, this is impossible. The first problem
you encounter is that there is no stories_done callback.
Todd Larason pointed out that this is mistaken, because (as I
mentioned in the article) the foot template is called once,
just after all the stories are processed, and that is just what I was
asking for.
My first reaction was "Duh."
My second reaction was to protest that it had never occurred to me to
use foot, because that is not what it is for. It is for
assembling the footer!
There are two things wrong with this protest. First, it isn't a true
statement of history. It never occurred to me to use foot,
true, but not for the reason I wanted to claim. The real reason is
that I thought of a different solution first, implemented it, and
stopped thinking about it. If anything, this is a credit to Blosxom,
because it shows that some problems in Blosxom can be solved in
multiple ways. This speaks well to the simplicity and openness of
Blosxom's architecture.
The other thing wrong with this protest is that it assumes that that
is not what foot is for. For all I know, when
Blosxom's author was writing Blosxom, he considered adding a
stories_done callback, and, after a moment of reflection,
concluded that if someone ever wanted that, they could just use
foot instead. This would be entirely consistent with the
rest of Blosxom's design.
M. Larason also pointed out that even though the head
template (where I wanted the menu to go) is filled out and appended to
the output before the article titles are gathered, it is not too late
to change it. Any plugin can get last licks on the output by
modifying the global $blosxom::output variable at the last
minute. So (for example) I could have put PUT THE MENU HERE
into the head template, and then had my plugin do:
$blosxom::output =~ s/PUT THE MENU HERE/$completed_menu/g;
to get the menu into the output, without hacking on the base code.
I briefly considered and rejected the spinning
wheel, on the theory that people have spun plenty of thread with
nothing but their bare fingers and a stick to wind it around.
Brad Murray and I had a long conversation about this, in which he
described his experience using and watching others use several kinds
of spinning tools, including the spinning wheel, charkha (an Indian
spining wheel), drop spindle, and bare fingers, and said "I can't
imagine making a whole garment with my output sans tools." It
eventually dawned on me that I did not know what a drop spindle was.
A drop spindle is a device for making yarn or thread. The basic
process of making yarn or thread is this: you take some kind of
natural fiber, such as wool, cotton, or flax fiber, which you have
combed out so that the individual fibers are more or less going the
same direction. Then you twist some of the fibers into a thread. So
you have this big tangled mass of fiber with a twisted thread sticking
out of it. You tug on the thread, pulling it out gently, while still
twisting, and more fibers start to come away from the mass and get
twisted into the thread. You keep tugging and twisting and eventually
all the fiber is twisted into a thread. "Spinning" is this
tugging-twisting process that turns the mass of combed fibers into
yarn.
You can do this entirely by hand, but it's slow. The drop spindle
makes it a lot faster. A drop spindle is a stick with a hook stuck
into one end and a flywheel (the "whorl") near the other end. You
hang the spindle by the hook from the unspun fiber and spin it.
As the spindle revolves, the hook twists the wool into a thread. The
spindle is hanging unsupported from the fiber mass, so gravity tends
to tug more fibers out of the mass, and you help this along with your
fingers. The spindle continues to revolve at a more-or-less constant
rate because of the flywheel, producing a thread of more-or-less
constant twist. If you feed the growing thread uniformly, you get a
thread of uniform thickness.
When you have enough thread (or when the spindle gets too close to the
floor) you unhook the thread temporarily, wind the spun thread onto
the shaft of the spindle, rehook it, and continue spinning.
A spinning wheel is an elaboration of this basic device. The flywheel
is separate from the spindle itself, and drives it via a belt
arrangement. (The big wheel you probably picture in your mind when
you think of a spinning wheel is the flywheel.) The flywheel keeps
the spindle revolving at a uniform rate. The spinning wheel also has
a widget to keep the tension constant in the yarn. With the wheel,
you can spin a more uniform thread than with a drop spindle and you
can spin it faster.
I tried hard to write a coherent explanation of spinning, and
although spinning is very simple it's awfully hard to describe for
some reason. I read several descriptions on the web that all left me
scratching my head; what finally cleared it up for me was the videos of
drop spinning at the superb The Joy of Handspinning
web site. If my description left you scratching your head, check out
the videos; the "spinning" video will make it perfectly clear.
The drop spindle now seems to me like a good contender for one of the
twenty most-important tools. My omission of it wasn't an oversight,
but just plain old ignorance. I thought that the spinning wheel was
an incremental improvement on simpler tools, but I misunderstood what
the simpler tools were.
The charkha, by the way, is an Indian configuration of the spinning
wheel; "charkha" is just Hindi for "wheel". There are several
varieties of the charkha, one of which is the box charkha, a
horizontal spinning wheel in a box. The picture to the right depicts
Gandhi with a box charkha.
Doug Orleans asked whether I had considered the key. I hadn't,
but I think it's exempt from consideration for the same kinds of
reasons as those I cited for the remote control: any particular key
serves not a general door-opening function but a specific one. Not
that keys aren't important, but rather, they seem to be outside the
scope of this particular discussion.
Mike Krell asked why I would list the radio on my third-tier list
and omit the computer. Obviously, the question is not one of
usefulness or of importance, but of whether the "computer" is a "tool"
in the original sense of the list, or whether it is disqualified by
reason of being too general, too abstract, too complex, or something
like that. I made several arguments, most of which I think he
refuted.
My initial answer was that computers are disqualified for the same
reason that remote controls are: people do carry calculators, cell
phones, personal organizers, handheld GPS devices, and (if they work
for FedEx) package tracking gizmos. All of these are tools that
incorporate computers, but they don't really seem to be merely
different forms of the same thing. Each one is a tool, but "computer"
is not, similar to the way that the microscope and the telescope are
different tools, and not merely variations of "the lens".
Perhaps so, but then I had a fit of insanity and asserted that people
do not walk around toting general-purpose computers in case they
happen upon some data that needs processing. M. Krell very
gently pointed out that yes, they do exactly that: "I see them every
time I fly on an airplane, breaking them out to process some data for
work (or play) as soon as the flight attendant says it's OK."
Whoops. Quite so.
M. Krell suggested that by restricting the definition of "tool"
to devices that perform a single specific function or a few such
functions, I have circumscribed the definition of "tool" in an
arbitrary way that "does not accurately reflect the realities of our
current Information Age". Okay.
Jon Evans said "Everyone always forgets the hoe." I did indeed
forget the hoe. Not quite a knife, not quite a shovel...
Regarding the start of the year prior to 1751, when it
was moved from 25 March to 1 January. You may recall that I had a
series of articles
(1234)
in which I was concerned that Benjamin Franklin
might be only 299 years old, not 300, because of confusion about just
what year was meant in discussions of the date "6 January 1706". (The
legal year 1705 ran from 25 March 1705 through 24 March 1706.)
This ambiguity
was confusing at the time as well. It makes little difference to my
life whether Franklin is 300 years old or only 299, but if you were a
person living in 1706, you might like to be sure, when someone said
they would pay you fifty shillings on 6 January 1706, when the payment
would be.
It seems that baroque authors had a convention to disambiguate such
dates. Here's a quote from William Derham's 1726 introduction to
Robert Hooke's notes on the invention of the barometer:
To this I W.D. shall add another Remark I find
in the minutes of the Royal Society, February 20.
!!167^8_9!!, viz.…
And similarly, from Richard Waller's summary of The Life of
Dr. Robert Hooke:
…they prosecuted their former Inquiries, their first meeting at
Arundel house being on the ninth of Jan. !!166^6_7!!.
* In an
earlier post, I referred to "Ramanujan's approximation to π":
$$
\cfrac{1}{1+
\cfrac{e^{-2\pi}}{1 +
\cfrac{e^{-4\pi}}{1 + \cdots}}}
=
\left(
\sqrt{\frac{5+\sqrt5}2} -
\frac{\sqrt5-1}2
\right)
e^{2\pi/5}
$$
But this isn't the formula I was thinking of; I showed the wrong
formula! It's obviously not an approximation to π. The
approximation formula I was thinking of is no less astonishing:
Finally, in my article on the
20 most important tools, I said that I didn't think I knew anyone
who had used a gas chromatograph; Geoffrey Young pointed out that he
had used one.
Regarding my
bad solution to the problem of preventing multiple simultaneous SMTP
connections from the same place, Chris Siebenmann suggests that a
better strategy is to centralize all SMTP access through a single
server that can manage the connections in any convenient way, without
IPC, and fork child processes to perform the actual SMTP transactions.
I had ended my post with "duh", but this suggestion requires an even
bigger "duh", because I am already running such a server and
modifying it appropriately would have been even easier than the
modification I did make to the SMTP program. Thank you,
M. Siebenmann. Duh!
Regarding the 3n+1
domain, I should mention first that my use of the word "domain" is
incorrect here. A domain, properly speaking, is required to have both
addition and multiplication; the 3n+1 system supports only
multiplication. Addition doesn't work because (for example) 1+1 is
undefined in this system, 2 having been omitted.
I may discuss this in more detail in a future post.
Regarding Perl's
accidental s/.../.../ee feature, John Macdonald remarks that
he thinks it was first discovered by Randal Schwartz, not Tom
Christiansen, as I said. M. Macdonald suggests that
M. Schwartz first used it in the form
s/.../.../eieio in a "Just Another Perl Hacker" signature,
and that M. Christiansen then invented the
s/(\$\w+)/$1/ee form as a way to make real use of it.
Regarding Robert Hooke's
mismeasurement of the frequency of G above middle C, I referred to
Benjamin Wardhaugh's suggestion that the error was in the length of
the pendulum he used to mention the time. Carl Witty points out that
this is unlikely, for two reasons. First, Hooke would have been quite
familiar with how to make a pendulum of the correct length to time a
one-second interval; indeed, he probably would have had such pendulums
sitting around, ready to be used. And second, the period of a
pendulum is proportional to the square root of its length, so to get
the error of a factor of √2 in the measurement of the frequency
of the brass wire, Hooke's pendulum would have had to be twice
as long as it should have been.
In reply, I suggested several possible causes of error:
Perhaps the initial wire was not vibrating at precisely 1 Hz.
Synchronization with the 1 Hz pendulum might have been done by eye.
Any error in the original frequency would have been multiplied by 136
in the final result.
The halving of the wire might not have been exact.
If the tension in the wire changed during the halving process, the
shortened wire would have a frequency different from twice that of the
unshortened wire.
The note produced by the one-foot wire might not have been
exactly G. it could have varied somewhat from true G without being
detected by the musical observers.
G in 1664 wasn't 384 Hz anyway. In fact, I haven't finished
finding out just what Hooke meant by it, since pitches weren't fully
standardized; I don't know what Hooke intended for the reader to
understand from his assertion that it was 272 Hz. See Wikipedia's
discussion, for example.
I don't yet know that the second was accurately measured. You
need a pendulum that strokes exactly 86,400 times per day. They would
have had to calibrate it against sandglasses and such things. How
accurate was that calibration?
Even if the second was accurately measured, was it the same second
that we use today? I'm not sure. I should be able to find this out
by reading Hooke's lectures on gravitation (which I have handy) and
seeing what he gives as the acceleration due to the earth's gravity.
There may be some other possible causes of error that I haven't
thought of. Which of these actually contributed, and how much, I do
not know.
M. Witty also wondered if the fact that apparent error in the
measurement was almost exactly √2 was a coincidence. I imagine
so, but I could easily be wrong.
Someone once told me that some famous scholar, I think perhaps Thomas
Aquinas, was the only one of his contemporaries to read non-orally,
that they were astonished at how the information would just fly from
the book into his mind without his having to read it.
Ricardo J. B. Signes has confirmed this, except that it wasn't
Aquinas. He says that Augustine wrote of Ambrose that "When he read,
his eyes travelled over the page and his heart sought the sense, but
voice and tongue were silent." Thanks, Ricardo.
. . . a certain bishop John Wilkins had invented a
language in which the meaning of each word would be immediately
apparent from its spelling.
(I don't have an example handy, so I will make one up. All words that
begin with "p" are animals. Words beginning with "pa" are birds,
those with "pe" are fish, and so forth. Words beginning with "pel"
are fish with fins and scales. Words for fin-fish that live in rivers
and streams all begin with "pela". "pelam" is a salmon.)
I have now obtained a copy of this book, and it uses "salmon" as an
example. Wilkins' word for "salmon" is "zana". The first two letters
always identify one of forty primary classifications for things;
animal words begin with "z", and fish with "za". Each major group is
divided into nine subgroups; the third letter identifies which of the
nine subgroups the thing is in, with "n" denoting the ninth. The
ninth subgroup of fish are "squamous river fish". Each subgroup is
then divided into (usually) nine species, and the fourth letter
identifies which of the nine species the thing is in with "a" denoting
the second. The "squamous river fish" are divided as follows:
Bigger fish
Voracious fish
With loose scales
With one fin, near the tail; wide mouths, and sharp teeth (1)
With two fins
Common to both fresh and salt water (2)
Common to fresh water only
Spotted (3)
Not spotted
More round (4)
More broad or compressed (5)
With close, compact scales (6)
Not voracious
Bigger
Those that live in standing waters (7)
Those that live in running waters
Those that are thick and round (8)
Those that are broad and deep (9)
Lesser (10)
Smallest river fish
In the lower parts of the water
With one fin on the back (11)
With two fins and a broad head (12)
In the upper parts of the water (13)
"If you ask me to name the proudest distinction of Americans, I would
choose--because it contains all the others--the fact that they were
the people who created the phrase 'to make money.' No other language
or nation had ever used these words before; men had always thought of
wealth as a static quantity--to be seized, begged, inherited, shared,
looted or obtained as a favor. Americans were the first to understand
that wealth has to be created."
I looked this up, and I found that it is not true. The OED has
citations back to 1472:
1472 R. CALLE in Paston Lett. (1976) II. 356, I truste be
Ester to make of money..at the leeste l marke.
1546 O. JOHNSON in H. Ellis Orig. Lett. Eng. Hist. 2nd
Ser. II. 175 Besides the monney that I shal make of the said
wares.
1583 T. STOCKER tr. Tragicall Hist. Ciuile Warres Lowe
Countries II. 64 [They] furnished him with all the money they
were able to make.
1588 R. PARKE tr. J. G. de Mendoza Hist. China 45 Then may the
husband afterwardes sell his wife for a slave, and make money
of her for the dowrie he gaue her.
I suppose it's possible that the phrase only became common in the
United States, but Rand's assertion that "No other nation had ever
used these words before" is mistaken.