Some of these are probably trivial . . .
- Why don't the Australian Aborigines seem to be related to the Polynesians?
Is the Northern shore of Australia so inhospitable that they never met,
or is there some history missing somewhere?
[One suggestion: too far, they
ate their livestock on the trip (as in NZ)]. [Another--NZ is `pointed to'
by a few chains of islands from the North. Australia is partly shielded by
other inhabited regions (New Guinea, Melanesia [where people are supposed
to be somewhat related to the Aborigines]) and is not `pointed at' by
a convenient chain of islands from NZ.] [An interesting sidelight is a
recent proposal that Aborigines colonized South America circa 50K years
ago. That would definitely imply a lot of missing history.]
- Did the American Indians have chickens? One would expect recent colonists
to bring along such useful animals . . .
It looks like the answer is NO. This could be because: A) They weren't
available yet or B) They didn't stand cooler weather well.
There were dogs..(the link that described this died).
- Why do no mammals have green fur? (I know Cecil had a go at this,
but I suspect a biochemical explanation.) The `Totalitarian Principle'
says that anything not forbidden or excessively improbable must happen.
Polar Bears can have green fur. Their fur is hollow, and algae can
take up residence there, giving their fur a greenish hue. So there
are mammals with green fur. And, of course, some sloths acquire moss.
- I know that the sine, cosine, tangent, etc of a quantity (N pi)/M is
`algebraic', in the sense that it is a solution of a polynomial with
integer coefficients. Is it true that all algebraic numbers can be
expressed as such a {sine,cosine,tangent,etc} of ((N/M) pi)? More
weakly, does each polynomial with integer coefficients have at least
one such solution?
A counter-example will suffice to answer the first question.
Let sin(x) =1/3. cos(x)=2sqrt(2)/3. In the series sin(x*2^n), if
x is of the form (N/M)pi then eventually (by the first n+1 terms)
there must be i and j such
that sin(x*2^i) = sin(x*2^2), with the cosines also being equal.
The first is sin(2x) = 4sqrt(2)/3^2, cos(2x) = 7/3^2. Then we get
sin(4x)=8*7*sqrt(2)/3^4, cos(4x)=17/3^4. sin(8x)=16*7*17*sqrt(2)/3^8,
cos(8x)=-5983/3^8, and so on. Notice that after j=0, the sine consists
of 2^(j+1) times sqrt(2) times an integer, divided by 3^(2^j). The cosine
consists of an integer (always odd) divided by 3^(2^j). That the cosine's
numerator is always odd is easy to see, since it is the difference of
an even and the square of an odd number. Thus the sine's numerator will
be that sqrt(2) times 2^(j+1) times a product of odd numbers.
If sin(x*2^j) = sin(x*2^i) for some j > i, then we have
2^(j+1) odd_j /3^(2^j) = 2^(i+1) odd_i /3^(2^i). Dividing both sides
by 2^(i+1)/3^(2^i) and defining b=j-i gives us
2^b odd_j /(3^(2^j-2^i)) = odd_i
However, since b is not zero, this can't be true (even != odd). Thus
1/3 cannot be a sine or cosine of (N/M) pi.
1/(2*sqrt(2)) is plainly an algebraic number, and letting it be tan(theta)
means sin(theta) = 1/3, and the above counterexample serves for the
tangent case also. (OK, not entirely fair, but true anyhow: see
the detailed document.
The second (nominally weaker) question is also answered
no, since we have the simple counterexample of 3X-1. This doesn't exclude
solutions of form of products of such sines with different N and M, but
I suspect that this will not work.
-
A related question deals with Pythagorean triples (the 3/4/5 triangle, for
example). Each triple is associated with a family of scaled triples, of
which one is the smallest; all at the same angle. For simplicity (believe
it or not) distinguish the 3/4/5 from the 4/3/5 and the -3/4/5, -3/-4/5,
etc. Adding any two of these family's angles results in an angle corresponding
to another integer triple, and thus another family. There are trivial
integer triplets at 1/0/1, -1/0/1, 0/1/1, and 0/-1/1, and we can consider
the first as representing an identity. When we are adding the angles for
the families, the angle for -3/4/5 when added to that for 3/4/5 results in
the inverse. These angles form a group under addition modulo 2pi.
Now consider the set of angles resulting from repeated addition of a
single one of the triplet angles (call it the seed) to itself.
Will this ever repeat, or is there an infinite set of angles for each seed?
If there are an infinite set of angles for each seed, can we find a finite
set of seeds? We almost certainly need an infinite set of seeds.
Partial answers seem to be that there is
an
infinite set of angles for each seed, and almost certainly we require
an infinite number of seeds. Composing the 4/3/5 triangle with itself
to add the angles results in an infinite number of primitive triples.
- What happened to the Prisoner's Aid Societies of the last century?
Did they just fade away as prisons became more humane?
- I'm told half of the price of American beer goes to pay for the advertising
to induce us to buy the stuff. What fraction of the price of a car
goes to the same end?
A salesman said it was $175. However, every
salesman has a relative who succeeded in getting 200,000 miles out of
whatever make of car you ask about.
- Is it still claimed that Berengia (the Bering Straits land bridge) was
unglaciated? Has anybody sent a ship out to take cores to see what is
under the mud? If in fact it was unglaciated, one would expect to see
river channels somewhere under the mud. (If there are river channels,
would one expect human settlements near the ocean? They'd be very hard
to spot even if you knew where to look, especially since the surf from
rising sea level would erode anything not buried deeply.)
Some people
are looking around off the west coast, and seem to have found a stone tool
or two. US News
- Tropical rainforests sometimes have laterite soils.
A laterite soil doesn't seem to be
an inevitable consequence of tropical heavy rain, though. What is the
chemical reaction(s) that hardens laterite soils? Weathering under
strong oxidizing and leaching conditions, with Al or HFeO2/FeO(OH)/Fe2O3
in silica-poor (porous and clay-like soils).
Is there something that
can, if added to the soil, prevent this set of reactions? Can this
improve fertility (volcanic ash? sand?)? Is there something that, mixed with
pulverized hardened laterite, will soften/restore it? What kind of
quantities are we talking about? (For example, if 1cm of extra stuff is
required over a square km, that's 10^4 cubic meters per km^2--over 200
railroad boxcars per km^2.)
How long does
the leaching take that deteriorates the soil?
- Take a white noise spectrum and subtract the spectrum due to playing
a tune on an instrument (normalize appropriately). Can the absence
of the tune be detected? Does it sound even remotely melodious? Obviously
one has to limit the spectrum so that it isn't too loud.
I tried the
experiment of playing Twinkle Little Star by playing the octave it was
in (white keys only) absent the melody keys. I'm not sure how to describe
the result. Something more or less
tuneful resulted; but I think a blind test is required, with a better
pianist.
- Why do I see so few `black' students in physics? Even granted that about
a third of black Americans endure elementary and high schools so bad that
it would be very hard to catch up academically to middle class students,
still I'd expect about 1 in 13 or 14. I have a
simple model but no real information.
- In some countries names are mutable and the true identification of
a person is family and place. How tough is it to make a mini-genealogy
into a user-id? It complicates databases somewhat :-)
- Birds fly, insects fly,
some mammals fly, some reptiles have flown, even a fish
can `fly' (sort of). How about amphibians? Any fliers? Gliders?
J.K. Battle pointed out that an Amazonian tree frog can glide.
- Which cultures created team sports? A number of different American
Indian tribes had team sports. I can't recall any indigenous team
sports from Liberia (a very complicated form of jumprope, though).
The Greek Olympics had individual events but no
team events. How old are cricket or soccer?
OK, found a reference:
the
Romans played some ball games.
- Out of the mouths of babes . . . My six-year-old asks "What eats fleas?"