There's no way to define left/right without reference to a physical object that
exhibits the convention.
merriamwebster.com defines left as "of, relating to, situated on,
or being the side of the body in which the heart is mostly located".
You could try to define left in terms of the curvature of an exponential function. While driving on a road whose path follows the graph of y = 2^{x}, headed in the direction toward higher values of x, you turn left continuously. Problem is, that depends on the layout of Cartesian coördinates, which place the positive y axis to the left of the positive x axis. That's arbitrary; we could just as easily use a coördinate system with the y axis directed the other way. dictionary.com defines left as "of, relating to, or located on or near the side of a person or thing that is turned toward the west when the subject is facing north". Implicit in that definition is that the person is upright. When I stand on my hands, west is to my left when I'm facing south. That definition makes reference to west and north, so how are those defined? North is "a cardinal point of the compass, lying in the plane of the meridian and to the left of a person facing the rising sun." Left is defined in terms of north and north is defined in terms of left. Let's say there was an episode of Sesame Street that taught kids left and right by reference to visual examples. Imagine an extraterrestrial civilization receives the broadcast. Our TV systems scan lines left to right, an arbitrary convention; without knowing that, aliens could just as easily render the frames in mirror image.
I
walked past this rabbit today and he hid under an
upside‑down wheelbarrow.
He's a youngster, perhaps not experienced
enough to know how readily he can run away from big slow bipeds.
I figured he'd come out once the coast was clear
but he hung out there for several hours. It was,
after all, a shady spot.
These things happened today:
An excerpt
from the first movement of Contrasts by Béla Bartók.
The upper staff is the violin part (nontransposing); the lower staff
is for clarinet in A (sounds a minor third lower than notated).
I like the (sic!) which seems to be telling the clarinetist: not a typo!—despite how dolce the rest of this passage was, this pair of notes really is one semitone lower than what the violin is playing. Roadrunner
from yesterday morning.
This one doesn't have its crest raised like most of the ones in pics I've posted before—kind of like the jackrabbit a couple weeks ago whose
Beware: mathintensive posting today.
The Wikipedia Vector space article gives the complex numbers as an example of a vector space over the reals and notes that it's isomorphic to the vector space of ordered pairs of real numbers. That's straightforward; what follows is less obvious. Consider the real numbers as a vector space over the rationals. A theorem that follows from the Axiom of Choice says that every vector space has a basis, thus this vector space does. Call such a basis B. B is an infinite set with the same cardinality as ℝ. Consider the set of all ordered pairs (b₁,0) where b₁∈B or (0,b₂) where b₂∈B; call this set BB. That is, BB = (B×{0}) ∪ ({0}×B). BB also has the same cardinality as ℝ and can be put into a one‑to‑one correspondence with B. With b₁ and b₂ interpreted as coefficients of 1 and i respectively, BB is a basis for the vector space of complex numbers over the rationals. A one‑to‑one correspondence between B and BB induces an isomorphism between their respective vector spaces. Thus the group (ℝ,+) is isomorphic to the group (ℂ,+). This is one of my favorite counterintuitive consequences of the Axiom of Choice.
xkcd #2184
by Randall Munroe used here
by kind permission under license
(CC BYNC 2.5)
/ converted to indexed PNG
