Archive for July, 2011
mathblogging.org’s weekly picks #22.
Consider a set of five objects.
In fact, consider the “canonical” set of five objects: {0, 1, 2, 3, 4}.
There are then ten ways to form pairs of these objects, as one sees from the following display:
04 03 02 01
14 13 12
24 23
34
.
(Note that by “pairs”, I refer to unordered pairs. For example, “40” doesn’t occur in the display because this represents the same pair of objects as “04”. In fact, since {4, 0} = {0, 4} as sets, one could formalize the notion of “pairs” as I’m using it here by referring to sets-of-two-objects from our original five-set. Indeed, I’d do it this way despite the extra typing if I wasn’t afraid it would then look harder to read.)
Now let’s call the objects of this ten-set
{01, 02, 03, 04, 12, 13, 14, 23, 24, 34}
the points of a “space” called D (for Desargues). “D” itself is then this set of points, together with a collection of certain (unordered) triples of points. The triples themselves are called the lines of D.
Here are the lines of D.
{01, 02, 12}
{01, 03, 13}
{01, 04, 14}
{02, 03, 23}
{02, 04, 24}
{03, 04, 34}
{12, 13, 23}
{12, 14, 24}
{13, 14, 34}
{23, 24, 34}
.
To form a line, take any three objects from {0, 1, 2, 3, 4} and form all three possible pairs.
One now has a geometry-free construction of the “space” I’ve been examining here in the blog. In particular, each point appears in exactly three lines (just as each line contains exactly three points).
I intended more of this today. But, to my delighted surprise, duty calls: I’m meeting a new tutee in a couple hours and have to go catch the bus.
yesterday’s more verbose version.
the MEdZ logo indicates where
the ten lines of the desargues diagram
fall. one has ten such lines.
also ten points. each line
can be considered as a set
of three points; similarly each
single *point* belongs to three *lines*.
in fact, we have a “duality” here…
theorems about points-and-lines
remain true when the words
“point” and “line” are interchanged.
for example, in a self-dual space
having the property… which this one
does *not*… that “any two points
determine a unique line”, one would
also have “any two lines determine
a unique point” (and one makes
adjustments to plain-english like
“any two points *lie on* a line”
becoming-replaced-with
“any two lines *meet at* a point”
or what have you).
it turns out in desargues-space
(let’s say, in D) each line has
three parallel lines, all meeting
at a point. the point is said to be
the pole of the line and
the line is said to be the polar
of the point. a choice of pole-and-polar
for the diagram is a *polarity*.
ten polarities are displayed in color here.
each white dot represents the pole
for its diagram; the polar is colored
with the three “secondary” colors
(green, purple, and orange).
the three lines through the pole
have matching “primary” colors:
a red line, a blue line, and a yellow line
(if you will).
it turns out that the primary-color points
can be arranged… in exactly one way…
into *two* red-yellow-blue “triangles”
(whose “edges” are along lines of D).
now we come to the payoff.
the two triangles are said to be
“perspective from the pole”
(“p”, say; call the polar line “l”
while we’re at it if you please):
one imagines shining a light
held at p through the vertices
of one triangle to produce the
other triangle… like a slideshow.
and what happens is that now
*either* red-blue line will “hit”
the purple point… and *either*
red-yellow line will hit the orange
point… and *either* blue-yellow
line will hit the green point:
the colors “mix like pigments”.
recall that the secondaries…
the “mixed” colors… all fall on
the polar of p. recall that this
is a *line* of D.
when, as in this case, the three
points-of-intersection for the three
corresponding-edge-pairs
of a pair of triangles happen to
lie on a single line, the triangles
are said to be “perspective from
the line”. in our RBY metaphor,
perspectivity from a line means
we color the vertices of the triangles
and form the secondary colors
by intersecting the lines.
perspectivity from a line means
that the secondaries all line up.
now
if P is a “space” (a set of points
together with certain subsets
called lines) satisfying certain
axioms (those of a projective
space), then we have
desargues’s theorem
any two triangles perspective
from a point are perspective
from a line; any two triangles
perspective from a line are
perspective from a point.
(d’s theorem is *almost* true in
the ordinary euclidean plane…
but alas special cases must be
written in to account for
parallel lines. parallels are
banned from projective spaces
which makes ’em easier to work with
algebraically but harder to visualize.
luckily “ordinary” planes can be
made to “sit inside” projective planes
so we can recover all of euclidean
geometry in a more-convenient-for-
-abstract-symbol-manipulation form.
PS
i still haven’t “solved”
ten-point reverse TTT,
by the way. but it’s very likely
only a matter of time.
somebody skilled in computer coding
could probably knock it out in a few hours.
a by-hand reduction
of my desargues-space poster,
with one less layer of redundancy
(and the colors of the triangles
reversed; also some cosmetic tweaks).
ten poles, ten polars,
and ten pairs-of-triangles:
ten ways to use one drawing
(from MathEdZine, of course)
to illustrate one theorem.
(desargues’ theorem at w’edia.)
performing calculations using
actual blobs of color rather than
alphabetical symbols *standing*
for colors is time-consuming
(and demanding of special tools)…
but one literally “sees” certain things
*much more readily* than with
symbol manipulation.
hey, i’m a visual learner.
whenever anybody says anything like
“this vision can only succeed if
we all get on board and do our parts”…
well, all i can hear is “this vision
is doomed to fail”. there’s *no way*
we’re all going to get on board…
we all never got on board with *anything*.
never have so far anyway. history is conflict.
and anyhow, what-you-mean-we, mister politician?
and this is true even for
solomon garfunkel on common core standards
(PDF; from the notices).
they screw you up, your uncle sam.
they ought not mean to, but they do.
the livingston review 