Archive for the ‘desargues’ Category
somewhere in all this “desargues” stuff
i claimed that i don’t have the skill to draw
the “symmetric” version on the diagonals of
a dodecahedron (or, equivalently, on pairs-
-of-opposite-faces of an icosa… as shown here).
well, now i can prove it. the notes are already
posted i guess. i’m going on not much sleep &
anyhow, the *really* awesome stuff is guitar-
-music mostly these days. lectures-without-
-words is just something to do in between
tabbing-out arrangements for simple tunes.
you know my methods, watson.
the other four ways can be found
by holding the page up to a mirror
(rotated 90 degrees from this
orientation).
this is a fairly old drawing.
here is my recent discovery
of the “five-way symmetric” version
of the same situation (the “ten
circles theorem” as i call it).
finding this stuff out… color
displays in projective geometry…
is my probably my proudest accomplishment
in mathematics-as-such. my classroom
work impressed me quite a bit, too,
but those days are over now most likely.
here’s fano’s cube from 2011.
i wish i’d stayed a grill cook.
by now, i’d’ve probably gotten
pretty good at it.
cf
the second, smaller, sketch is from
desargues’ theorem in color (but let’s call it
desargues’ rainbow from here on
to match fano’s rainbow,
posted the next day).
in the first, newer, bigger sketch,
i’ve used my mystical “projective
geometry” powers to bend all the lines
into circles. so we now (as you can see)
have a red *circle* (at the “omega” point)
along with (arcs of) blue and yellow
*circles* (replacing the red, blue, and
yellow *lines* on the original [textbook]
drawing).
again, as one should expect from the
names-of-colors aspect of all this,
we find certain yellow-and-blue
point-pairs occurring on the arcs
of certain circles… and two such
circles meet in the *green* point
(green is the “blend” of yellow
and blue). and then likewise for
the other “secondary” colors: an
*orange* point at the intersection
of two red-and-yellow circles, and
a “purple” point where two red-and-
-blue circles meet.
desargues’ theorem is then that the
secondary colors are on one of the
“lines” of the system at hand.
[in this case, (an arc of) another
“wide circle” (our system consists of
ten circles; the “narrow circles”
appear as circles in the diagram
and arcs of three of the “wide”
circles are indicated by three-
-point arcs).]
the diagram is, essentially, traced from
bruce e.~meserve’s _fundamental_concepts_-
_of_geometry_ (dover reprint of 1983;
originally addison-wesley 1959). but
i added in the colors. with which, one
has as follows.
there’s a red line, a blue line, and a yellow line.
to start with. all sharing a point.
then two “triangles” are constructed:
each of these is to have a red, a blue, and a yellow
“vertex”.
(on the diagram, the “red”, “blue”, and “yellow”
vertices of one of the triangles is actually black;
this helps [maybe… it helps *me*] in determining
which vertices belong to which triangles.)
next we can think of the *edges* of these two triangles
as “blending two primary colors”; for instance we can
think of the red-and-yellow edge of either triangle as
determining an *orange* line… and go on to construct
an “orange vertex” at the intersection of these lines.
likewise, the yellow-and-blue edges of the two triangles
will determine a *green* point (at the intersection of
two green lines) and the blue-and-red edges will determine
a *purple* point.
desargues’ two-triangle theorem then says that the
orange, green, and purple points will “line up”.
proof: meserve (or any of *many* other sources)
footnote: alas, one has *exceptional* cases
in “euclidean two-space”, i.e., the ordinary
(two-dimensional) *plane* of high-school
geometry. one remedies this by working
instead in a (so-called) *projective* plane
(in such a plane, there are no “parallel” lines;
nevertheless, much of “ordinary” plane geometry
becomes *easier* in projective geometry [example:
this theorem]).
putting color names on things is a common trick in math,
but if anybody else is using blends-at-intersections in
anything resembling this way (elementary projective
geometry, i suppose i mean), i don’t know about it.
(priority claim; if you use this amazingly good idea,
remember where you got it. please.)
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.
the livingston review 