### with crazy and cool in the same molecule

here’s the page where i decided
i’d been looking at things inside-
-out… that the “primary color”
points of Rainbow Space should be
displayed as (midpoints of) *sides*
of the “color triangle”, rather
than as *vertices* (as i’ve been
doing for a couple of years or more
now… how long *was* it since i
discovered “color”, again?).

anyhow. *four* (rather than one)
of the lines on this display
now appear to the eye as “circles”:
the line-at-infinity (still) and
the three “blends” (newly). also
the line at infinity shows up
*outside* the rest of the figure

the primaries R, B, Y can now be seen,
as it were, “emerging from the Mud”
(the “ideal” point in the center),
with the secondaries then
“following” (as it were in *time*;
evolving or big-banging or what have
you) by “forming the blends” (ROY,
BPR, & YGB of course; if this isn’t
clear one should break out the finger-
paints).

the “blurs” take us back to the Mud
(and the whole thing starts over again…
or not… the metaphysics isn’t clear
at this point…)

yesterday’s post follows this (and, as i type,
doesn’t have the “finite geometry” category-tag on it;
there’s *much* more on this topic to be found by clicking
that category).

### more from the projective space coloring book

$P^2({\Bbb F}_5)$, i’d’ve called it today.
i drew this thing over twenty years ago as a page
in a geometry test. the coloring-in is much more
recent.

anyhow, two triangles perspective from a point
are perspective from a line. you could look it up.

### then love died

product endorsement:
triangle-graph paper.
better than “hex”, even.
anyway, for my purposes now.
you scribble away contentedly
for a couple hours. *then*
(maybe) you might as well
get online to get *in* line
and make a digital copy & post.
which is increasingly difficult.
and not *only* because i’m getting
stupider and more stubborn…
though both of these faults of mine
do play their parts in the whole mess.

### what can you do with this?

what *i’m* gonna do… now that i’ve spilled on it
and smeared some of the equations… is throw away
the hard-copy. one man’s treasure is the same man’s
trash.

### the mod-three projective plane

“modular arithmetic” was introduced to me
somewhere around 1968 as “clock arithmetic”
(among other things). it turns out—
very interestingly, as it turns out—
that one can do Additions and Multiplications
very much like the familiar operations on
${\Bbb Z} = \{ 0, \pm1, \pm2, \ldots\}$
(the integers), using any of the sets
${\Bbb N}_2 = \{0, 1\}$
${\Bbb N}_3 = \{0, 1, 2\}$
${\Bbb N}_4 = \{0, 1, 2, 3\}$
${\Bbb N}_5 = \{0, 1, 2, 3, 4\}$

${\Bbb N}_{12} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \}$

(the last-named, of course, gives
“clock arithmetic” its name).

and a multiplication (with the addition assumed
commutative and with the multiplication “distributive
over” the addition) are (for some reason) called
rings. the rings we care about today are
called fields; examples include whichever
“clock arithmetics” you care to name that have
a prime number of elements.

fields are “good” because we can *divide* in them.
(not just add, subtract, and multiply). the “prime”
criterion ensures this by ruling out the possibility
of solutions to “xy=0”
having “x” and “y” both *non*-zero.
for example, “on a mod-12 clock”, one has 3*4=0,
so we have a zero-product formed by two nonzero factors.
this is a “bad” thing and doesn’t happen in fields.

once we can do divisions, we can carry over most of
the “analytic geometry” from intro-to-algebra into
any of the settings ${\Bbb N}_2, {\Bbb N}_3, {\Bbb N}_5, {\Bbb N}_7, {\Bbb N}_{11}, \ldots$
and much else besides.

the value for me in replacing the so-called real field ${\Bbb R}$
with the finite fields ${\Bbb N}_2, {\Bbb N}_3, {\Bbb N}_5, \ldots$ is inestimable:
one can show *every single instance* of a certain phenomena
directly, without appeal to such weird notions as “continuity”.

in the example at hand, we have
“projective space over ${\Bbb N}_3$
(which the colorful image calls ${\Bbb F}_3$).
each line has four points. four of the lines…
including the “line at infinity”… have been
color-coded (and given equations as names).
the “finite points” here have z=1;
the line-at-infinity has z=0;
the “point at infinity” has x=z=0.

the rest is commentary.

### econ 101

i’m browsing as hard as i can.
but i can’t find any images
as pleasing to my eyes as these.
and yet these are sloppy first takes.
it’s almost as if reality
were more beautiful than the net.
this belongs in the other blog.
who said life is fair.

### square one

$\sum_{i=1}^\infty (1/2)^i = 1$, “without words”.

### crossposted from a numberwarrior thread

*At least when I was at school we were correct in writing “Answer = “, even though the teachers hated it!*/— howardat58, upthread.

i’m more likely to’ve encouraged this behavior
than to’ve “hated” it.

but “Answer” is a pretty awkward variable-name
so, given a chance, i’m also likely to’ve made it
as plain as i could find a way to do that what i’d
*really* like to see is a clear

A = …. messy expression to be simplified

right at the beginning and *then* the

bit at the end. which gives a presentation
clearer than one is likely to find on the
blackboards unerased by the previous
class. alas.

because “define variables (with units) precisely”
is a *major* sticking point for *many* students
and i’m not just talking about Remedial Algebra.
one of my favorite-ever calculus tutees
refused my excellent advice on this subject
*many* times.

but without it, we simply *cannot* organize
our presentations coherently.

she finally… same calc ii student here…
couldn’t endure my continual insistence
on keeping equations balanced as she
wrote out her calculations. we broke up
over it.

the attitude seems to be “it’s all just
ritual-process calculation anyway
until i can get the Answer”, whereas
of course one seeks to instill instead
something like “the Answer is itself
a collection of equivalent statements
(leading to the value of a variable)”.

“scratch” work is *obviously* the enemy of clarity
and not just clarity of *presentation*.
having calculated out some expression,
let’s say correctly, one is in the position
of having to *do something* with the result.

but without the whole A = Answer format…
a “proof”, if you will… one is left with a
bunch of area-on-the-page with certain
code-strings (and scattered english)
bearing no particular *stated* relation
to one another at all.

and if Answer = “the thing i want to see”
i’m very likely to give ’em full credit.
but that won’t make it good work.

### i probably won’t be able to find it here either: random comment crosspost

I have briefly mentioned that the alternative to explicit instruction may be described as ‘constructivist’ teaching. I don’t want to become bogged-down in this – I am aware that constructivism is actually [a] theory of learning and not of teaching and I have no problem with it in this regard; we link new knowledge to old etc. If it is true then, no matter how we teach, our students will learn constructively. However, some educationalists clearly do see implications for how we should teach.

i don’t want to become bogged-down in this either.
and yet i have been, deeply, many times, for years.
not so much these days. i just, you know, despair
of anything useful being said or done and check out.

all educational philosophies are useless in practice
until particular special cases are to be discussed
in carefully constructed contexts… so all we readers
ever seem to get is atrocity stories and suchlike
ill-disguised partisan politics.

“carefully constructed contexts” would include, for
example, a lot more attention than i’m usually able
to find about who the heck “we” are supposed to be.
this annoying pronoun is used as if it’ll mean all things
to all people. but it usually means nothing to me.

(in the passage at hand, i take “we teachers”
readily enough, so this isn’t a good example of
what bothers me… but hints at it. in electoral
politics, “we” can mean we-voters, we-americans,
we-patriots… and, often enough, two or three meanings
must be inferred to make any sense out certain
passages at all.)

angels dance on pinheads and owen leaves the room.﻿

### an invitation to vlorbik’s library

i pulled my (dover edition of) cantor’s epoch-making
contributions to the founding of the theory of transfinite numbers
yesterday to show tony from church;
he’d noticed my (prominently displayed)
copy of god created the integers
(hawking’s anthology of great math by math greats)
and mentioned “infinity” a few times in
my hearing, so it seemed like a natural.
and maybe it is… anyway, one does *not*
need a lot of high-tech “advanced math” to
read cantor’s stuff… and be just as mystified,
most likely, as most of the mathematicians
of cantor’s time (and many long after).

but i *should* have broken out the fourth dimension (w’edia),
by rudy rucker (w’edia).
tony’s *also* mentioned “the fourth dimension”
(as a concept) and *this* thing is bound to be
a whole lot more accessible than cantor.

i don’t know this particular rucker book at all well…
but i used his infinity and the mind in a class
long ago and’ve read some of his stories and whatnot.
rucker’s one of math’s best “popular” writers ever,
with a “transreal” SF-like vibe all his own.

i’d post more but my mouse is acting up again. damn it.

Next Page »

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)