## Archive for the ‘MathEdZine’ Category

- There are 4 quarts to a gallon, 2 pints to a quart, and 16 ounces to a pint. There are 4 ounces to a gill. There are (about) 26.4 gallons in a hectoliter. How many gills are in a hectoliter? Display the unit fractions for your calculation.
- Write as a single number in scientific notation: ( 1.5 * 10^{-17} ) / ( 7.5 * 10^8 ).

3. Consider sets X, Y, and Z in a universal set U, where

U = {b, i, g, f, a, t, c, o, d, e}

X = {e, g, b, d, f}

Y = {f, a, c, e}

Z = {b, e, a, t}.

Use the operations \prime, \cap, \cup—complement, intersection, and union—to find the sets represented by the given expressions.

X \cap Y = { f, e}

Y \cup Z =

Z’ =

(Y \cup Z)’ =

(X \cap Y) \cup Z’ =

X \cap (Y \cup Z)’ =

4. Sum the sequence: 88 + 99 +110 + . . . + 462.

5. Expand the expression (x-3)^4.

6. 250 beverages are tested for Purity, Body, and Flavor.

65 pass the Purity test.

95 pass the Body test.

90 pass the Flavor test.

20 pass on both Purity and Body.

25 pass on both Purity and Flavor.

40 pass on both Body and Flavor.

5 pass all three of the tests.

How many of the beverages pass *none* of the tests?

starting with the mathy stuff

on what used to be a door

(

at the corner of what used to

be a street: in “the livingston

library”, or, more precisely,

as i think of it, at the living-

ston “branch of the UUCE” libe,

where for “UUCE”, read th’

UU church in reynoldsburg

[ohio; do i have to tell you

*everything*?]… g-d willing

i might make it back to the

*main* branch while it still

stands…):

a bunch of taped-up drawings (and

reproductions of same) by me.

the entire RHS (right-hand side,

natch) is given to various versions

of “desargues’ theorem in color”;

there’s another of these at upper-

-left (& in between, “vlorbik’s

seven-color theorem” [in one of its

many versions]).

then the three big (eight-&-a-half-by-

-eleven) 16-point thingums; these are

versions of the “hurwitz tesseract”

(as i hereby dub it); the two 8-point

“cosets” of the normal 8-group in the

unit integral quaternions. a-four-hat,

as i like to call it. anyway.

and, illegible here more or less of course,

the back cover of an em-ed-zed (M Ed Z —

“math ed zine” to the acronym-averse).

featured here are *more* covers of MEdZ

(what else?): specifically, this very one

(#1—the “hip pocket vocab”, 2010; reissued

in digest size with new [-ly reprinted]

graphics & hand-lettered comments [same year,

i think]); K_n -slash- K_4 (a “remix” of

two “microzines” [eight pages on one side

of an 8 1/2 by 11 each] into one such);

P_2(Z_2) & P_2(F_n) (“projective planes”);

& , , & (“number sets”).

but enough about me. some of my *other* stuff.

r.~crumb’s _art_&_beauty_ (cover shot of mrs.~

~crumb). two “books” with comic-book-swearing

for “titles”; also clowes’ _modern_cartoonist_.

that poster of magazine covers from _starlog_.

you could look at that thing alone for quite

a while. in the right company. (alas.)

what you can’t see covering up part of this

poster (lower left) is the _comic_book_artist_

cover by i-know-not-who showing “magnus

robot fighter” clobbering an evil droid. i did

a song about fighting robots and have a soft

spot for this character.

two pics of fitzgerald & a bunch of his books; you

might also be able to see burroughs. in the brick

under the pez-head of winnie-der-pooh is a period

shot of dad early 60s. you can tell the images

of auden are *there* in my version but not who

they are images of; likewise langston hughes.

this is quite a satisfying way of making the time

go by as long as you don’t wish for somebody else

actually to *read* it…

… but really it’s the *same* d-mn thing,

over and over.

i can now say for sure that some of the drawings

in these old Math Ed Zines scattered all around

the place could be adapted for coloring books.

the B&W drawing underlying both colorations

is from a zine of about ten years ago and shows

graphically (“lectures without words”) that

. you’d see this code

on the actual display if i could get a decent

shot. a poor workman blames his tools. and i

intend to make the most of it.

from blank file-folder (and no idea)

to conceived, drafted, penciled, and inked

before finishing my third cup of coffee:

behold: MEdZ # (1+i+j+k)/2!—

th’ G-mod-H issue!! in which we can see

no less than *four* (count ’em) more-or-less

familiar examples of “modding by a subgroup”.

namely,

(-+) {E,O}—even & odd: the ol’ cosmic yin-yang…

(++) time considered as an endless spiral of half-days…

(- -) the “unit circle” & the “periodic functions”…

(+-) time considered as an endless spiral of weeks.

one can easily look up the (- -) diagram expanded

at arbitrary length in textbooks considering “trig”.

and the helices of endless time are too familiar

to say much more about. the “clock face”, though,

is a bottomless well of shorthand examples—there’s

a car trying to run us off the road at three o’clock—

and so might be worth some further development

if it were useful in, say, fixing notations.

but (-+) one is prepared to go on about at any length.

is one of the most useful finite sets there is,

after all. for example, the 64 hexagrams of the i-ching

amount at some level to displayed pleasingly.

in the zine by me about “the 64 things” the hexagrams are

replaced with “subgraphs of K_4” (where of course K_4 is

the “complete graph on four points”; a diagram having

six edges [giving the six “lines” of the ching in that

version]; the 64 things are then subsets-a-six-set

[say {y, b, r, p, o, g} for the “full color” version]).

for example. i hope to continue in this vein later.

hello out there. ☰☱☲☳☴☵☶☷

the seven black triangles are

the blends

Mud Yellow Purple

Mud Red Green

Mud Blue Orange

the blurs

Yellow Blue Green

Yellow Red Orange

Blue Red Purple

and

the ideal

Purple Orange Green.

the “theorem” in question is then that

when the “colors” MRBGPYO are arranged

symmetrically (in this order) around a circle

(the “vertices”of a “heptagon”, if you wanna

go all technical), these Color Triples will

each form a 1-2-4 triangle.

but wait a minute, there, vlorb. what the devil

is a 1-2-4 triangle. well, as shown on the “ideal”

triple (center bottom), the angles formed by these

triangles have the ratios 1:2:4. stay after class

if you wanna hear about the law of sines.

note here that a 1-4-2 triangle is another beast altogether.

handedness counts. (but only to ten… sorry about that.)

anyhow, then you can do group theory. fano plane.

th’ simple group of order 168. stuff like that.

all well known before i came and tried to take

the credit for the coloring-book approach.

with, so far anyway, no priority disputes.

okay then.

the tetrahedral group at left: A_4, to the group-theory geeks.

up top, the “yrb” labeling of the vertices of a cube,

with the bit-string digital code and a 2-D projection.

the seven-color theorem… concerning the simple group

of order 168 & MRBGPYO… is hinted at.

under that, as one can *kind of* read on the blurry photo,

is “desargues theorem in color” — ten “points”

(one Mud, two Yellow, two Blue, two Red, one Green,

one Purple, one Orange) in abstract “space”.

the best version… i’m not technically up to drawing it…

is to put the colors on the ten diagonals of a dodecahedron.

next best is the five-point star version taking up

the biggest part of the file-folder.

next to that on the right: the vertices of the cube

colorized again. pretty much the same way if memory serves.

the points-to-lines “duality” is colorized better here, i think.

at the bottom, several versions of the seven-point star version

of the MRBGPYO theorem… and other stuff about heptagons.

there’re some books in there, too. that’s it for today.

here one has labeled the vertices of a cube

with (“euclidean”) 3-space co-ordinates.

the “origin”… we can think of it as

our “point of view”… is at 000.

[the “point” in 3-space usually

denoted (x, y, z) is here written

as “xyz”; we restrict our attention

to the values in {0, 1} for these

variables (since the (8) points

000, 001, 010, 011,

100, 101, 110, 111

then form the vertices of our cube).]

putting the primary colors (Red, Yellow, & Blue)

“on the x, y, & z axes (respectively)”, i.e., putting

Red—>001

Yel—>010

Blu—>100,

we may then conveniently put the *secondaries*

(Green, Orange, & Purple) at the “third vertex” of

the back left wall (“G = Y + B”),

the front left wall (“P = R + B”), and

the floor (“O = R + Y”)

(respectively).

the last vertex is the “ideal” point 111;

all the colors blend here to form Mud.

cool algebra ensues. but not till after dinner.

or you could look it up.

here’s a hex board

with seven icons on it;

each icon has seven colors;

the seven permutations of

colors-into-hexes (each icon

has seven hexes) can be

considered as the objects

of a cyclic group.

specifically

{

() = \identity

(1234567) = \psi

(1357246)= \psi^2

(1473625)=\psi^3

(1526374)=\psi^4

(1642753)=\psi^5

(1765432)=\psi^6

}.

(of course one has \psi^7=\identity,

etcetera… the group here is

essentially just integers-mod-7:

{ [0], [1], [2], [3], [4], [5], [6]},

with addition defined by

“cancelling away” multiples

of 7 (forget about the fancily

denoted “permutation structure”

displayed in defining \phi

and just look at the exponents).

the “cycle notation” used here

is *much* under-used, in my opinion.

but we’ll really only need it

for future slides.

my latest… i thought i was quarterly

when i issued it but maybe i’m semi-annual…

looks like this.

1876

2345

in the upper left are the “covers”:

these become the front and back

of the zine in folded form.

i discussed the title a few hours ago

(along with some other stuff).

suffice it to say here that our

subject is *self-dual spaces*.

*under* the title (in folded form;

above it here) are two

representations of the *simplest*

example: duality in P^2(F_2).

the space P^2(F_2) is itself

displayed (without a duality)

on p.5 (the lower right of the

photo) in the usual “triangle” way.

the colors are used to show

how these two objects…

*and* a certain subobject

of the space on pp. 7 & 8

(the 21-point P^2(F_4))…

can be considered “the same”.

the lower left (pp. 2 & 3)

are the desargues configuration.

i’ve discussed it recently

(here, for example)

and’m starting to hurry.

finally then (for now),

the cube-and-cross sections

bits are much the most vivid

visual motivator i’ve found

for the concept of “projectivity”.

the colors are surprisingly

helpful here. i’ll try to convince

you later. company coming. bye.

**introductory ramble**

i’ve been passing around the “spring 2011”

issue of MEdZ pretty freely and’ve posted

quite a bit *about* it.

but it isn’t very well-organized.

so in hopes of finding a curious reader

hoping to learn about the pictures in the zine

(MEdZ S’11 is in pictures-only “lecures

without words” format… an 8-page

micro-zine “unstaplebook” made by copying

all 8 images [including the front & back

covers of course] onto a single standard sheet

and cut-and-folded into tiny-booklet form;

this is the first issue in color and very much

my personal favorite [of the moment]), i decided

to poke around in the ol’ Archive for the ‘Graphics’ Category

(and so on) and write out some connecting remarks.

**prehistory of MEdZ S ’11**

the title is .

i pronounce this “ess is isomorphic

to the dual of ess” if i read it

symbol-for-symbol; one might also

read it (more freely) as

“S is a *self-dual* space”.

in some sense, this zine is about

spaces isomorphic to their own duals.

whatever that might mean.

in Some Finite Projective Spaces (01/07/11)

are several lectures-without-words shots

from earlier zines. including this one:

.

this was something of a breakthrough drawing for me.

(*Self-duality of the Fano Plane*, one might call it

[were it not a lecture-without-words].)

somehow i’d finally stumbled on a “visual” representation

for duality. this inspired a great deal of graphical

fiddling around by me. the same algebraic tricks

used in constructing the 7-point-to-7-line duality

for fano space could *also* be used on any n-point “space”

having n = 1 + q + q^2,

where “q” is some power-of-a-prime.

fano space is the case q = 2. so i did q=3 and q=5

(and the results are in the post i’ve linked to above)

and published ’em. by february 1, i’d developed

a much cleaner-looking graphical presentation

for these (projecive-planes-of-small-order).

but the next real “breakthrough” ideas

came with the case of q=4. in this early draft

i went a little bit “deliberately weird looking”.

a later version of *Self-duality of the Projective
Plane Over the Field of Four Elements*

became the last display of the (black-and-white)

edition of Spring 2011 and i announced it

on april 17.

but now i have to backtrack. i haven’t done any of the math.

okay. enough for today.

PS. fall quarter finds me doing two sections

of Calc !V at big-state-u. i met the lecturer

wednesday and met my students thursday. whee!