seven signs in seven positions.

PALEGAS and MRBGPYO (aka ROYGBIV)

and days-of-the-week and ages-of-

-man and whatnot… the trivium &

quadrivium… may be superimposed

in various interesting ways.

the seven principles of UU will

eventually be invoked if this

is ever (again) the basis of a

sermon by me. “window crayons”

on cardboard box; 2015.

three newish guitar-stands arranged in such a way

that any two will fall down without the third.

and madeline’s “three women” statue, having a

similar property. blessings from our happy home

to yours if you’ve got one; double blessings if

you’re doing without. happy “black friday”.

the linking and not-linking rings

(blogpost of 05/03/014).

w’edia.

flickr shot.

that 7-space has seven-way symmetry is obvious

(rotate the drawing by 2\pi/7—about 51.43\degrees—).

but the *three*-way symmetry isn’t so obvious.

here it is in its 7-color wonder.

we’ve chosen to fix the “Mud” point at the top.

we then chose the “secondaries” line

{Green, Purple, Orange} and permuted;

the “primaries” permute accordingly;

voila.

*************************************************

a 2-way symmetry can be displayed by “swapping”

each primary with its “opposite”:

(RG)(BO)(YP).

i call this one “reflection in the Mud”.

(one may also… of course… reflect

in any of the other colors;

for each (there are three)

“line” through a given color,

interchange the positions for

the other two colors.

the lines-on-blue are

{bgy, bmo, bpr}, so

“reflection in the Blue” has

(GY)(MO)(PR)

as its permutation-notation.)

exercise: draw the other six lines.

hint.

about time these guys got some names.

ROY (this one) is obvious.

let’s see.

green blend.

bgy byg gby gyb ybg ygb;

GABY, then.

secondaries.

gop gpo ogp opg pgo pog;

PEGGIE-OH, then, maybe.

the orange blur.

bmo, bom, mbo, mob, obm, omb.

JIMBO comes to mind.

submissions welcome. yes, it’s the great

name-the-lines contest of ought-fifteen!

i’ll be forced for the sake of my own peace

of mind to type out the rest of the exercise

and make up three more names. “the purple

blur” isn’t much of a *name* for a line.

more like a *secret identity*.

or, the fano plane presented symmetrically.

each of the three triangle-edges

found along any of the “long lines”

(joining vertex-to-vertex

on the biggest 7-point “star”)

is a “line” of rainbow-space.

check it out. the “points” are

Mud Red Blue Green Purple Yellow Orange

the lines are MRG, RBP, BGY, GPO, PYM, YOR, OMB.

here’s a version from last year.

this new one’s much cooler.

formed by my right hand. i got a new shipment

of strings today… thanks, madeline!… so i

(finally!) tuned up one of my guitars “lefty”.

if i try to *play* it that way, it sounds like

day one. but if i play it *right* handed, i

can make it sound like music. the chords and

the dynamics are different, though. so, cool

trick, it sounds like somebody else playing

(while still sounding a lot like me).

or maybe that’s the cough syrup talking. whee!

consider ordinary (x,y,z) space.

co-ordinatize a “unit cube” in the all-positive octant.

put “primary colors” on the axes. next slide.

from the edges of the cube, “cut out” the three that

pass through the Origin of our system—i.e., (0,0,0).

next.

distort the resulting diagram so that the “top face”

(and the “missing” bottom face) remain *square*. i’ve

shown this “flattening out” in two steps: once as a

truncated-pyramid in a “3-D” view, and then as a

fully-flattened 7-vertex “graph”.

meanwhile, introduce the secondary colors… in the

“natural way”.

all this is pretty old hat around here. you could

look it up.

the novelty here is the stick-figure iconography

(each of the “icons” has the “top of the cube”

represented by the square-in-the-middle; the

three nonzero vertices of the “bottom” of the

cube appear along the top and right-hand “sticks”

of a given icon).

each of these 7 icons now represents a

*linear equation*; these are precisely

the equations of the 7 2D-subspaces-

-through-the-origin of the vector-space

{(0,0,0), (0,0,1), … , (1,1,1)}

having exactly eight vectors.

one can calculate directly on the icons

(rather than the triples-of-numbers or

the colors) using “set differences”.

but that’s it for today.

saturday night i colored in the corners of this cube.

the underlying black-and-white is based on a work of

the great dutch artist m.~c.~escher. the cardboard

cut-out version is from a collection by the american

mathematician doris schattschneider.

(_m.c._escher_kaleidocycles_).

anyhow, i’ve had the whole “5 platonic solids” set

from this work on display in the front room at home

for a while. the others are in color already, right

out of the book. i’ve had *another* set of these,

too: it’s a great “book” and might still be in print

for all i know. i had two editions, from years apart,

years ago.

i took this one to church on sunday and used it in my talk.

there wasn’t time to explain why i’d colored it the way i

did. but i *did* count the symmetry group of the cube,

two ways. any talk by me should have a theorem in it;

i’m happy to count that as a theorem.

24 because any of the 6 faces can be “face down”,

and each such choice-of-face allows for any of 4

remaining faces then to “face front” (all but the

“face-up” one *opposite* to our “chosen” face).

but also 24 because

1 identity

6 180-degree “flips” that fix two edges

(one for each pair of opposite edges)

8 120-degree “corner-turns”

(fix a pair of opposite corners;

there are 4 such; one may “turn”

right or left)

6 90-degree “face turns”

(fix opposite faces—3 ways; again,

one can go “right” or “left”)

3 180-degree face turns.

and this messy version is actually quite clear when

one is actually holding up an actual cube and pointing

at the drawing on the board. or in this case, at one’s

own sweatshirt. canvas makes a good whiteboard.

you can do it all without even mentioning “group theory”…

and i *sure* didn’t get to prove that this set of 24

“moves” gives a version of “the symmetric group on 4

objects”. anyway, part of the point is that one need

not have introduced any “math code” into the discussion

at all to arrive some some *very* useful results.

i learned the symmetry-groups-of-solids trick from an

old master. i mentioned this book in the service.

but mostly i talked about stuff like bible studies and music.

stuff that people actually show up at church *for*. bring what

you love to church and share it. food and money are particularly

welcome.