(And they’re peculiar
too, in-particular..
..
{But yet, though,
there’s a..
pair of parallel l’s
in ‘parallel’!..}*)
..
*(Go figure!^..
{.. with a straight-edge
and compass, of course}.)
Blog-post # 644:
(644 = 2 * 7 * 23 * 2.)
Seven and two-halves
art-inanimations
(plus two more halves
below):
“Mobius-Tube-esquely
Klein-Bottlefly-esque”
has the alternative name:
“Inside One Ear..
And Outside The Other”..
(!)
It is more math than art.
(And it’s not even math.)
See below for info on what I
was attempting to accomplish
with this image.
–
“It Is Empty Ruin” is just
a line-drawing version of
“In Time’s Purity”.
These two images’ names
are anagrams of each other.
(I posted that anagram in
last week’s post.)
—————————
—————————
Anagrams, 37:
..
.
This reality rotates
and orbits weird nodes.
=
It alternated or is
so stirred beyond
what is.
—
Inside Earth,
clock-towers rise.
=
Their circles too
wander skies.
—
The symmetry is
as the machines.
=
Mathematics yet
rhymes/shines.
—
Universes were as
knotted as dimensions.
=
A stained torus is
never skewed in
no mess.
—
Time has an edge.
Its horn spews.
=
Weirdness is
that megaphone’s.
—
Mathematics’ helices
rotate/spin.
=
Hemispherical tones
tame static.
—
Inert colors do
form hues’ space.
=
A corner of those
is so crumpled.
—
As in such shape
or motion,..
=
This is an
amorphous cone.
—
The occult is
in odd science.
=
Such coincidence
does tilt.
—
Trust/belief:
=
Butterflies.
—
Televisions:
=
To sieve nils,..
=
Evil is stone,..
=
Is evil’s tone.
=
(Lens is to vie..
=
In lies’ votes.)
—
Colors skew.
=
Looks screw.
—
In parallels
to glassiness,..
=
All is stone,
angels, spirals.
—
Death’s mushroom was..
=
A hot summer’s shadow.
—
Thorned fires or deaths:
=
Hot froth is as rendered.
—
Rude morons..
=
.. Soon murder.
—
As Mammon has been cast
inside Its void,..
=
That demonic sin is
so massive/mean/bad.
—
Hatred’s evil asses win
..
=
..
Via this era’s lewdness.
—
God hates/hit
all losers.
=
That Hell’s goals
do rise.
—
As do racist ruses,
wings, or knives,..
=
Our vast rising
wickedness soars.
—
That is a
fastest helicopter.
=
As it lifts/reaches
at the top.
—
On a knot’s edge,..
=
God takes none.
—
The rapist:
=
That spire.
—
A spire..
=
.. Is rape.
—
Pathetic assholes:
=
Hate spoils/cheats.
=
Each’s hate so spilt.
=
Each is the apt loss.
=
Cheats’ hate is slop.
—
Our racist and
asinine con-games:
=
Arousing sanctions
end America.
=
(Sanctions:
Ending America or USA,
..
=
Sanctioning
aroused Americans,..
=
.. Arousing
sanctioned Americans.)
—
The smite is enraged.
=
This agreement dies.
—
Anger was inside us.
=
A dire gun is as sewn.
=
A dire gun sins as we.
—
As divine
as brutalizing:
=
Nazi evil
again disturbs.
—
Brutal designs..
=
Disturb angels.
—
The urinal’s
sinful design is
what ends.
=
It aligns and flushes
within rudeness.
—
Nasty alien stain:
=
An entity is nasal.
—
Chunks folded.
=
Ends hold/fuck.
—
We escape no spilling..
=
In sweeping collapse,..
=
In collapse’s weeping.
—
Nine-Eleven is
sad/ridiculous.
=
Asinine evil’s coil
endured us.
—
A Nine-Eleven rids us.
=
Insane evil endures.
—
Stellar ice is folded.
=
All coldest fire dies.
—————————
—————————
Palindrome:
‘No omen indeed: Nine moon.’
—————————
—————————
What smells funny?
..
A clown’s nose!..
—————————
(Not nearly as funny,..
but at least just as
funny-smelling..)
..
.
Maybe the new version of
NAFTA, which (if it even
ever happens) Trump is now
renegotiating, should be
called NASTIA*.
As in,..
Hey, the old NAFTA was bad.
But the new Trump version
is even
NASTIA’!..
..
.
–
*(North American
Super-Trump-Initiated
Agreement?)
—————————
—————————
But, now for a much more
intellectual game, though..
(If not really that
intellectual.)
..
.
Perpendicolor
————-
For 2 players.
Start with a square grid
with a large number of
grid-squares.
Then label the grid’s
middle four squares
like so:
A B
C D
Or, to better fit the
game’s name, use colors
instead.
(Best only if using a
computer, though.)
The game consists of a
predetermined (and not too
large) number of “super-
moves”, where each super-
move consists of, in total,
8 “sub-moves”, 4 sub-moves
per player per super-move.
Players alternately take
turns making sub-moves
within each super-move.
On a player’s sub-moves
within the nth super-move,
he/she repeats the letter
pattern in the large square
(“Large square” here refers
to a collection of grid-
squares) of 4*9^(n-1)
letters (with 2*3^(n-1)
letters on each side) that
was made during the (n-1)th
super-move (ie. during the
previous super-move).
(See image below.)
A player can rotate this
replicated large square by
either 0, 90, 180, or 270
degrees (no flipping).
And then he/she places it
into any of the (8 at most)
still-empty large squares
of 4*9^(n-1) grid-squares
located immediately above,
left of, below, right of,
or catercorner to the large
already-filled square (also
of 4*9^(n-1) grid-squares)
that was formed by the
(n-1)th super-move.
As so:
(After only 3 super-moves.^
The left half of the image
shows the layout of the
large squares’ placement.
And the image’s right half
shows a sample game, played
without much strategy —
I use colors instead of
letters here.)
A player gets a point for
each grid-square’s letter/
color along the edge of
his/her newly-placed large
square that matches up with
and that is just next to
a grid-square of the same
letter/color along the
adjoining edge of any
adjoining already-placed/
lettered large square.
(To score, the previously-
placed large square may have
been placed by either player
and during either the current
or most-recent super-moves.)
Diagonally-touching matching
letters net no points, though.
The winner is the player with
the largest score, after the
predetermined number of super
-moves have been played.
(Note: Given the rapidly-
growing size of the large
squares and the fact that the
interiors of large squares
do not affect the game’s
scoring, players thus need
only to label the edges of
the newly-placed large
squares — an especially good
idea if this game is played
by-hand and not by using a
computer.)
—————————
—————————
Regarding some math:
Consider the fractally
function, which I’ve
discussed before,
f(x) =
sum (-1)^floor(k*x)/k,
where k runs from 1 to
infinity, and where
floor(k*x) is (k*x)
rounded down to the
nearest integer.
I once proved that if
x is a rational n/d,
where n and d are coprime
integers, then the sum
converges to a finite
value whenever n is an
odd integer, and d can
be either an odd or even
integer.
(I think I proved this.)
But if n is an even
integer and d is an odd
integer, however, the
sum diverges to plus or
minus infinity.
(I think.)
(Both n and d can’t be
even, of course, because
then they would not be
coprime.)
Though the new no-big-deal
thing I figured out (if it
is true) is that if x =
y + 1/y + 1, where y can
be any _rational_ number
that you choose, then the
sum converges to a finite
value.
This is because, for y =
any rational number,
y + 1/y + 1 is a rational
number with (in reduced
form) an odd numerator.
But note that y + 1/y + 1
can still equal a rational
number that has an even
numerator and an odd
denominator, and so the
sum diverges.
Or it can equal an
irrational number. (But I
have not proved anything
regarding convergence of
the sum f(x) if x is
irrational.)
But in these cases where
y + 1/y + 1 does not
equal a rational with an
odd numerator, y must be
an irrational number.
However, the plot of
f(x +1/x +1)
looks to me to be just
as much fractally as
f(x) looks to be.
(Though f(x +1/x +1) when
x is in the vicinity of 1
seems to be much less
fractally.)
–
–
Between any two distinct
values of X_1 and X_2,
no matter how close to
each other, there should
be an infinite number of
x’s where the sum f(x)
diverges to +-infinity
and an infinite number
of x’s where the sum
f(x +1/x +1) diverges
to +-infinity.
(Note: I use “fractally”
not only to mean the
adjective form of fractal,
but too to imply that f(x)
looks fractal-LIKE, and
might not actually be a
true fractal — though I’m
pretty sure, without proof,
that it indeed is.)
—————————
Regarding the non-art image
“Mobius-Tube-esquely
Klein-Bottlefly-esque”:
I thought it a good idea to
represent a geometrically
symmetrical version of a
Klein-bottle.
(I am surprised I’ve never
seen such a form before, or
at least I don’t recall.
..
Oh, whoops, now I see that
Boy’s{/Morin’s} surface has
3-fold symmetry, is 3D, is
smooth, is “non-orientable”,
and has no singularities.)
Though my form, which is 3D,
is not a Klein-bottle in the
strictest sense, or maybe
not even at all.
(In a relaxed sense, it is
indeed easy to construct
the traditional Klein-
bottle in only three
dimensions.)
Unlike the strict type of
Klein-bottle, the surface
here is not a simple “sheet”,
but it instead branches off
at certain places, such as
where the surface around
either of the two openings
connects directly to the
tube that is wrapped around
that opening.
Topologically, this 3D
surface is basically just
a torus with a knotted
hole.
It could be transformed
(by expanding the outer
surface — which is
continuous and could be
made smooth despite the
sharp creases in its
representation here, and
which is considered to
be “glued” together
wherever it abuts itself
{so that any two parts of
the surface that touch
are considered to be
joined}) into a punctured
sphere, where this
punctured-sphere’s/torus’
hole forms a simple knot.
So maybe it isn’t accurate
even at all to refer to
this surface here as being
‘Klein-bottle-esque’.
I mean, Klein-bottles are
not simply just 3D tori,
not even tori with knotted
holes.
(Though knotted holes are
still otherwise cool, at
least..)
Anyway,..
Maybe someone (other than
Boy and Morin) can conceive
of other symmetric-looking
Klein-bottles, creating
something more advanced
and interesting, and more
strictly a Klein-bottle
than what I’ve crudely
represented here.
—————————
—————————
More strictly a Leroy Quet
than what I’ve crudely
represented here,..
Leroy
Hyperthetically 