Equilateral Triangle Tiler Conjecture; possible proof #514 Correcting Math
Archimedes Plutonium
conjecture yesterday by using "any shape in 2D". Muddy waters because,
obviously if that shape is a unit square itself that the unit
equilateral triangle is not going to tile it. So what I meant to say
was that given a choice of a unit tiler, because a large number of
shaped 2D objects will be given to you to tile, the question is what
"unit tiler in 2D" is the best choice to make? Do we pick a unit
square? or a unit area circle?
or a unit right-triangle? This conjecture believes that the best
choice to make for a unit tiler is the equilateral triangle. For the
reason that two triangles can be situated to form a parallelogram and
thus imitating the tiling ability of a square. And if the object is
roundish, the equilateral triangle is best capable of tiling a rounded
object since its long pointed end acts as spokes on a wheel, so that a
circle is better tiled by equilateral triangles than is square tiling.
So let me try to write out the Conjecture as clear and
logical as possible:
Equilateral Triangle Tiler Conjecture: In 2D, given a large number of
2D objects to tile by unit tilers, the unit tiler that achieves the
maximum tiling is the unit equilateral triangle.
Now I am going to offer some proof of this Conjecture, but before I do
that I want to remark that in 3D, I wonder if the tetrahedron is the
maximum unit tiler? I wonder if that is true since, unlike the unit
sphere that is always going to have gaps and holes between tangent
spheres, we can pack tetrahedrons as parallelograms in vast reaches of
3D as solidly as unit cubes without any gaps in between and then when
we reach the outer surface of the 3D object with its irregular shapes,
the pointed ends of the tetrahedron are more likely to fit into those
irregular shapes.
So I do not know if I can generalize this Conjecture from 2D to 3D
with equilateral-triangles to tetrahedrons.
And also, the idea in 3D is that a wedge is the most
versatile tiling 3D object, for a tetrahedron resembles either a wedge
or a fulcrum.
And let me touch on the topic of how and why I think this conjecture
is the equivalent of Least Action Principle in Physics. I believe they
are equivalent because the residue left over in the packing by
equilateral triangles is considered to be the residue of
maximum energy applied to the overall system. So that
if I can pack the object with maximum unit areas by the equilateral
triangles, then the residue portion is the reversal of what the "least
path in physics" is. So the equivalency is the idea that the reverse
of one is the other.
Possible proofs: of this conjecture of Equilateral Triangle Tiler:
(A) I already noted that if we are given a fractal geometry of a item
that is the same throughout as it becomes larger and larger, and if
the equilateral-triangle unit is the maximum tiler of this fractal
geometry, implies the proof.
(B) Let me offer an alternative proof scheme. The maximum tiler is
going to be the maximum tiler of various diameters of circles. So here
we focus on just
one object shape but of various diameters. It is an easy proof that
the hexagon composed of equilateral-triangles is the maximum tiler of
any given diameter circle. Unit squares or unit circles or unit
rectangles never come close to matching that 6 unit equilateral-
triangles tile a circle and all other unit objects will be less than
6.
Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
Archimedes Plutonium
Let me offer a possible third alternative proof (C).
We know in Calculus that the area of integration can be viewed as
picket fences summed up
over the graph of the figure. Now picket fences are long slender
rectangles, and in Calculus those rectangles are made as skinny as
possible. And sitting atop those skinny rectangles
forming a picket fence is a triangle. Now noone in math history ever
bothered to focus on those triangles, whether they are equilateral-
triangles or whether they are right-triangles. I have always, in my
mind, assumed they were tiny right-triangles. But what if they were
actually, really better off in being equilateral-triangles.
So here, if equilateral-triangles are better served than right-
triangles for the picket-fences
in the integral of Calculus, would be the fastest and most simple
proof that the equilateral
triangle is the maximum tiler in 2D.
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped)
>
> Now I am going to offer some proof of this Conjecture, but before I do
> that I want to remark that in 3D, I wonder if the tetrahedron is the
> maximum unit tiler? I wonder if that is true since, unlike the unit
> sphere that is always going to have gaps and holes between tangent
> spheres, we can pack tetrahedrons as parallelograms in vast reaches of
> 3D as solidly as unit cubes without any gaps in between and then when
> we reach the outer surface of the 3D object with its irregular shapes,
> the pointed ends of the tetrahedron are more likely to fit into those
> irregular shapes.
>
> So I do not know if I can generalize this Conjecture from 2D to 3D
> with equilateral-triangles to tetrahedrons.
> And also, the idea in 3D is that a wedge is the most
> versatile tiling 3D object, for a tetrahedron resembles either a wedge
> or a fulcrum.
>
Apparently I do not have enough tetrahedral dice to actually "seeing
is believing"
and where my mind-images are untrustworthy. I can see in 2D that the
equilateral
triangle forms parallelograms which can be packed solid with no gaps
in between in
2D. But I cannot see whether tetrahedron packed in 3D can end up with
no gaps and
be a solid packing. I can see where triangle on triangle of the
tetrahedron fit together, but
it appears as though, becuase the angles are 60 degrees that there is
a cleave-gap sooner
then later. So apparently tetrahedrons do not make a good tiler in 3D.
Apparently cubes or rectangular solids are better tilers in 3D.
Archimedes Plutonium
as the best tiler
because of obvious inability to form a solid packing with no gaps in
between.
I found this website to show me the densest tetrahedron packing:
--- quoting from wolfram ---
http://demonstrations.wolfram.com/DensestTetrahedraPacking/
The cover article for the 13 August 2009 issue of Nature published a
packing method for tetrahedra with a packing density of 0.782021, a
new record. For complex packings, space is divided into an orderly
arrangement of identical cells. In this packing, each cell has 72
tetrahedra, shown here.
--- end quoting ---
So that is proof that tetrahedron packing in 3D is not going to be the
maximum tiler.
So what is?
Well, I wonder what the figure is called if you split apart a cube or
a rectangular solid down
one of its diagonals. Just sliced the diagonal plane and leaving
behind two pieces. What
are they called? Are they a wedge? The cube division looks like a 3D
isosceles right triangle
and the rectangular-solid divided looks like a 3d right-triangle-
solid.
Now I wonder if these figures have a formal name in math already,
rather than simply calling
them wedges?
Now the reason I want them is because two of them put together form
the cube or rectangular
solid and for which I can tile without any holes or gaps between. I
believe one of these is going
to be the maximum tiler in 3D.
Archimedes Plutonium
thing leads to another.
So I am not going to dive into this Kepler Packing, but only outline
what I think the
answers are.
The trouble with Kepler Packing was the trouble with Number theory in
there was no
clear precise definition of Finite versus Infinite. If you do not know
what infinity is, well
you darn sure am not going to have a proof of Kepler Packing, just as
you are never
going to prove Goldbach, perfect numbers, Riemann Hypothesis and
thousands of other
number theory problems. If you do not have a precise definition of
finite and infinite
then Kepler Packing is just muddy muck.
I have defined Finite precisely as below the largest Planck Unit of
coulomb interactions inside
an atom of element 109 which is 10^500. Infinite is 10^500 and below
is finite.
But that number is too large to work with and we can abbreviate our
job of Kepler Packing
for we can use the convenient 100-Model. In this Model, with fractions
99.99 is the the last
and largest finite number with 100 starting the infinite numbers and
it is written as 900 =
-100 = 100 and where 999 = -1.
So the question in Kepler Packing is given a cube that is 99.99
length, how many unit
spheres can be maximally fit inside this cube. The answer is
surprizing for it is not the
hexagonal closed packing but rather a modification which I called
Oblong Hexagonal
Closed Packing, dealing with the last rows.
You see, the problem with the original Kepler Packing is that they
went into it without a
clear understanding of "infinity" and thus never a proof and never the
insights that packing
may require an end-row adjustments which gives a larger packing than
what was previously
known.
But now since I have precision defined finite with infinite, we can
have similar Packing Problem Conjectures. We can have a Conjecture
that asks for the maximum-tiler of
Euclidean geometry and Elliptic geometry and Hyperbolic geometry in 3D
and in 2D.
In 3D we use the inside volume of a cube of 99.99 cm for the Euclidean
and for Elliptic
we use 99.99 cm diameter of the inside of a sphere and for Hyperbolic
we use the same
diameter of the inside of a pseudosphere. By using these we can find
out if a rectangular
wedge or a cube-wedge is the maximum tiler. Both, if you remember must
be a unit volume
which keeps this math game fair. Now I happen to believe without
checking that the
rectangular solid sliced in half from its diagonal is the maximum
tiler rather than a cube
sliced in half at its diagonal. Because when two of these wedges are
put together they form
a cube and thus there are no gaps or holes between. Only at the edges
of the figure will there
be holes or gaps because of the irregular shape. But as for the
question of cube or rectangular
wedges, I believe the rectangular wedges will win over cube wedges
because we can make these rectangular wedges long and thin as needles
whereas the cube wedges are short and fat. And obviously for small
cubes or spheres or pseudospheres, we can tile some cube wedges but no
long thin needle wedges since they are longer than the diameter
itself. But once
the size of the container gets larger and larger, the thin and long
needles not only form a solid
interior but as packing in the edges of the hollow sphere we can fit
more of these long and thin
needle like wedges. Now we use the Calculus to tell us the ideal
rectangular-wedge of unit volume to work for the 100-Model or the
10^500 Model
Now for 2D, we again have infinity as 99.99 cm long in the 100-Model
and we use the
cube, circle and pseudosphere cross section to account for 2D. Here,
as I have already
explained that the maximum tiler is the unit equilateral triangle. For
not only does it form
a solid interior of no gaps with its parallelogram matrix, but that
its wedge shape can fit
into boundary ends.
porky_...@my-deja.com
<plutonium.archime...@gmail.com> wrote:
> I sort of made the waters muddy when I first blurted out this
> conjecture yesterday by using "any shape in 2D". Muddy waters because,
cause I'm your Hoochie Coochie man ...
Ostap S. B. M. Bender Jr.
<plutonium.archime...@gmail.com> wrote:
> I sort of made the waters muddy when I first blurted out this
> conjecture yesterday by using "any shape in 2D". Muddy waters because,
> obviously if that shape is a unit square itself that the unit
> equilateral triangle is not going to tile it. So what I meant to say
> was that given a choice of a unit tiler, because a large number of
> shaped 2D objects will be given to you to tile, the question is what
> "unit tiler in 2D" is the best choice to make? Do we pick a unit
> square? or a unit area circle?
> or a unit right-triangle? This conjecture believes that the best
> choice to make for a unit tiler is the equilateral triangle. For the
> reason that two triangles can be situated to form a parallelogram and
> thus imitating the tiling ability of a square. And if the object is
> roundish, the equilateral triangle is best capable of tiling a rounded
> object since its long pointed end acts as spokes on a wheel, so that a
> circle is better tiled by equilateral triangles than is square tiling.
>
> So let me try to write out the Conjecture as clear and
> logical as possible:
>
> Equilateral Triangle Tiler Conjecture: In 2D, given a large number of
> 2D objects to tile by unit tilers, the unit tiler that achieves the
> maximum tiling is the unit equilateral triangle.
>
Wouldn't the answer depend on the nature of these "2D objects"?
> Archimedes Plutoniumhttp://www.iw.net/~a_plutonium/
Archimedes Plutonium
Ostap S. B. M. Bender Jr. wrote:
>
> Wouldn't the answer depend on the nature of these "2D objects"?
In the 100-Model,
if we had a circle of diameter 99.99, and a hyperbolic triangle of
area
99.99, and a square of area 99.99 and asked what single unit figure
tiles each of these three objects the best, and by best meaning the
highest percent of density, then the conjecture of Equilateral
Triangles
posits that it is the unit equilateral triangle that wins the
contest.
One may find a rectangle that specifically wins for the square of
99.99 but
loses to the equilateral triangle in the circle and Hyperbolic
triangle.
The meaning of the conjecture is that there is a single, best tiler in
2D
and in 3D. For 3D, I am guessing it is a rectangular-solid-wedge
shaped
object.
In 3D there is a analogy in restaurant eating of sugar cubes, coffee
cup and
square ended chop sticks. So in a restaurant, have a waiter get you
these
three items and the coffee cup is sort of semisphere. Try seeing how
many
sugar cubes fit into the cup. Then try seeing how many chop stick
where
each chop stick is equal in volume to a sugar cube can tile the inside
of the
coffee cup.
These type of tiling conjectures were never possible in Old Math where
they
never had a handle on what infinity versus finite meant.
Archimedes Plutonium
Archimedes Plutonium
illogical
verdict in a case where they now allow corporations and labor unions
to spend
as much as they want on campaign financing.
It relates to this book and to what I call Old Math from New Math.
Whenever
math, the science of precision has a crumby or lousy or imprecise
definition,
or the lack of a definition, then math is in trouble. The lousy
definition or lack of
one was "finite number versus infinite number". Since Pythagorus in
ancient Greek
times to the 1990s, mathematics never precisely defined what it means
to be
a "finite number" and also, never distinguished between what is finite
and what
it means to be infinite. This book does precisely define Finite as
less than 10^500.
Even a theorem was proven in this book, using geometry, that in order
to define
finite from infinite, a number has to be selected with prudence, and
that number
is the boundary between finite and infinite.
But, now, why bring up this USA Supreme Court debacle of a decision?
Because
it again shows what happens when a body or organization lacks
precision of meaning
or definition. It is wildly disgraceful to think that the spending of
money and finances
is a form of "freedom of speech". If Roberts, Alito, Scalia, et al
Court thinks that
money in political campaigns is a freedom of speech act, then the
jails ought to let
out all the murders, rapists, especially bribers, as denying them of
their "freedom of
speech".
In other words, the error of Roberts, Alito, Scalia, is that they do
not know what
freedom of speech is in the first place. Money in political campaigns
is not a act
of freedom of speech but is more of a act of finance and contract law,
not speech.
So every act of bribery that occurs in the world, to Roberts or Scalia
or Alito is not
bribery but exercise of freedom of speech.
What I recommend is that the grave deficiency of the USA Supreme Court
is that
its members are vastly lacking in logic and clear thinking because few
if any are required
to have Symbolic Logic and Calculus under their belts before being
appointed to the bench.
Few judges are required to have College courses that teach them to
think with clarity and
with logic.
What I would recommend is that the Congress pass a law that says that
the Supreme Court
when deciding a case that involves Science, such as global warming, or
biotechnology
or energy or has some deep seated science in the case, that the
American Academy of Sciences be required to vote in those cases and
their vote counts as 2 votes, whereas Roberts vote counts as only 1
vote.
And in this case of travesty of justice of campaign financing, have it
in the law that the
American Academy of Sciences have a retro say on the decision because
the subject of
Logic is a science, and clearly, the Roberts court has no handle on
the meaning and definition
of Freedom of Speech. To Roberts court, those guys think money and
freedom of speech are
one and the same.
So if the AAS stepped into the picture and voted there 2 votes in nay
of the recent decision
would negate that horrible mistake of the Roberts Court.
And our Supreme Court is woefully lacking in science wisdom for which
many of our most
important judicial decisions are centered on getting "science right
and correct".
Ostap S. B. M. Bender Jr.
> whole entire Universe is just one big atom
> where dots of the electron-dot-cloud are galaxies
>
Wouldn't the answer depend on the nature of these "2D and 3D
objects"?
For example, if you objects were all polytopes, the best fit would be
one thing. Nut if your objects were all collections of spheres, the
answer could be quite different.
In fact, do you assume that these are "solid" objects, or are these
totally random sets in R^2 and R^3?
How do you define the sample space of and the probability distribution
on these "objects"?
Archimedes Plutonium
This recent Supreme Court Ruling is so awful and so offtrack of the
society
in general, that we legislated over the past decades of keeping "money
bribery"
out of campaigns and campaign financing. The McCain campaign finance
legislation.
And here the Supreme Court turns back the tide. Talk about "judicial
legislation" at
its worst.
And when those justices were appointed, they were grilled over and
over again that
they would not be engaged in "judicial legislating from the bench" and
leave law
making to the Congress. But in one case, the Court turned back all of
the Congress
work on controlling campaign financing.
So I am tempted to think that the Supreme Court does not have 4 or 5
members who
are too stupid to differentiate between "spending of money" and
"freedom of speech".
That all the justices are knowledgeable enough to know that money and
speech are
not in the same kettle of fish. That spending money is an action that
is not the same
as "freedom of speech".
So I am of the opinion that the justices all know the difference
between money and
freedom of speech. But that rather draws me to a ugly suggestion. That
about 4 or
5 of the justices colluded to give this verdict. And the best way to
hide behind the
collusion was to try to call spending money as a "freedom of speech".
So what would motivate such a collusion? Easy. That the Republican
party stands to
gain in a big way by having unlimited campaign financing. So that 4 or
5 of the justices
are merely making judicial legislation that favors the Republican
Party, and it comes
at an auspices time when the Democrats won so heavily in 2008 and
2006.
So either I have two explanations to the ready:
(1) the 4 or 5 justices are just ignorant about the difference between
money and
freedom of speech
(2) the justices are fully wise as to the differences but they wanted
to have a law
that allows huge money flow to the Republican Party, and the way to
hide behind
such a lousy verdict is to call money a act of freedom of speech.
Now I would think that the Executive branch should have the FBI
investigate the
supreme court justices, simply because of this rather astonishingly
ugly new
law passed by the Supreme Court. Not only are most Americans taken
aback by the
new law, but foreigners are also confounded as to how such ugly court
decision came
out of the sky blue yonder.
I suspect there has never been a case in which the president called
for the FBI to investigate
the Supreme Court members for impropriety. But there is always a first
time.
It seems that no-one should be above the law in the USA, even the
supreme court members.
But this ugly new campaign law has a smell or odor of impropriety. It
is hard to imagine the
justices so dumb as to think freedom of speech is equivalent to
spending of money. That means bribery is just an act of freedom of
speech. Corruption would then be a form of
freedom of speech.
So it is hard for me to think the Supreme Court was not familar with
this differentiation of
money spending and the act of freedom of speech. Rather, the more
plausible picture is that
the Supreme Court wanted a free licence of corporations to spend
unlimited money in
campaigns for that favors the Republicans.
So I think the president should have the FBI investigate the Supreme
Court, and I think
that Frontline and other investigative reporters should delve into the
actions leading up to
this new ugly law passed by the Supreme Court.
Archimedes Plutonium
There are too many restrictions involved:
(a) restricted in that there are only 3 geometries, so a square and
cube,
a circle and sphere, and hyperbolic triangle and pseudosphere cover
all the geometries
(b) restriction that these models are all 10^500 plus inverse fraction
of 10^-500
so the boundary lines of those models in (a) are 999..9-9d9999..-9
where that number
is 0.0000..0-1 shy of 10^500 and where the -1 is in the 10^-500
decimal place value.
(c) restriction on the tilers as all being "unit tilers"
> For example, if you objects were all polytopes, the best fit would be
> one thing. Nut if your objects were all collections of spheres, the
> answer could be quite different.
>
I have clarified my Conjecture to mean that in the three and only
three
geometries there is a unique tiler that is a maximum.
Since there is a maximum number as finite, there is a maximum tiler.
> In fact, do you assume that these are "solid" objects, or are these
> totally random sets in R^2 and R^3?
>
Originally, when I blurted out the conjecture, it was too murky and
clouded with
error. I do not need to have "every and any figure in any geometry."
This is what
you are confused over now.
I revised my conjecture and have confined it to the three geometries
with "related
models". The rectangular wedge tiler in 3D would give a specific
answer to
a hollow sphere of 10^500(subract that 0d00..0-1) volume, with the
pseudosphere of volume 10^500(subtract that 0d00..0-1)
and with the cube of volume 10^500(subtract that 0d00..0-1)
So all those restrictions are going to pull out a unique tiler.
> How do you define the sample space of and the probability distribution
> on these "objects"?
You are still settled on my first murky statements, for which I
revised, yet
you fail to comprehend that the revised statements eliminates those
probability
sample spaces. The restrictions pull out a unique tiler.
Archimedes Plutonium
Deadrat
news:69158f45-bde1-415b...@m37g2000yqf.googlegroups.com:
> I want to make mention of a recent USA Supreme Court decision of its
> illogical
> verdict in a case where they now allow corporations and labor unions
> to spend
> as much as they want on campaign financing.
<snip/>
> But, now, why bring up this USA Supreme Court debacle of a decision?
> Because
> it again shows what happens when a body or organization lacks
> precision of meaning
> or definition. It is wildly disgraceful to think that the spending of
> money and finances
> is a form of "freedom of speech". If Roberts, Alito, Scalia, et al
> Court thinks that
> money in political campaigns is a freedom of speech act, then the
> jails ought to let
> out all the murders, rapists, especially bribers, as denying them of
> their "freedom of
> speech".
Your problem is that you understand even less about First Amendment law
than you do about mathematics. In the Citizens United decision, the
Supreme Court did not equate the spending of money with speech. They
considered whether a governmental action (forbidding corporate
sponsorship of certain political ads) restricted political speech. A
literal interpretation of the First Amendment prohibits such action.
> In other words, the error of Roberts, Alito, Scalia, is that they do
> not know what
> freedom of speech is in the first place. Money in political campaigns
> is not a act
> of freedom of speech but is more of a act of finance and contract law,
> not speech.
The fact that spending money is a financial act is irrelevant. The
question is whether that act by the government impinges on speech. It's
the impinging that's the issue.
> So every act of bribery that occurs in the world, to Roberts or Scalia
> or Alito is not bribery but exercise of freedom of speech.
If you'd taken the trouble the read a little about First Amendment law,
then you'd understand that in spite of the wording of the Amendment,
freedom of speech does not mean the freedom to speak anything, anywhere,
and at any time. Bribery and contract murder do not fall within the
sphere of protected speech. Political advocacy does.
<snip/>
> What I would recommend is that the Congress pass a law that says that
> the Supreme Court
> when deciding a case that involves Science, such as global warming, or
> biotechnology
> or energy or has some deep seated science in the case, that the
> American Academy of Sciences be required to vote in those cases and
> their vote counts as 2 votes, whereas Roberts vote counts as only 1
> vote.
This would, for obvious reasons, require an amendment to the
Constitution.
<snip/>
> Archimedes Plutonium
<snip/>
Archimedes Plutonium
Deadrat wrote:
>
> Your problem is that you understand even less about First Amendment law
> than you do about mathematics. In the Citizens United decision, the
> Supreme Court did not equate the spending of money with speech. They
> considered whether a governmental action (forbidding corporate
> sponsorship of certain political ads) restricted political speech. A
> literal interpretation of the First Amendment prohibits such action.
>
Not only are you a coward that cannot post with a real name, but an
ignorant coward.
To try to mince words that "political speech" is anything different
from "freedom of
speech" points out how silly stupid you are.
The facts are, the USA Supreme Court of Roberts lead, has never
defined "freedom
of speech" and is using that as a hazard of all of American society.
The Roberts
lead USA Supreme Court can rule on anything that comes before it, rule
any way
they wish to rule, and cite as a reason "freedom of speech".
Roberts should have first off, when receiving this case, should have
"defined"
what is Freedom of Speech. None of those who voted in favor of this
case did that.
They were severely negligent of the Constitution itself.
At least the Founding Fathers gave us clues as to what freedom of
speech means.
Because they also cite freedom of religion and freedom or right to
bear arms.
(1) Freedom of religion-- to belief in a faith or religion of whatever
without being attacked
by others who have a different belief
(2) Freedom or right to bear arms-- the right to own a gun
(3) Freedom of Speech -- the right to say or write whatever is on ones
mind without be
attacked by others for saying or writing such.
The Roberts court should have at least done what I have done in 1
through 3. Sociology
includes law and courts and constitutions but sociology does not have
the precision
of definitions that mathematics or logic has. That is why it is
important that judges
have math and logic in their college training and that why Supreme
Court judges are unfit
if they had no logic courses.
There are limits and restrictions on 1-3 because we cannot have school
kids bearing guns
in schools, nor owners pointing rifles at other people, nor can we
tolerate libel and slander
for (3) nor shout "fire" in a crowded theater, nor tolerate those that
repeatedly shout lies
and profanity as "disturbing the peace". So all three have limits and
restrictions, but all
three have some vital core definition.
And it is apparent that the "spending of money" is not a act of
freedom of speech. Nor is it
an act of "freedom of religion" nor is the spending of money an act of
"right to bear arms".
So the USA Robert led Supreme Court has committed a huge error and is
leading the
USA Supreme Court vastly astray.
By their actions one can see that they were premeditated into taking
this case into the
Twilight Zone, because they knew that freedom of speech does not
equate in any way shape
or form to spending of money. And that leads to the suspicion that the
Roberts court premeditated on getting a law that corporations can
spend as much as they want to help
the Republicans in future elections.
In fact, if a case comes before the Roberts court dealing with
abortion. Then the Roberts
Court must rule that a women's right of choice is her freedom of
speech right and that
abortion can never be outlawed.
I do not know? Does anyone know, if Roberts, Alito, Scalia, ever had
any math or logic course
in their college or high school training? At the University of
Cincinnati in the 1970s, all becoming lawyers were required to take
Symbolic Logic in hopes that such a course would
help them think clearly and correctly. I doubt that Roberts, Alito, or
Scalia ever had a course
in logic, judging from this horrendous decision. And I suspect that in
all of the history of the
Supreme Court that this decision is one of the worst to ever be set.
For it is so unreasonable
and it mangles the very Constitution itself.
If anything in the Constitution that the Roberts court should have
looked at was the Freedom
to Vote in an Election. Of course the Constitution is cheqchered on
this freedom since it
lacks the freedom for women until later and it gave slaves some
fractional vote. But nonetheless the Constitution outlines a freedom
of voting:
Freedom to Vote in Elections: is a participation to choose a
representative to form a government.
So the freedom of speech is different from freedom of religion and
different from
freedom to vote. And all of these three freedoms are different from
the act of
spending money.
So how in the world that Roberts, Alito, and Scalia, messed all that
up, is very
alarming and dangerous to all of American society. Are we to expect
that Roberts
court can judge every case it wants and cite as "freedom of speech" as
their
excuse?
Ostap S. B. M. Bender Jr.
You are a very rude and clueless troll. Goodbye and good luck in
finding anybody wiling to have a conversation with you.
> that the revised statements eliminates those
> probability
> sample spaces. The restrictions pull out a unique tiler.
>
Deadrat
>
>
> Deadrat wrote:
>
>> [...]
>
> Not only are you a coward that cannot post with a real name,
Would it help if I told you I had my name legally changed to Deadrat?
> but an ignorant coward.
I may be a coward, and on certain subjects I'm ignorant as well. But not
this one. I know what I'm talking about, and you don't. Let's count the
ways below.
And at least I don't snip your posts when I respond to them.
> To try to mince words that "political speech" is anything different
> from "freedom of speech" points out how silly stupid you are.
That's one. Different types of speech gets different types of
protection. Only someone who knowss nothing about First Amendment law
would think that this is "stupid." Restriction of political speech
receives the toughest scrutiny.
> The facts are, the USA Supreme Court of Roberts lead, has never
> defined "freedom of speech"
That's two. The Court in its history has defined "freedom of speech"
over the years, and that definition has changed over the years. That
*you* don't understand the case law or that you don't agree with the
current Court doesn't mean that the Justices don't understand the
definition.
> and is using that as a hazard of all of American society.
That's three. The Court does not take potential hazard into
consideration. Some aspects of living in a free society may be dangerous
to your health. Too bad.
> The Roberts
> lead USA Supreme Court can rule on anything that comes before it, rule
> any way
> they wish to rule, and cite as a reason "freedom of speech".
That's four. The Court doesn't hand down First Amendment rulings on
matters that don't have First Amendment issues, and you can't name one
time that it has.
> Roberts should have first off, when receiving this case, should have
> "defined"
> what is Freedom of Speech. None of those who voted in favor of this
> case did that.
> They were severely negligent of the Constitution itself.
That's five. The Court is perfectly cognizant of the case law. I know
that because I actually perused the opinion you object to. You haven't,
have you?
> At least the Founding Fathers gave us clues as to what freedom of
> speech means.
I'm guessing that's six. I doubt you have the slightest idea as to what
the Founding Fathers meant by "freedom of speech." Look up the Alien and
Sedition Acts.
> Because they also cite freedom of religion and freedom or right to
> bear arms.
Thanks for the nonsequitur.
> (1) Freedom of religion-- to belief in a faith or religion of whatever
> without being attacked
> by others who have a different belief
That's seven. Freedom of religion is far broader than this. Look up the
Lemon test.
> (2) Freedom or right to bear arms-- the right to own a gun
That's eight. As of right now, there's no such right except in DC,
although this is likely to change soon.
> (3) Freedom of Speech -- the right to say or write whatever is on ones
> mind without be attacked by others for saying or writing such.
That's nine. The First Amendment does not apply to individuals; only to
the gov.
> The Roberts court should have at least done what I have done in 1
> through 3. Sociology
> includes law and courts and constitutions but sociology does not have
> the precision
> of definitions that mathematics or logic has. That is why it is
> important that judges
> have math and logic in their college training and that why Supreme
> Court judges are unfit
> if they had no logic courses.
If you don't mind, I won't take the opinion of an ignoramus on the
qualifications of Supreme Court Justices.
> There are limits and restrictions on 1-3 because we cannot have school
> kids bearing guns
> in schools, nor owners pointing rifles at other people, nor can we
> tolerate libel and slander
That's ten. We do tolerate libel and slander, at least in the sense that
we don't criminalize it, and we make it nearly impossible for public
figures to successfully sue for defamation.
> for (3) nor shout "fire" in a crowded theater, nor tolerate those that
> repeatedly shout lies
> and profanity as "disturbing the peace". So all three have limits and
> restrictions, but all
> three have some vital core definition.
>
> And it is apparent that the "spending of money" is not a act of
> freedom of speech.
No one says it is; certainly not the Supreme Court. But the act of
prohibiting corporations from sponsoring speech is an abridgement of
speech. And right now, it's illegal.
> Nor is it
> an act of "freedom of religion" nor is the spending of money an act of
> "right to bear arms".
Thanks for the nonsequitur.
> So the USA Robert led Supreme Court has committed a huge error and is
> leading the USA Supreme Court vastly astray.
I actually agree, but at least my opinion isn't based on ignorance and
foolishness.
> By their actions one can see that they were premeditated into taking
> this case into the
> Twilight Zone, because they knew that freedom of speech does not
> equate in any way shape
> or form to spending of money. And that leads to the suspicion that the
> Roberts court premeditated on getting a law that corporations can
> spend as much as they want to help the Republicans in future elections.
So what?
> In fact, if a case comes before the Roberts court dealing with
> abortion. Then the Roberts
> Court must rule that a women's right of choice is her freedom of
> speech right and that abortion can never be outlawed.
That's eleven. Only an ignoramus would believe this. Abortion is not
protected by the First Amendment, and Citizens United does not change
that.
> I do not know? Does anyone know, if Roberts, Alito, Scalia, ever had
> any math or logic course
> in their college or high school training? At the University of
> Cincinnati in the 1970s, all becoming lawyers were required to take
> Symbolic Logic in hopes that such a course would
> help them think clearly and correctly. I doubt that Roberts, Alito, or
> Scalia ever had a course
> in logic, judging from this horrendous decision. And I suspect that in
> all of the history of the
> Supreme Court that this decision is one of the worst to ever be set.
> For it is so unreasonable and it mangles the very Constitution itself.
That's twelve. Logic courses are not a Constitutional requirement for
Justices. And Citizens United pales in comparison to Dred Scott or
Plessy.
> If anything in the Constitution that the Roberts court should have
> looked at was the Freedom to Vote in an Election.
Wow! You're actually close on this one. I'm amazed.
> Of course the Constitution is cheqchered on this freedom since it
> lacks the freedom for women until later and it gave slaves some
> fractional vote.
That's thirteen. The Constitution never gave slaves a fraction of a
vote. Slaves were always property and had next to no rights at all.
They counted as 3/5 of a person for purposes of the census. But that
helped their masters.
> But nonetheless the Constitution outlines a freedom of voting:
>
> Freedom to Vote in Elections: is a participation to choose a
> representative to form a government.
>
> So the freedom of speech is different from freedom of religion and
> different from freedom to vote. And all of these three freedoms are
> different from the act of spending money.
Thanks for the nonsequitur.
> So how in the world that Roberts, Alito, and Scalia, messed all that
> up, is very
> alarming and dangerous to all of American society. Are we to expect
> that Roberts
> court can judge every case it wants and cite as "freedom of speech" as
> their excuse?
Only an ignoramus like you would expect it.
> Archimedes Plutonium
<snip/>
Archimedes Plutonium
Deadrat wrote:
>
> > The facts are, the USA Supreme Court of Roberts lead, has never
> > defined "freedom of speech"
>
> That's two. The Court in its history has defined "freedom of speech"
> over the years, and that definition has changed over the years. That
> *you* don't understand the case law or that you don't agree with the
> current Court doesn't mean that the Justices don't understand the
> definition.
>
That you do not state the definition used by the Roberts/Alito/Scalia
court
shows you to be either a liar or a ignoramus.
I showed my definition, so which is it for you: a liar or fool?
--- quoting from Wikipedia ---
http://en.wikipedia.org/wiki/Citizens_United_v._Federal_Election_Commission
Scalia stated that Stevens dissent was “in splendid isolation from the
text of the First Amendment. It never shows why “the freedom of
speech” that was the right of Englishmen did not include the freedom
to speak in association with other individuals, including association
in the corporate form.” He further considered the dissent’s
exploration of the Framers’ views about the “role of corporations in
society” to be misleading, and even if valid, irrelevant to the text.
Scalia’s principally argued that the first amendment was written in
”terms of speech, not speakers" and that "Its text offers no foothold
for excluding any category of speaker“.
--- end quoting from Wikipedia ---
The trouble with Scalia, and which I have heard others say he is
"pigheaded" is that Scalia
is not fit to be a USA Supreme Court Justice for all he can do, as the
above seems to show is
critize others and tear down others, yet Scalia could never construct
a definition of
"freedom of speech" and how it relates to spending of money, which
this case is begging
for. Anyone can tear things down, but the onus on Scalia was to
provide a definition of
freedom of speech that the founding fathers put in the Constitution
and for which the link
to spending of money does or does not exist.
Pigheaded, because if one has their mind made up and then uses the
court to fish for excuses as to why to vote in favor is a disservice
to the court and the country.
Scalia could never construct a definition of "freedom of speech" as
given by the USA
Constitution. Of course, Scalia and Alito and Roberts can fly off the
handle in criticism of
others discussing "freedom of speech" but that none of these justices
was able to
define freedom of speech in this entire case. And, then, most
importantly show and
convince that "spending of money" is anyway related to "freedom of
speech".
Because all in all, this case is just a Republican party windfall, and
the notion of freedom
of speech just a oblique excuse.
If someone was judging the supreme court justices in the case of
Citizen's United, and that
judge would have Logic and ruled by logic, would simply ask Scalia et
al these questions:
Logic Court: "Justice Scalia, and Roberts and Alito, where is your
definition of Freedom
of Speech as per the USA Constitution. That which, you went into these
deliberations?"
Scalia: "Hack, hack, hack (trying to clear his throat), your honor,
have you got a problem
with that?"
Logic Court: "Pardon me, problem? Just answer the questions. We are
asking you what your definition and understanding of freedom of speech
is."
Scalia: "You have a problem with freedom of speech?"
Logic Court: "This case revolves and pivots on the spending of money
in campaign elections.
The US has moved strongly, in the past decades in the direction of
curbing money contributions because it is becoming increasingly aware
to all citizens that their
political system is not representative of the people but of those who
bought the politicians
for favors, and yet you
opened the floodgates of money spending in political campaigns. You
have based your
decision of Citizens United on "freedom of speech". So what is your
definition of "freedom of speech"?"
Scalia: "Your honor, get over it."
Logic Court: "Justice Alito what is your definition of freedom of
speech, that you used
in Citizens United?"
Alito: "(Going through his famous Alito eye-rolling). I did not know I
needed a definition
of "freedom of speech.""
Logic Court: "Chief Justice Roberts, can your court even decide on a
case of freedom of
speech when none of you have a definition of "freedom of speech" and
how that relates
to the act of "spending money"? Apparently, you and Alito and Scalia,
see freedom of
speech as one and the same as spending of money. And here you preside
over a case
involving "speech" and "money spending" and none of you can tell us
the difference between
the two."
Roberts: "Your honor we do not do "judicial activism" of legislating
from the bench, for I
game my word of honor to the Congress, before being appointed."
Logic Court: "Roberts, your court is in shambles. You lack the logic
and reason to even
hear cases. You and Scalia and Alito can not even define first terms
of "freedom of
speech". Your court is a danger to this country, not a asset. Your
court wants to set the
clock backwards on corruption in campaign finances and the loss of
accountability
of politics to the people instead of money lobby interests. Your
Citizens United
case is "judicial activism on steroids.""
Logic Court Decision: "This logic court has decided that the USA
Supreme Court lacks
the logic and science background for its members to even define first
terms and first
principles such as "freedom of speech". This lack of ability is
probably due to the fact
that so few of you justices ever had a course in logic or math in
college, but shyed
away from subjects that would make you think clear and straight.
Therefore, it is the opinion of the Logic Court that so many cases in
the future involve
science and technology for which you guys are ill equiped to face and
handle. And therefore
it is best for the country and nation of the USA to append the
American Academy of Sciences
with a 2 vote count in all cases in which the AAS deems they should
vote on a decision, and
allow them to retrograde vote if they choice to.
In other words, the USA Supreme Court is not knowledgeable or
expertise enough as stands
and needs an additional appendage such as the +2 vote of a body such
as the
American Academy of Sciences. They would have entered this Citizens
United case and seen
the lack of logic of defining what freedom of speech actually is and
that it is unrelated to
spending of money and so the AAS with their +2 vote would have
defeated the case.
The USA faces future tough problems of global warming, species
extinction, energy,
nuclear weapons, that which the supreme court of present has little to
no expertise about,
and thus a appendage of the AAS with +2 votes or maybe even +3 votes
is needed
in future court cases."
Archimedes Plutonium
Deadrat wrote:
>
> > The facts are, the USA Supreme Court of Roberts lead, has never
> > defined "freedom of speech"
>
> That's two. The Court in its history has defined "freedom of speech"
> over the years, and that definition has changed over the years. That
> *you* don't understand the case law or that you don't agree with the
> current Court doesn't mean that the Justices don't understand the
> definition.
>
That you do not state the definition used by the Roberts/Alito/Scalia
of "freedom of
speech". Your court is a danger to this country, not a asset. Your
court wants to set the
clock backwards on corruption in campaign finances and the loss of
accountability
of politics to the people instead of money lobby interests. Your
Citizens United
case is "judicial activism on steroids.""
Logic Court Decision: "This logic court has decided that the USA
Supreme Court lacks
the logic and science background for its members to even define first
terms and first
principles such as "freedom of speech". This lack of ability is
probably due to the fact
that so few of you justices ever had a course in logic or math in
college, but shyed
away from subjects that would make you think clear and straight.
Therefore, it is the opinion of the Logic Court that so many cases in
the future involve
science and technology for which you guys are ill equiped to face and
handle. And therefore
it is best for the country and nation of the USA to append the
American Academy of Sciences
with a 2 vote count in all cases in which the AAS deems they should
vote on a decision, and
allow them to retrograde vote if they choice to.
In other words, the USA Supreme Court is not knowledgeable or
expertise enough as stands
and needs an additional appendage such as the +2 vote of a body such
as the
American Academy of Sciences. They would have entered this Citizens
United case and seen
the lack of logic of defining what freedom of speech actually is and
that it is unrelated to
spending of money and so the AAS with their +2 vote would have
defeated the case.
The USA faces future tough problems of global warming, species
extinction, energy,
nuclear weapons, that which the supreme court of present has little to
no expertise about,
and thus a appendage of the AAS with +2 votes or maybe even +3 votes
is needed
in future court cases."
alie...@gmail.com
You know, real objects tend to behave more like spherical-ish sponge
balls that like to stick together (think Velcro patches on the balls'
surfaces representing bond geometry) and can be squished to fill in
any voids between them; their packing fraction can always approach
unity, but there's a compression force penalty depending on how
they're arranged.
> In fact, do you assume that these are "solid" objects, or are these
> totally random sets in R^2 and R^3?
Must the exponent be an integer?
> How do you define the sample space of and the probability distribution
> on these "objects"?
Natural sponges are limited-range fractals...
Mark L. Fergerson
Ostap S. B. M. Bender Jr.
What's real to you, may be surreal to me. But if you define your
object space and the probability measure on them, I am willing to play
by your rules.
> > In fact, do you assume that these are "solid" objects, or are these
> > totally random sets in R^2 and R^3?
>
> Must the exponent be an integer?
>
Should all answers be in the form of a question?
> > How do you define the sample space of and the probability distribution
> > on these "objects"?
>
> Natural sponges are limited-range fractals...
>
"Natural sponges are limited-range fractals" is not a responsive
answer to the question "How do you define the sample space of and the
probability distribution on these objects?"
But I am willing to play along. So what's the bottom line: do "natural
sponges" satisfy Archimedes Plutonium's "Equilateral Triangle Tiler
Conjecture": "In 2D, given a large number of 2D objects to tile by
unit tilers, the unit tiler that achieves the maximum tiling is the
unit equilateral triangle"?
Deadrat
>
>
> Deadrat wrote:
>>
>> > The facts are, the USA Supreme Court of Roberts lead, has never
>> > defined "freedom of speech"
>>
>> That's two. The Court in its history has defined "freedom of speech"
>> over the years, and that definition has changed over the years. That
>> *you* don't understand the case law or that you don't agree with the
>> current Court doesn't mean that the Justices don't understand the
>> definition.
>>
>
> That you do not state the definition used by the Roberts/Alito/Scalia
> court shows you to be either a liar or a ignoramus.
Oh, look! An ignoramus calls me a liar.
>
> I showed my definition, so which is it for you: a liar or fool?
Your definition is meaningless in a legal context. The boundaries of
freedom of speech are defined by precedent.
>
> --- quoting from Wikipedia ---
<snipped: who cares?/>
> --- end quoting from Wikipedia ---
>
<snipped: attack on Scalia/>
> yet Scalia could never construct a definition of
> "freedom of speech" and how it relates to spending of money, which
> this case is begging for.
Sez you? Bwahahahahahaha!
> Anyone can tear things down, but the onus on Scalia was to
> provide a definition of freedom of speech that the founding fathers
> put in the Constitution and for which the link
> to spending of money does or does not exist.
Scalia carries no such burden to educate ignoramuses like you. He
discusses the relevant precedent in his opinion. It's not his fault if
you can't understand it.
> Pigheaded, because if one has their mind made up and then uses the
> court to fish for excuses as to why to vote in favor is a disservice
> to the court and the country.
This is a different quesiton. You're welcome to your opinion, about
which nobody much cares.
<snipped: more ignorant whining/>
> Because all in all, this case is just a Republican party windfall, and
> the notion of freedom of speech just a oblique excuse.
Could be. So what?
<snipped: fantasies/>
> Archimedes Plutonium
<snip/>
Archimedes Plutonium
To be expected, failure to produce a definition of "freedom of speech"
> > Anyone can tear things down, but the onus on Scalia was to
> > provide a definition of freedom of speech that the founding fathers
> > put in the Constitution and for which the link
> > to spending of money does or does not exist.
>
> Scalia carries no such burden to educate ignoramuses like you. He
> discusses the relevant precedent in his opinion. It's not his fault if
> you can't understand it.
>
Scalia, just like HH can tear things down but never build anything.
They
talk about freedom of speech, but never are able to give a definition
of
freedom of speech. When
asked to furnish his working definition of "freedom of speech" to show
that
it is no way connected to "spending of money", both Scalia and HH are
failures.
Here is my definition, again, of freedom of speech:
(3) Freedom of Speech -- the right to say or write whatever is on
ones
mind without being attacked by others for saying or writing such.
The opposite of freedom of speech is suppression or repression.
Obviously
money spending is no part of "freedom of speech" as cited in the USA
Constitution. Everyone in corporations in the USA or the world for
that matter
had the same "freedom of speech" in the last elections that everyone
else had.
And there is a reason that "freedoms" are multiple types such as:
freedom of religion
freedom to bear arms
freedom to vote in elections
freedom to procreate
freedom to move
freedom to quit a job
freedom to go to school
freedom of artistic expression
To name but a few freedoms. And the reason all freedoms are not one
and the same
and not all are "freedom of speech" is because they are all different.
The freedom to
bear arms is quite different from freedom of speech.
So the trouble with the USA Supreme Court is that it has become a
extension of the
Republican Party, and in the past, it could have been a extension of
the Democratic
Party but I was not around or interested in following the court in the
past.
But when the Supreme Court judged Bush rather than Gore as president
in 2000, is
clear sign that the Supreme Courts primary job is to further and
enhance the Republican Party.
When the Republican Party faltered badly in the 2006 and 2008
election, what happened
was the Republican Party got together which included Scalia, Roberts,
Alito and what was
considered is that the Democratic Party outshined in money in the
presidential election of
2008 and so the Republican Party needed and wanted more money to elect
more Republicans
in 2010. So what was hatched or brewed in 2009 was to get rid of the
limitations of
corporation on campaign spending. This meant the McCain Feingold
limits had to be
made unconstitutional. So that is where the Supreme Court steps into
the arena to deliver
a "back favor" to the Republican Party. So a campaign case was looked
for and tracked down
and it was the Citizens United.
Now in the case of letting Bush win over Gore, the Supreme Court could
easily squeek by that
decision because most people felt that we may have violence in the
streets unless some quick resolution accrued to the 2000 election.
That we were verging onto a banana republic
type of democracy if we could not decide who won the election and that
violence would break out. So the Supreme Court was looked upon as
rather a helper in that case. But still, it was
a Republican Party kickback favor.
And now, this unlimited campaign spending by corporations is another
Republican Party kickback favor. And the hiding of it under the guise
or disguise or ruse, is the hiding of it
under "freedom of speech".
So when the Democrats won the 2006 and 2008 elections, this Supreme
Court was mustered
to deliver a stricking down of the McCain Feingold legislation that
curbs corporation spending.
The Court could not strick it down in plain view without any reasons,
so they ratted around and
finally came up with the idea that they can sneek and hide it under a
"freedom of speech" sort of nonsense. Saying that corporate spending
of money is a form of freedom of speech.
Now that would be agreeable to unintelligent people like HH, who
probably is a Republican in the first place and whatever is good for
Republicans, who cares how it was achieved.
So what was done with Citizens United was a Supreme Court favor
kickback to the Republican
Party.
PROOF: noone on the USA Supreme Court of Alito, Roberts, Scalia, have
a definition or
working definition of "freedom of speech" during the Citizens United
deliberation. Since they
had no working definition of "freedom of speech" means they had no
link to the act of
"spending money" in campaigns. That meant they had prejudged Citizens
United and were
only delivering a favor to the Republican Party.
This is why I say, that to correct and improve the USA Supreme Court
is to append a independent and objective body of the American Academy
of Sciences and give them a
+2 or +3 vote in cases in which they have a stake in. In the case of
Citizens United, the
AAS would see that the justices of Alito, Scalia, Roberts failed in
the basic logic skills
of defining freedom of speech and showing that it has nothing to do
with money spending
and the AAS would have overturned the "wily bards".
Archimedes Plutonium
To be expected, failure to produce a definition of "freedom of speech"
> > Anyone can tear things down, but the onus on Scalia was to
> > provide a definition of freedom of speech that the founding fathers
> > put in the Constitution and for which the link
> > to spending of money does or does not exist.
>
> Scalia carries no such burden to educate ignoramuses like you. He
> discusses the relevant precedent in his opinion. It's not his fault if
> you can't understand it.
>
Scalia, just like HH can tear things down but never build anything.
They
talk about freedom of speech, but never are able to give a definition
of
freedom of speech. When
asked to furnish his working definition of "freedom of speech" to show
that
it is no way connected to "spending of money", both Scalia and HH are
failures.
Here is my definition, again, of freedom of speech:
(3) Freedom of Speech -- the right to say or write whatever is on
ones
Archimedes Plutonium
alie...@gmail.com
I was considering a "middle way" between spheres and polyhedrons,
and the fact that real objects seldom exhibit "pure" characteristics
of ideal objects.
Tiling the plane (or higher spaces) as a mathematical exercise is
somewhat boring to me. Since I'm reading this thread from sci.physics
I'd rather consider real objects than abstractions.
> > > In fact, do you assume that these are "solid" objects, or are these
> > > totally random sets in R^2 and R^3?
>
> > Must the exponent be an integer?
>
> Should all answers be in the form of a question?
Is there a reason they shouldn't?
> > > How do you define the sample space of and the probability distribution
> > > on these "objects"?
>
> > Natural sponges are limited-range fractals...
>
> "Natural sponges are limited-range fractals" is not a responsive
> answer to the question "How do you define the sample space of and the
> probability distribution on these objects?"
A sponge "occupies" varying amounts of "space" depending on the
granularity of your sampling technique. Their final distribution need
not resemble their original distribution, given the intervening
compression. This rather increases the sample space over the use of
typical mathematically abstract objects like rigid polytopes. How the
sponge spheres rearrange themselves may not always be predictable.
> But I am willing to play along. So what's the bottom line: do "natural
> sponges" satisfy Archimedes Plutonium's "Equilateral Triangle Tiler
> Conjecture": "In 2D, given a large number of 2D objects to tile by
> unit tilers, the unit tiler that achieves the maximum tiling is the
> unit equilateral triangle"?
AFAICT his problem is that equilateral triangle don't necessarily
maximally tile certain geometric shapes, not the infinite plane.
Two-dimensional circular fractal analogs of natural sponges (or
appropriate slices through natural sponges) should, no matter how
they're initially arranged given enough compression to fill in the
voids between them.
Compression may also be unnecessary. Do you consider fractals as
valid objects with which to attempt to tile the plane? Remember, they
may interpenetrate; also the space a fractal *doesn't* occupy is also
a fractal, hence the space between several touching fractals is
another fractal. Consider it an unusual type of Penrose tiling.
Mark L. Fergerson
Transfer Principle
> > revised, yet you fail to comprehend
> You are a very rude and clueless troll. Goodbye and good luck in
> finding anybody wiling to have a conversation with you.
And here we go again. Someone makes an interesting conjecture,
and here comes Bender to call that poster a "troll."
And here I come, to make AP's conjecture more rigorous in order
to discuss whether it's true or false, or possibly equivalent
to a known theorem, as was the case with JSH. (Of course, at
least AP, unlike JSH, had enough sense to call his claim a
"conjecture" and not an "axiom.") So here goes.
Let's start with AP's words themselves and proceed from there:
> > Equilateral Triangle Tiler Conjecture: In 2D, given a large number of
> > 2D objects to tile by unit tilers, the unit tiler that achieves the
> > maximum tiling is the unit equilateral triangle.
We begin by defining "unit tilers." Mathematicians often use
the phrase "unit square" to denote a square with side-length
(and hence area) 1, but the phrase "unit circle" usually means
a circle of _radius_ 1 (and hence an area of pi). To me, the
best thing to do would be to define a "unit tiler" as having
an _area_ of unity. Thus the "unit equilateral triangle" would
be an equilateral triangle of area 1. (Its side length works
out to be 2/sqrt(sqrt(3)).)
How general a shape can our "unit tilers" be? Obviously, we
want them to have area 1, so they must be Lebesgue measurable
at the very least, with finite nonzero area. We usually think
of "tiles" as being bounded by simple closed curves. I think
that it's best to require the regions to be convex. I don't
find it necessary to require the boundaries to be _regular_
polygons, or even _polygons_ at all. So the "unit" circle
(which here means a circle of _area_ 1, meaning that its
radius would be 1/sqrt(pi)) would be acceptable here.
So we define a "unit tiler" to be a simple closed curve,
along with its interior region, where said region is convex
and has an area of unity. Actually, we want it to be the
equivalence class of all such regions that are congruent, so
that we can use many of them to "tile" a larger region.
Now a "tiling" of a region by a "unit tiler" denotes a finite
subset of this equivalence class, such that no two regions in
this class intersect except at their boundaries, and whose
union is a subset of the region to be tiled.
Finally, we need to figure out what AP means by maximize. Now
AP states that the regions that we're trying to tile are
squares, circles, and hyperbolic triangles. So, perhaps we
can take the square of side-length s (i.e., we take the set
[0,s]x[0,s]) and a unit tiler T. (Recall that T is a set all
of whose elements are congruent and have area 1.) Then we
define f_T(s) to be the cardinality of the largest set x such
that x is a T-tiling of the square of side-length s.
We notice that the theoretical maximum of f_T(s) is obviously
floor(s^2), since we can't ever expect to tile a square of
area s^2 with more than s^2 tiles of area 1, no matter what
shape T has. We want to know which shape T takes us closest to
that theoretical maximum, for the most values of s. We might
take the set of all s less than some real number M (and AP
takes this M to be 10^500-10^-500) such that there is a
T-tiling of cardinality floor(s^2), and find the Lebesgue
measure of such set (provided that it exists), in order to see
which shape T maximizes said measure.
At this point, of course, it doesn't seem obvious that the
equilateral triangle should maximize. For if s is an integer,
then obviously there is a tiling of unit squares whose
cardinality is s^2 exactly. But the set of integers has
measure zero, and so this tells us nothing about what happens
for all of the uncountably many non-integral values of s. It
could be the case that for many non-integral values of s, we
can do better than we can with unit squares.
I did a Google search on this subject, but I wasn't able to
find any information on AP's conjecture -- though I'm probably
not looking in the right place. Even if what I'm writing here
isn't what AP has in mind, it's certainly more constructive
than asking AP for more information and then calling him a
"troll" once he provides that information, as Bender has done.
Mark Ferguson, at least, made more of an attempt to solve the
problem than Bender has done.
Transfer Principle
> On Mar 15, 4:15 pm, "Ostap S. B. M. Bender Jr."
> problem than Bender has done.
Corrected spelling -- he gives his name as Mark L. Fergerson. I
apologize for the misspelling.
Deadrat
<snip/>
>> > yet Scalia could never construct a definition of
>> > "freedom of speech" and how it relates to spending of money, which
>> > this case is begging for.
>>
>> Sez you? Bwahahahahahaha!
>>
>
> To be expected, failure to produce a definition of "freedom of speech"
Asking for a "definition" in this case is like asking why no one in a
chess match ever shouts, "Bingo!" There is no such definition that can
capture the body of case law about freedom of expression.
>> > Anyone can tear things down, but the onus on Scalia was to
>> > provide a definition of freedom of speech that the founding fathers
>> > put in the Constitution and for which the link
>> > to spending of money does or does not exist.
>>
>> Scalia carries no such burden to educate ignoramuses like you. He
>> discusses the relevant precedent in his opinion. It's not his fault
>> if you can't understand it.
<snip/>
> Here is my definition, again, of freedom of speech:
> (3) Freedom of Speech -- the right to say or write whatever is on
> ones mind without being attacked by others for saying or writing such.
I'm sorry that can't recognize what an epic fail this is. The First
Amendment does not give you the right to say whatever is on your mind.
For instance, if you're a grand juror, and grand jury testimony is on
your mind, you still may not legally communicate it. The First Amendment
does not protect you from attacks by others. It only operates to
constrain the state.
>
<snip/>
> Obviously
> money spending is no part of "freedom of speech" as cited in the USA
> Constitution.
The only thing obvious here is your complete ignorance. The First
Amendment says that the state may not "abridge" freedom of speech. And
one way to do that is to prohibit someone from spending his money on the
dissemination of his ideas. Suppose the state forbade a publisher from
paying an author? That's spending money. The author may still write and
pass out his book for free. Do you think such a ban on spending abridges
the author's freedeom of speech?
<snip/>
> But when the Supreme Court judged Bush rather than Gore as president
> in 2000, is
> clear sign that the Supreme Courts primary job is to further and
> enhance the Republican Party.
> When the Republican Party faltered badly in the 2006 and 2008
> election, what happened
> was the Republican Party got together which included Scalia, Roberts,
> Alito and what was
> considered is that the Democratic Party outshined in money in the
> presidential election of
> 2008 and so the Republican Party needed and wanted more money to elect
> more Republicans
> in 2010. So what was hatched or brewed in 2009 was to get rid of the
> limitations of
> corporation on campaign spending. This meant the McCain Feingold
> limits had to be
> made unconstitutional. So that is where the Supreme Court steps into
> the arena to deliver
> a "back favor" to the Republican Party. So a campaign case was looked
> for and tracked down
> and it was the Citizens United.
All of this may be true, but that does not make Citizens United v FEC an
irrational decision. It may be wrongly decided or if may be bad public
policy, but it's not absurd. It comes down on the side of the
proposition that the cure for bad speech is more and better speech.
> Now in the case of letting Bush win over Gore, the Supreme Court could
> easily squeek by that
> decision because most people felt that we may have violence in the
> streets unless some quick resolution accrued to the 2000 election.
> That we were verging onto a banana republic
> type of democracy if we could not decide who won the election and that
> violence would break out. So the Supreme Court was looked upon as
> rather a helper in that case. But still, it was
> a Republican Party kickback favor.
This is utter nonsense. No one was afraid of riots in the street, and
Constitutional provisions would have kicked in to ensure a smooth
transition in 2001.
>
<snipped: more rants about Republicans/>
>
<snipped: more nonsense about some mythical definition/>
<snipped: more nonsense about the AAS/>
>
> Archimedes Plutonium
<snip/>
Look, maybe I shouldn't be so harsh in my posts, but you really have no
idea how the Constitution works and the Supreme Court's role in the
workings.
Ignorance is curable, though.
Ostap S. B. M. Bender Jr.
Is there a reason Celine Dion shouldn't sing?
>
> > > > How do you define the sample space of and the probability distribution
> > > > on these "objects"?
>
> > > Natural sponges are limited-range fractals...
>
> > "Natural sponges are limited-range fractals" is not a responsive
> > answer to the question "How do you define the sample space of and the
> > probability distribution on these objects?"
>
> A sponge "occupies" varying amounts of "space" depending on the
> granularity of your sampling technique. Their final distribution need
> not resemble their original distribution, given the intervening
> compression. This rather increases the sample space over the use of
> typical mathematically abstract objects like rigid polytopes. How the
> sponge spheres rearrange themselves may not always be predictable.
>
I am not asking on an object's distribution in the 3-D or 2-D space.
What I am asking is what probabilities you assign to various objects
to appear in the SET OF OBJECTS:
http://en.wikipedia.org/wiki/Sample_space
Sample space
In probability theory, the sample space or universal sample space,
often denoted S, Ω, or U (for "universe"), of an experiment or random
trial is the set of all possible outcomes. For example, if the
experiment is tossing a coin, the sample space is the set {head,
tail}. For tossing a single six-sided die, the sample space is {1, 2,
3, 4, 5, 6}.
http://en.wikipedia.org/wiki/Probability_distribution
In probability theory and statistics, a probability distribution
identifies either the probability of each value of a random variable
(when the variable is discrete), or the probability of the value
falling within a particular interval (when the variable is continuous).
[1] The probability distribution describes the range of possible
values that a random variable can attain and the probability that the
value of the random variable is within any (measurable) subset of that
range.
>
> > But I am willing to play along. So what's the bottom line: do "natural
> > sponges" satisfy Archimedes Plutonium's "Equilateral Triangle Tiler
> > Conjecture": "In 2D, given a large number of 2D objects to tile by
> > unit tilers, the unit tiler that achieves the maximum tiling is the
> > unit equilateral triangle"?
>
> AFAICT his problem is that equilateral triangle don't necessarily
> maximally tile certain geometric shapes, not the infinite plane.
>
And that was exactly my point: the equilateral triangle doesn't
necessarily maximally tile certain geometric shapes. I wroteL
> > > > > > Wouldn't the answer depend on the nature of these "2D objects"?
So, what is YOUR point?
> Two-dimensional circular fractal analogs of natural sponges (or
> appropriate slices through natural sponges) should, no matter how
> they're initially arranged given enough compression to fill in the
> voids between them.
>
> Compression may also be unnecessary. Do you consider fractals as
> valid objects with which to attempt to tile the plane? Remember, they
> may interpenetrate; also the space a fractal *doesn't* occupy is also
> a fractal, hence the space between several touching fractals is
> another fractal. Consider it an unusual type of Penrose tiling.
>
How does this relate to my discussion with Archimedes Plutonium?
Ostap S. B. M. Bender Jr.
>
>>>> Wouldn't the answer depend on the nature of these "2D objects"?
>
>>>> on these "objects"?
>
> > > revised, yet you fail to comprehend
>
> > You are a very rude and clueless troll. Goodbye and good luck in
> > finding anybody wiling to have a conversation with you.
>
> And here we go again. Someone makes an interesting conjecture,
> and here comes Bender to call that poster a "troll."
>
I didn't call him a "troll" for making his conjecture. I called him a
"troll" for refusing to answer my question and blaming me for not
"comprehending" what he was saying.
> And here I come, to make AP's conjecture more rigorous in order
> to discuss whether it's true or false, or possibly equivalent
> to a known theorem, as was the case with JSH.
And you devoting your life to interpreting the works of these giant
and profound thinkers will surely make you immortal in the memory of
the grateful Humankind.
I thought that AP had a precise formulation of his problem in mind and
would give it to me. That's why I asked him for the definition of what
objects he planned to tile and with what relative probability. But
what you seem to say here is that AP has never bothered to define his
conjecture precisely but instead was blowing hot air. And it took you
to come and to give some meaning to his ramblings. And I was wrong in
calling him a "troll". Right?
> Mark Ferguson, at least, made more of an attempt to solve the
> problem than Bender has done.
I made a perfectly valid attempt to solve the problem. I did the first
thing that I always do when I hear a new problem/conjecture from
another person: I asked for a precise formulation.
Archimedes Plutonium
Transfer Principle wrote:
(snipped)
>
> And here I come, to make AP's conjecture more rigorous in order
> to discuss whether it's true or false, or possibly equivalent
> to a known theorem, as was the case with JSH. (Of course, at
> least AP, unlike JSH, had enough sense to call his claim a
> "conjecture" and not an "axiom.") So here goes.
>
> Let's start with AP's words themselves and proceed from there:
Thanks, I appreciate any thoughtful look at my work.
>
> > > Equilateral Triangle Tiler Conjecture: In 2D, given a large number of
> > > 2D objects to tile by unit tilers, the unit tiler that achieves the
> > > maximum tiling is the unit equilateral triangle.
>
Lwalk, I made some modifications to the above listed Conjecture, but
the pursuit of a maximum tiler is maintained. What is different about
these
conjectures is that in the past, it was the Kepler Packing Conjecture
with
an infinite space. Here infinity is well defined as meaning 10^500 or
larger.
So unlike Kepler's conjecture, what is the densest packing of unit
spheres
in Euclidean 3D is tempered by the upper bound of 10^500 which is
considered
infinity. So that the number "9"999..99d999..9"9" where the "9"
signifies 10^500 and
10^-500 decimal place value and where d represents decimal point.
So where Kepler Packing could never be solved or proven because of
infinity
cloud. Here we have a strict boundary of this large number. In the 100-
Model
instead of the 10^500 Model, the question would be what is the maximum
number
of unit spheres that can fit inside a 99.99 cubic box? It is the 0.99
that causes
concern and leads to a surprizing answer that the hexagonal close
packing is not the
most dense packing but a oblong hexagonal close packing.
So in these conjectures, unlike Kepler Packing we have an outer
boundary and we
have to deal with a fraction that goes no smaller than 10^-500. If the
container were
to be smack exactly 10^500 with zero fractions, then it would be
solidly packed by
unit cubes and 100% density. But because infinity is 10^500 and thus
this number
"9"999..99d9999..9"9" is the maximum volume of Finiteness, that the
question of
what is the maximum tiler in finite geometry becomes wide open
question. And it makes
the conjecture of Kepler Packing a nonsensical question.
By maximize, I mean what shape of a unit tiler, fills in the densest
area of a square (in 2D) or a cube (in 3D) of 99.99 in 100-Model
(likewise
in the 10^500 Model). So that a unit sphere in 3D would fill in less
than
80% of the volume of 99.99 cube but that a unit cube would fill in
perhaps
95% of the volume but that a unit rectangle may fill in perhaps 97%.
So by maximum, I mean to what percent of the 99s number, whether
the 100-Model or the 10^500 Model is filled by the unit tiler of its
99s area
or its 99s volume.
Please keep in mind that in the 10^500 Model there is no number larger
than
this 99s number nor smaller than the 10^-500 9s. In the 100 Model,
the number
99.99 is the largest finite number and 0.99 is the largest finite
fraction.
It is these restrictions, of unit tiler, and of the boundary of 99.99
numbers that
forces there to be a unique maximum tiler in 2D and 3D.
>
> We notice that the theoretical maximum of f_T(s) is obviously
> floor(s^2), since we can't ever expect to tile a square of
> area s^2 with more than s^2 tiles of area 1, no matter what
> shape T has. We want to know which shape T takes us closest to
> that theoretical maximum, for the most values of s. We might
> take the set of all s less than some real number M (and AP
> takes this M to be 10^500-10^-500) such that there is a
> T-tiling of cardinality floor(s^2), and find the Lebesgue
> measure of such set (provided that it exists), in order to see
> which shape T maximizes said measure.
>
LWalk, the 10^500 and 10^-500 are not the boundary. Those are infinite
numbers. The boundary in the 100Model is 99.99 so the boundary
in the 10^500 Model is 999..99d999..99. So the unit tiler of a cube or
square
is not going to be 100% density for it has to reckon with that 0.99s
fraction.
> At this point, of course, it doesn't seem obvious that the
> equilateral triangle should maximize. For if s is an integer,
> then obviously there is a tiling of unit squares whose
> cardinality is s^2 exactly. But the set of integers has
No, the 10^500 is infinity itself, so the boundary is not at this
number
but a 99s number slightly below, so the cube unit tiler is not a 100%
density.
I am reckoning that in 2D, the equilateral triangle is the maximum
tiler. And to be
that winner, it must be the maximum in tiling a Euclidean square of
999..99d999..99
area and a circle of 999..99d999..99 area and a hyperbolic triangle of
999..99d999..99
area. If the equilateral triangle is able to be the maximum tiler of
at least two of those
three in 2D, then the equilateral triangle is the maximum tiler in 2D
geometry.
For 3D geometry, I suspect the rectangular wedge is the maximum tiler
since it will
be the maximum tiler of a cube of 999..99d999.99 volume and a sphere
of
999..99d999..99 volume and a pseudosphere of 9999..99d999..99 volume.
So if the
unit rectangular wedge wins two of those three contests in density of
tiling then
the maximum tiler in 3D geometry is the rectangular wedge.
LWalk, I have a sneeky suspicion that the rectangular wedge shape in
2D may win
over the equilateral triangle. So I am a little nervous about whether
the equilateral
triangle can win over the long and skinny unit rectangle wedge. In 2D
this wedge is
a right triangle. And it probably relates to the picket fence in
integration in Calculus.
The unit tilers require that all tilers be of unit area or unit volume
to give all shapes a
fair play in this game. But the shape that is most competitive is a
shape when two
are fit together that no gaps or holes exist. So that eliminates
circles or spheres as
acceptable tilers. And only tilers that have a straight edge can
eliminate holes and
gaps and come closest to a 100% density.
So is it the long and skinny rectangular wedge that comes nearest to
100% density
or is it the short and fat cube wedge or the equilateral triangle in
2D??
Sometimes it is harder to make a precision Conjecture statement than
it is to prove the Conjecture statement once it is precision cast.
Archimedes Plutonium
Archimedes Plutonium
for the 100-
Model where 99.99 is the largest finite number and where 100 is
infinity, that the
square box of sides 99.99 can be 100% tiled by rectangles of length
99.99 and width
0.01 (where we make allowances that such is a unit area rectangle).
So the mistake here is that, yes, those rectangles of unit area 100%
tile the Euclidean
100-Model and likewise the 10^500 Model, but, the problem begins,
because in the
Conjecture, the maximum tiler has to tile not just Euclidean geometry
but at least one
of the other two geometries of Elliptic and Hyperbolic and be a winner
in one of those.
So when we transfer this rectangle of 99.99 by 0.01 to that of the
circle of diameter
99.99 we get a 0% tiling in Elliptic geometry and a 0% tiling in
Hyperbolic geometry.
So this tells us that the unit rectangle tiler has to be shorter and
wider than 99.99 by
0.01.
So to be the winning tiler, the best tiler or the maximum tiler, it
has to win in two of the
three geometries. And the restrictions that the tilers are unit area
tilers, and that the
containers in the three geometries is the 10^500 Model (or we can use
the 100-Model
since it is more convenient then writing out all those 9s).
So the above also is an arguement in the favor of the equilateral
triangle, because if you
make the rectangles too skinny and long, they cause too many gaps and
holes in the
Elliptic and Hyperbolic geometry containers and thus lose out.
Archimedes Plutonium
winner of this contest
has to win in at least two of the three geometries presented so that
in 2D the square box
container in Euclidean, and the hyperbolic triangle in Hyperbolic, and
the circle in
Elliptic. And where I misleadingly said it has to win in at least two
of those geometries
is misleading. What is better to say is that if the equilateral-
triangle unit tiler is the
maximum tiler in 2D would mean that it is able to tile perhaps 97% of
Euclidean, 96%
of Elliptic and perhaps 95% of Hyperbolic leaving it with a contest
score of
97 + 96 + 95 = 288%.
And that let us say the nearest rival to the equilateral unit triangle
is the unit right-triangle of
2 by 1 sides which tiles 97% + 95% + 94% for a total of 286%.
So the maximum tiler in 2D will have a score at the end of how well it
did in the three geometries.
And this makes it obvious that tall and skinny rectangles can tile
100% or 99% of the Euclidean but when put into the Elliptic and
Hyperbolic have a hard time of achieving any placement. So the
Elliptic and Hyperbolic containers favor the short and fat unit tilers
to fit into those curved edges.
After fiddling with this problem late last night I am confident that
the Equilateral unit
triangle is the maximum tiler in 2D
And that the rectangular unit wedge from a 2 by 1 by 1 that is split
in half at the diagonal
is the maximum tiler in 3D.
If the unit tiler is tall and skinny boosts the Euclidean container
but hinders the Elliptic and
Hyperbolic containers. So the calculus is needed to compute the
maximum for the end result
of the highest added percentage of the three geometries.
Archimedes Plutonium
whether tall and skinny
rectangular wedges in 3D or tall and skinny right triangles in 2D are
better than short and
fat unit cubes in 3D or unit equilateral triangles in 2D. The answer
is that although tall
and skinny can tile nearly 100% of the Euclidean geom of 99.99 in 100-
Model, the trouble
is that the 99.99 diameter (or we can do area for the 99.99) in
Elliptic and Hyperbolic geometry decrease the percentage of tiling
density. So the short and fat figures may not achieve a
near 100% tiling in Euclidean, that the short and fat excell when it
comes to the Elliptic and
Hyperbolic.
So to win the award as the maximum tiler in either 2D or 3D, the unit
figure has to achieve
the highest sum total percentage of density in the three geometry
containers.
So I am now confident that the unit equilateral triangle is probably
the maximum tiler in 2D
and the rectangular unit wedge for 3D.
I prefer to talk about the 100-Model rather than the 10^500 Model
because of all those
9s. I hate writing out 9999..99d999..99 and explaining the place
value. Easier to say
in the 100-Model that 99.99 is the last and largest finite number and
that 100 is infinity
and beyond. So in the 100-Model the unit equilateral triangle is about
1.5 for each of its
three sides, giving a total area of 1 square unit. And the right-
triangle of 30-60-90 for a
unit square is going to be of sides 1.1, 1.9, 2.2. So the right
triangle unit is going to be
a 1.9 subtract 1.3 or 0.6 more "pointyier".
As I said, I could fetch a right triangle that was enormously tall and
skinny and would
tile nearly 100% of the Euclidean but when you put that figure over
onto the Elliptic and
Hyperbolic, those tall and skinny become a detriment.
Now there is a good question as to why the tilers are all Euclidean
tilers, yet two of the
testings are for Elliptic and Hyperbolic models? Would it not be
warranted to ask for
elliptic triangles or hyperbolic triangles to use in Elliptic and
Hyperbolic models?
My response is that Euclidean geometry actually is Elliptic unioned
with Hyperbolic
in that the concavity of one cancels the concavity of the other. And
the reason
I use the elliptic and hyperbolic models is that the contraints of the
geometry of
these three anticipates all types of containers. So if I tap into
these three geometries
then I have anticipated all universal and all general figures as
containers. So if I was
thrown a fractal container with all its weird bends and curves, well,
that figure was
anticipated by either the Elliptic or Hyperbolic or Euclidean three
models. So the fact that
the Universality or Generality of geometry in that it has three and
only three geometries,
removes my need to consider elliptic equilateral triangles or
hyperbolic equilateral triangles
for those geometries. That if the Euclidean equilateral triangle is
the maximum tiler, there is
no need to pursue elliptic and hyperbolic equilateral triangles.
Archimedes Plutonium
Archimedes Plutonium
bare bones and the
proof to bare bones.
I suspect the conjecture is true for Equilateral Triangles in 2D.
Simplification makes the statement easy and the proof easy.
The conjecture requires unit-figures of unit area as tilers. The
Conjecture
requires a Euclidean, Elliptic and Hyperbolic representation. Now
previously
I said a square, a circle and a hyperbolic triangle as
representatives. But that is
not uniform enough. So to make it more uniform. I am applying a
Euclidean
equilateral big triangle then a Elliptic triangle from that
equilateral big triangle
and finally a Hyperbolic triangle thereof.
And I need uniformity in the dimensions so if the 999..99d999..99 is
area or side length
or diameter length, it must be uniform. So the only sure uniformity is
that the total area
of these three geometry triangles is for their total area to be
uniformily the same.
Now to write out the number 999..99d9999..99 which is 10^-500 short of
10^500 is painful.
So I usually use the 100-Model which is 99.99 as the largest finite
number rather than the
10^500 Model. Or, to simplify even more I could use the 10-Model where
9.9 is the last and
largest finite number and where 10 and beyond is infinity.
So, in the 10-Model, the equilateral unit triangle is 1.5 by 1.5 by
1.5. And the 60-30-90 unit
right-triangle is 1.1 by 1.9 by 2.2.
So the Euclidean big triangle is in the 10-Model of total area of 9.9
The Elliptic big triangle in the 10-Model has total area of 9.9
The Hyperbolic big triangle in the 10-Model has total area of 9.9
Now that means the Elliptic triangle is begot from the Euclidean
triangle but then
we shrink the Elliptic so its area if 9.9 and the Hyperbolic is
magnified so its area
is 9.9.
Now I have it set up so that I can do overhead projector type of
transparencies.
I draw upon one transparencies filling the entire page with
equilateral triangles tiled
so as to have no gaps. I draw horizontal lines for the 10-Model spaced
1.3 apart.
Then I draw lines at a 60 degree angle from the horizontal also spaced
1.3 apart.
I repeat the 60 degree angle lines in the opposite direction. What I
have remaining
on the page is a matrix of equilateral triangles all of which whose
sides are 1.5.
So the entire page is tiled in equilateral triangles.
I do another transparency page tiled in 60-30-90 unit area right-
triangles.
I could do another in various forms of unit rectangles.
Now I set aside three special transparencies. One is a Euclidean
Equilateral triangle whose
area is 9.9. A second transparency is a Elliptic triangle derived from
the Euclidean triangle
previous and then shrunk in size so the area is 9.9. Finally, the
third transparency is the
Hyperbolic triangle derived from the Euclidean and magnified so its
area is 9.9.
Now the fun begins. I overlay the three big triangles onto the unit
equilateral triangles and see
what the percentage of tiling is for the three geometries. Only "whole
unit figures" can count as density. So the beauty of this proof is
that we can twist and turn the transparencies of the
big triangles to see what the maximum number of unit figures we can
manage to enclose inside the big triangles.
If my conjecture is true, then the winner in 2D is the equilateral
unit triangle.
I wish I had a simple proof for 3D as this 2D, but 3D is more
challenging.
Archimedes Plutonium
Now I have a preliminary result, which is highly fallible, so do not
go away
with any certainty. I say fallible because I am picturing it in my
head and
a rough pen and paper sketch.
If I use the 10-Model where 9.9 is the last and largest finite number
and where
the unit equilateral triangles are thus1.5 sides would give me a 9
square unit
tiling for a 9.9 square area in Euclidean, since the Euclidean model
is the bigger
equilateral triangle, that no matter how I would try to rearrange
anything there is that
remaining gap of 0.9 area loss. So the maximum density for Euclidean
is 9/9.9 = 90%
And for the Elliptic is also 90% because the shrinkage of the concave
outward sides
absorbs the 0.9 extra of Euclidean. And for Hyperbolic, the
magnification since the
sides are concave inwards is compensated by the 0.9 of Euclidean and
so the
tally for equilateral unit triangle is 90% + 90% + 90%.
Now what is the maximum tiling for a 30-60-90 right-triangle? And to
be fair, let me
make the big triangles that of 30-60-90 with its accompanying elliptic
and hyperbolic.
From my sketches I seem to be able to fit only 6 of these right-
triangles in the big
right triangle and also for the elliptic and hyperbolic leaving me
with the percentage
of 6/9.9 = 60% and with 7 for the elliptic but 5 for the hyperbolic
with a end total
of 60% + 70% + 50%.
Now for the square unit tiler, for the 9.9 as a big square it is
obvious that 9 is
the maximum in Euclidean but only 5 unit squares for a circle of 9.9
area. And
even worse for Hyperbolic. So the formula for a square tiler looks
like this:
90% + 50% + 40%
All of the above needs checking for I am likely to have made several
mistakes.
Archimedes Plutonium
Now I am going to check my numbers above as they were done crudely and
from a sketch and from my mind. So this is often fallible, so I need a
way to
check it. It is a nasty algebraic problem of fitting triangles or
squares or rectangles
into hyperbolic triangles. I want a method that even High School
students can do this
project.
Lately I have been burning old books in the woodstove but I do save
the book cover
if in decent shape of its mylar or plastic see through. And with some
of these book covers
I am going to draw the big equilateral triangle the big square and big
circle and big
elliptic and hyperbolic counterparts.
Then I am going to tile a sheet of blank paper with the unit tilers to
test. Then I overlay
the big representative Euclid, Elliptic and Hyperbolic. I twist and
turn the clear plastic to
the point where I can achieve the maximum tiling. And thus check my
numbers data above.
Archimedes Plutonium
Archimedes Plutonium wrote:
(some snipping)
> >
> > Now I have a preliminary result, which is highly fallible, so do not
> > go away
> > with any certainty. I say fallible because I am picturing it in my
> > head and
> > a rough pen and paper sketch.
> >
> > If I use the 10-Model where 9.9 is the last and largest finite number
> > and where
> > the unit equilateral triangles are thus1.5 sides would give me a 9
> > square unit
> > tiling for a 9.9 square area in Euclidean, since the Euclidean model
> > is the bigger
> > equilateral triangle, that no matter how I would try to rearrange
> > anything there is that
> > remaining gap of 0.9 area loss. So the maximum density for Euclidean
> > is 9/9.9 = 90%
> > And for the Elliptic is also 90% because the shrinkage of the concave
> > outward sides
> > absorbs the 0.9 extra of Euclidean. And for Hyperbolic, the
> > magnification since the
> > sides are concave inwards is compensated by the 0.9 of Euclidean and
> > so the
> > tally for equilateral unit triangle is 90% + 90% + 90%.
> >
It looks like cellophane, clear cellophane but it is called mylar.
Maybe the
same plastic, only different names. And I got out the old French-
rulers,
the straight edge and compass. Now the ink does not stay well on mylar
so I put some see through tape over the mylar and the ink stays well
on
that tape.
So I performed the above and it checks out as 90% + 90% +90%
Of course, if this were the 100-Model it would be 99% + 99% + 99%
and if this were the 10^500-Model the percentages would be even
larger.
And where the difficult for the tall and skinny triangles or the tall
and skinny
rectangles comes into play is the trouble that they block space in the
elliptic
and especially hyperbolic representations.
So it appears that the conjecture is panning out to be true, that the
equilateral triangle
is the maximum unit tiler in 2D.
I will need a similar geometric modeling to prove the rectangular
wedge shape is the
maximum unit tiler in 3D.
Archimedes Plutonium
conjectures of tiling
in 2D and 3D of what is the maximum unit tiler. But let me not lose
sight of what brought
me into these fields of exploration.
When mathematics makes precision definition of finite versus infinite,
it all boils down
to the selection of the largest finite number. Physics makes that
selection as 10^500.
Thence, the number (i) takes on a numeric form. As the below Cauchy
sequence of
sqrt 9s shows, the pattern 3162.... becomes the numeric value of (i)
whether we use
the 10-Model, the 100-Model or the 10^500-Model.
sqrt9 = 3
sqrt99 = 9.94
sqrt999 = 31.60
sqrt9999 = 99.99
sqrt99999 = 316.22
sqrt999999 = 999.99
sqrt9999999 = 3162.27
This is the first time in the history of mathematics that (i) was
recognized for its
numeric value and thus is able to use in the Euler Identity e^(pi)(i)
= -1
But the old math still held onto a prejudice or a falsehood of
thinking that (i) is a
90 degree rotation. The number 3162..... pattern contests that
assumption of old
math. If all the numbers range from 9999....9999 to 0, then the
3162.... pattern is
approx 1/3 of all the numbers and so 1/3 of 180 degrees as pi is 60
degrees.
So the new math with (i) being the 3162.... pattern invokes the
meaning of (i) as
a 60 degree rotation, not a 90 degree.
So that has lead me into this 2D and 3D conjectures. For the 2D, it
appears that the
maximum unit tiler is in fact the equilateral-triangle and the
equilateral triangle has
all its angles as 60 degrees. Now this is a pretty confirmation with
the Least Action
Principle in Physics that all physical interactions end up with a path
of least action.
But the 3D conjecture also has an astounding or remarkable conclusion.
Now I have
not done the proof experiment on the 3D, but only in my mind and a few
sketches.
What I am finding out is that the unit-rectangular-wedge is the
maximum unit tiler in
3D, but there is a pretty uniqueness involved in that the unit-
rectangular-wedge has to
be a 90-60-30 right triangle and no other unit right triangle is the
maximum. It involves
the Hyperbolic and Elliptic geometry containers as counterparts to the
Euclidean container.
If it was simply maximum tiling in Euclidean geometry various other
right triangle wedges
would do, but, since, the testing has to be for the maximum in all
three geometries, the
Hyperbolic and Elliptic forces a uniqueness upon the 90-60-30 right
triangle 3D unit wedge.
In physics, when talking about the Least Action Principle, it is
always cast in the 2D as
with "path integrals", but it can be generalized into 3D. And although
the right-triangle has
just one of those angles as 60 degrees in 3D, while all the angles are
60 degrees in
2D, does not diminish the argument.
Ostap S. B. M. Bender Jr.
> whole entire Universe is just one big atom
> where dots of the electron-dot-cloud are galaxies
You are too easily astoundable.
alien8er
Gack!
> > > > > How do you define the sample space of and the probability distribution
> > > > > on these "objects"?
>
> > > > Natural sponges are limited-range fractals...
>
> > > "Natural sponges are limited-range fractals" is not a responsive
> > > answer to the question "How do you define the sample space of and the
> > > probability distribution on these objects?"
>
> > A sponge "occupies" varying amounts of "space" depending on the
> > granularity of your sampling technique. Their final distribution need
> > not resemble their original distribution, given the intervening
> > compression. This rather increases the sample space over the use of
> > typical mathematically abstract objects like rigid polytopes. How the
> > sponge spheres rearrange themselves may not always be predictable.
>
> I am not asking on an object's distribution in the 3-D or 2-D space.
I am aware of that. I'm not sure it's possible to give a direct
answer in the terms you're asking given that I'm changing some of the
assumptions.
> What I am asking is what probabilities you assign to various objects
> to appear in the SET OF OBJECTS:
>
> http://en.wikipedia.org/wiki/Sample_space
>
> Sample space
>
> In probability theory, the sample space or universal sample space,
> often denoted S, Ω, or U (for "universe"), of an experiment or random
> trial is the set of all possible outcomes. For example, if the
> experiment is tossing a coin, the sample space is the set {head,
> tail}. For tossing a single six-sided die, the sample space is {1, 2,
> 3, 4, 5, 6}.
>
> http://en.wikipedia.org/wiki/Probability_distribution
>
> In probability theory and statistics, a probability distribution
> identifies either the probability of each value of a random variable
> (when the variable is discrete), or the probability of the value
> falling within a particular interval (when the variable is continuous).
> [1] The probability distribution describes the range of possible
> values that a random variable can attain and the probability that the
> value of the random variable is within any (measurable) subset of that
> range.
My point was that the shapes of proposed tilers and their
interactions having the freedom to change *as they interact*
influences whether they can tile a plane and if so, what tiling will
be observed (the outcome of the experiment).
The fundamental assumption of variables being either discrete or
continuous can be challenged. If it is, say by using non-rigid fractal
objects, tiling any bounded form becomes less troublesome in some
senses, more so in others.
ISTM the relevant sample space for arrangements of isotropic 2D
fractal sponges depends more strongly on their resilience vs. the
applied compression than it does their initial arrangement.
Example; dump real sponge spheres randomly into a cubical container;
they will fill a finite percentage of the volume. Apply compression
along one axis, and as the balls compress they will fill more of the
available volume. Eventually, depending on details of the balls'
structure and compressibility, they will fill the whole (compressed)
volume.
Instead, carefully pack the spheres in close-packed cubic packing,
then apply compression. What is the difference in required compressive
force to achieve the same packing fraction as above?
Now, consider that some initial arrangements of the balls are
forbidden. Remember the velcro patches? There are only certain allowed
ways to pack say carbon atoms, and changing the pressure forces
changes in those packings. How does this affect the final packing
fraction? It increases dependence on the initial packing fraction.
How to say this formally? I have no idea.
> > > But I am willing to play along. So what's the bottom line: do "natural
> > > sponges" satisfy Archimedes Plutonium's "Equilateral Triangle Tiler
> > > Conjecture": "In 2D, given a large number of 2D objects to tile by
> > > unit tilers, the unit tiler that achieves the maximum tiling is the
> > > unit equilateral triangle"?
>
> > AFAICT his problem is that equilateral triangle don't necessarily
> > maximally tile certain geometric shapes, not the infinite plane.
>
> And that was exactly my point: the equilateral triangle doesn't
> necessarily maximally tile certain geometric shapes. I wroteL
>
> > > > > > > Wouldn't the answer depend on the nature of these "2D objects"?
>
> So, what is YOUR point?
I agree. I was pointing out that idealized rigid polytopes aren't
the only choice, and that in real-world examples, they are applicable
only in limited sample spaces.
> > Two-dimensional circular fractal analogs of natural sponges (or
> > appropriate slices through natural sponges) should, no matter how
> > they're initially arranged given enough compression to fill in the
> > voids between them.
>
> > Compression may also be unnecessary. Do you consider fractals as
> > valid objects with which to attempt to tile the plane? Remember, they
> > may interpenetrate; also the space a fractal *doesn't* occupy is also
> > a fractal, hence the space between several touching fractals is
> > another fractal. Consider it an unusual type of Penrose tiling.
>
> How does this relate to my discussion with Archimedes Plutonium?
*Your* discussion?
I meant to offer a little discourse on "the nature of these "2D
objects"".
Mark L. Fergerson