the historical misconception of the 4-Color Mapping Problem
a_plu...@hotmail.com
strings to tie. One of them was the 4 Color Mapping theorem.
I conceded there was a valid 4 Color Mapping theorem, and also my own 2
Color Mapping theorem.
The error spot, or the spot in which it was easy to make a error is
that 4 Color Mapping is not a geometry problem but strictly that of
Topology. This is a mistake and error not of me but of the entire rest
of the math community for never really clearing up that error. Once the
error is cleared up, then there is a simple one page proof of the 4
Color Mapping which ignores borderlines and is strictly in topology and
not geometry, as well as the 2 Color Mapping that includes borderlines
and is in geometry and not topology.
So I have the rest of the mathematics community to blame for never
really making this 4 Color Mapping clear and is the reason why so many
mathematicians in the past history have abandoned working on 4 Color
Mapping is because it was not a clear problem.
The Jordan Curve theorem, JCT, proves both the 4-Color Mapping which
ignores borderlines and resides purely in topology theory and JCT
proves the 2-Color Mapping which resides in geometry and includes
borderlines.
The 4 Color Mapping is simply a statement that when you have 4
countries that fill 1 continent and that continent is surrounded by
ocean water (5th color is blue for water) that it is impossible to have
all 4 countries own ocean beachfront and that at least one of those
countries is landlocked and thus colored blue. And the Jordan Curve
theorem easily proves this problem because the countries have to be
closed figures which raises a contradiction.
The 2 Color Mapping is not topology but geometry and the Jordan Curve
theorem applies also. Only 2 colors are necessary and sufficient to
color all maps in the plane where the borderlines are color blue and
the interiors are white. Proof: JCT.
So, yes, I conceded my mistake of thinking that 4-Color Mapping was
fakery. Because I did not realize until 2005 that it was all in
topology and not geometry. Yet the rest of the mathematics community
needs to stand up and admit their mistake in that they never really
made 4Color Mapping a clear and concise problem of mathematics by
indicating it was not a geometry problem but a topology problem and
that its cousin-- 2 Color Mapping where borderlines are not ignored is
in the realm of geometry and not topology.
So I conceded my mistake, yet the rest of the mathematics community has
not conceded their mistake of a fuzzy and unclear 4 Color Mapping. They
simply needed to state it was topology and that it is connected to 2
Color Mapping with the Jordan Curve Theorem as the engine of both
proofs.
Considering how awfully arrogant are the people of the mathematics
community, I suspect they will never concede or recognize their big and
awful mistakes about 4 Color Mapping.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
Proginoskes
a_plu...@hotmail.com wrote:
> Before I departed from sci.math a week ago, I had a lot of loose
> strings to tie. One of them was the 4 Color Mapping theorem.
>
> I conceded there was a valid 4 Color Mapping theorem, and also my own 2
> Color Mapping theorem.
>
> The error spot, or the spot in which it was easy to make a error is
> that 4 Color Mapping is not a geometry problem but strictly that of
> Topology. This is a mistake and error not of me but of the entire rest
> of the math community for never really clearing up that error. Once the
> error is cleared up, then there is a simple one page proof of the 4
> Color Mapping which ignores borderlines and is strictly in topology and
> not geometry, as well as the 2 Color Mapping that includes borderlines
> and is in geometry and not topology.
>
> So I have the rest of the mathematics community to blame for never
> really making this 4 Color Mapping clear and is the reason why so many
> mathematicians in the past history have abandoned working on 4 Color
> Mapping is because it was not a clear problem.
Actually, if you had checked a REAL graph theory book, as opposed to
"popular" math books, you would see that this distinction was made. I'm
talking about textbooks like those written by Bollobas and Deistel.
So it's not the math community's fault; it's the fault of the author
who didn't bother to include all the details in the problem.
> The Jordan Curve theorem, JCT, proves both the 4-Color Mapping which
> ignores borderlines and resides purely in topology theory and JCT
> proves the 2-Color Mapping which resides in geometry and includes
> borderlines.
>
> The 4 Color Mapping is simply a statement that when you have 4
> countries that fill 1 continent and that continent is surrounded by
> ocean water (5th color is blue for water) that it is impossible to have
> all 4 countries own ocean beachfront and that at least one of those
> countries is landlocked and thus colored blue. [...]
That's part of it, but not all of it. Just because the one possible
counterexample you think of doesn't work doesn't mean there aren't
other ones.
It's like the reasoning behind the following theorem.
THEOREM. If a map has three countries which are mutually adjacent, then
you need at least 3 colors to color the map. (Proof: You need 3 colors
to color those 3 countries alone.)
However, if you don't have three countries which are mutually adjacent,
you still may need 3 colors. Think of a pie chart cut into 5 pieces:
You need 3 colors, but you can't find 3 slices such that each is
adjacent to another (along a boundary of course).
--- Christopher Heckman
Proginoskes
If you want a _real_ historical misinterpretation of the 4 Color
Problem, here's one. It's well-documented as well.
Around the 1930s or 1940s, someone found an old result of Augustus
Moebius that proved that you cannot have five regions in the plane
which are pairwise adjacent. This person (I don't recall his name right
off) therefore concluded that Moebius had solved the 4 Color Problem.
This mistake persisted until the early 1960s, when it was set straight.
However, if you check "The World of Mathematics" by James Newman, which
came out in 1952, it still gave credit to Moebius.
I found out about this a few years ago when someone at ASU was going to
give a talk on mathematics and asked me (with a straight face) whether
the 4 Color Problem had been solved.
--- Christopher Heckman
a_plu...@hotmail.com
--- Christopher Heckman
A.P. writes:
And another fault of the math community is its unclear telling of the
history of the 4 Color Mapping that the computer proof of Appel and
Haken were the only proofs and that there was no easy and short and
quick and understandable proof.
As far as your comment about my outline of a proof of 4 Color Mapping
using the Jordan Curve theorem, where you say that mine is a
"counterexample" and the outline does not solve the sufficiency. I say
Chris Heckman is simply wrong.
The way I have the outline set up is for the general case logic. Given
any 4 countries adjacent to one another filling a continent surrounded
by a blue ocean, that by Logic, you cannot have those 4 countries own
beachfront land and thus one of those 4 countries is landlocked and
thus colored blue. It is the general case. Given any 5 countries
filling a continent, is a superfluous exercise since 4 covers all
numbers. Somehow, Chris, you are lacking the logical abilities to see
this thing clearly.
What I have outlined is a one page proof of the 4 Color Mapping using
Jordan Curve theorem, and it is not a specific case but a general case.
The 4 Color Mapping, although it is purely a topology problem is
equivalent to the Jordan Curve Theorem because the only way that 3
colors sufficient or that 5 colors are necessary would be if the Jordan
Curve theorem was not what it is. Because 4 colors are necessary and
sufficient is because a Closed curve has an inside and outside. So the
Jordan Curve theorem is able to be derived from the 4 Color Mapping
given it is a true theorem and vice versa, for the theorems are
equivalent.
But another feature of the history of mathematics that has not been
made clear and should be made clear is that 2-Color Mapping is geometry
and that it is proven by the Jordan Curve Theorem and that 4-Color
Mapping is topology and proven also by Jordan Curve Theorem. This is an
important aspect of these theorems is that JCT is the engine that
drives both of them. And the history of this problem has never
mentioned the central role that JCT is in all of this.
Not to brag, Chris, but you spent much of your life on 4-Color Mapping
and still unable to ever have digested and offered a proof of 4 Color
Mapping that any High School student could understand. And here on the
Internet in about 50 posts of some hours of my time I am able to give a
proof of the 4 Color Mapping that is a mere one page long and easily
understood by even a bright Junior High School student. Plus the fact
that I corrected your bad logic. So I think you need a vacation from
academics Chris, because you are burnt out. You see Chris, there is a
huge difference between you and me. You are a mere professor of
mathematics whereas I am a genius of science. And whereas you can only
play around in mathematics, I play around in every science and math is
merely my minor sport.
Proginoskes
Well, there are quick and understandable proofs, but they have been
shown to be wrong, and wrong proofs count for nothing.
And as far as the history of the 4 Color Problem/Theorem goes, have you
bothered to check out the 4CT's official webpage? Yes, there is one, at
http://www.math.gatech.edu/~thomas/FC/fourcolor.html
I almost wrote that short proofs of the 4CT are "invariably" wrong, but
I'm still checking out the proof via Spiral Colorings.
> As far as your comment about my outline of a proof of 4 Color Mapping
> using the Jordan Curve theorem, where you say that mine is a
> "counterexample" and the outline does not solve the sufficiency. I say
> Chris Heckman is simply wrong.
And I say that AP does not know the difference between a planar
embedding and a 5-coloring.
> The way I have the outline set up is for the general case logic. Given
> any 4 countries adjacent to one another filling a continent surrounded
> by a blue ocean, that by Logic, you cannot have those 4 countries own
> beachfront land and thus one of those 4 countries is landlocked and
> thus colored blue. It is the general case. Given any 5 countries
> filling a continent, is a superfluous exercise since 4 covers all
> numbers. Somehow, Chris, you are lacking the logical abilities to see
> this thing clearly.
And you refuse to accept that there might be more possibilities.
Using the vertex-coloring version, what you've shown is that a
particular graph K5, which consists of 5 vertices, each of which is
adjacent to each other (which are your countries + the ocean), and
which needs 5 colors, is not planar.
There are other graphs which need 5 colors to color them. In fact,
there are even graphs which do not contain triangles (that is, if you
look at any three vertices u, v, and w, at least one of the edges uv,
uw, and vw is missing), which need 5 colors.
5 vertices, connected in a ring, form a pentagon. If you apply the
Mycielsky construction to the pentagon, you get a graph called
Grotzsche's graph, which needs 4 colors to color its vertices. If you
apply Mycielsky's construction to Grotzsche's graph, you get a graph
(which I'll call M5) that needs 5 colors.
Again, I have a link for you to check out, and don't worry, it has
pictures on it: http://mathworld.wolfram.com/MycielskiGraph.html
It has a graph called K2, the pentagon, Grotzsche's graph, and the
graph M5 with no triangles which needs 5 colors.
Now if M5 was planar, then it would provide a counterexample to the
4CT, in a way that avoids your proof.
> The 4 Color Mapping, although it is purely a topology problem [...]
It is NOT purely a topology problem, since determining whether you can
color a map with 4 colors is a combinatorial problem, since you are
assigning a finite number of colors to a finite number of countries.
Now there is a theorem by Kuratowski which tells when a graph (with
vertices and edges) is planar, and this can be checked without actually
embedding the graph. So with this theorem, the problem becomes entirely
combinatorial.
> But another feature of the history of mathematics that has not been
> made clear and should be made clear is that 2-Color Mapping is geometry
> and that it is proven by the Jordan Curve Theorem
The 2-Color Mapping doesn't need the JCT for a proof. Once you draw the
boundaries of a map, you've colored the boundaries. QED.
> Not to brag, Chris, but you spent much of your life on 4-Color Mapping
3 years is not "much of my life". I've spent more time teaching than on
the 4CT. I've spent more time on music than on the 4CT.
> and still unable to ever have digested and offered a proof of 4 Color
> Mapping that any High School student could understand. And here on the
> Internet in about 50 posts of some hours of my time I am able to give a
> proof of the 4 Color Mapping that is a mere one page long and easily
> understood by even a bright Junior High School student.
Yes, but it is wrong. This "proof" was discovered by Moebius in 1840
and has been rediscovered hundreds of times by amateurs like yourself.
And it is always refuted in the same way: If a graph contains K5, then
it's not 4-colorable, but the reverse is not necessarily true, and this
is what the proof hangs on. Furthermore, if it were a valid proof, then
Appel and Haken would never have bothered with their computer proof.
This misuse of logic is very fundamental, and in another form, it would
run like this:
Every person who lives in Denver lives in Colorado. TRUE
Every person who lives in Colorado lives in Denver. FALSE
> Plus the fact that I corrected your bad logic.
I must have stepped into a parallel universe, then. I don't remember
this.
> So I think you need a vacation from
> academics Chris, because you are burnt out.
If I was burnt out, would I bother wasting my time by refuting the
incorrect arguments that you post? On the contrary, if I burned out, I
would drop off of Usenet for a LONG time. Again, your logic is
backwards here.
(Another thing. For the past 5 years, "academics" has been teaching; I
haven't done any real research in 5 years, something I want to get back
to doing. I've run across some neat problems, but haven't done anything
deep recently.)
> You see Chris, there is a huge difference between you and me.
Yes. I've got one paper that has been peer-reviewed and published, and
two on the way, and you don't have any.
> You are a mere professor of
> mathematics whereas I am a genius of science.
Just because Mozart had a funny laugh doesn't mean you can play the
piano.
> And whereas you can only
> play around in mathematics, I play around in every science and math is
> merely my minor sport.
And I can play around in mathematics which you will never see, or
recognize as existing.
I read through part of your autobiography, and one incident in your
life seems to explain a great deal about your thought processes. I'm
talking about when you were collecting coins, and you had the
"realization" that it wasn't the year, or the pictures and numbers on
the coins that mattered, that it was the material that they were made
out of. The same theme showed up when you found a periodic table with
pictures of the elements. The theme is that everything you do has to
relate to the material world, that if there's no application to this
particular universe, the idea is meaningless. (Some people believe that
ghosts are the remnants of people who refuse to leave the physical
universe. If that's true, you may be in for a very long haunting ...)
I, too, was inspired by the Mathematics book in the Time-Life series.
My story is different, in that I ran across it while I was in grade
school and I was introduced to a world beyond what was called
mathematics at that time (namely arithmetic -- I hated doing long
division as much as anyone else). The existence of thigns like topology
and geometry gave me hope that what I was learning would eventually
have meaning, and that once I got past the mundane stuff, I'd get the
really fun stuff.
--- Christopher Heckman
a_plu...@hotmail.com
Yes, but it is wrong. This "proof" was discovered by Moebius in 1840
and has been rediscovered hundreds of times by amateurs like yourself.
And it is always refuted in the same way: If a graph contains K5, then
it's not 4-colorable, but the reverse is not necessarily true, and this
is what the proof hangs on. Furthermore, if it were a valid proof, then
Appel and Haken would never have bothered with their computer proof.
This misuse of logic is very fundamental, and in another form, it would
run like this:
Every person who lives in Denver lives in Colorado. TRUE
Every person who lives in Colorado lives in Denver. FALSE
A.P. writes:
I am curious now, did Moebius use the Jordan Curve Theorem, and I guess
not since it was discovered when? 1890s???
So how can mine be a Moebius replica?
And another fault of your argument Chris is that you constantly inject
Graph theory into every 4 Color Mapping reply. Whereas everyone else
never uses Graph theory in these discussions, you depend on them. So
you are not onboard and is in a different sea from the rest of us.
And I disagree with your Colorado Denver living analogy for it has
nothing to do with my proof argument. Let me restate my proof argument
instead of landlocked and surrounding ocean for a 4 country continent.
Let me outline the proof with the concept of Adjacency. Is it possible
to have 5 countries who are all adjacent to one another? No! So 4
colors are necessary and sufficient. Adjacency also uses the Jordan
Curve theorem, becuase in order to prove that there cannot be a 5
countries all adjacent to one another involves the inside and outside
of closed curves.
So again, Chris, your logic is at fault and you keep ascribing me with
failed logic when it is you who is not understanding of my arguement.
Perhaps the reason for your failure to understand my arguement is that
you have a closed and shut mind that there cannot be a "simple proof"
of 4 Color Mapping, yet which I am revealing that there is this simple
one page proof of 4 Color Mapping.
I have shown you the outline in two ways now. You can do it with 4
country continent where all 4 countries have beachfront property and
thus one country has to be landlocked. And my Second method is consider
5 countries adjacent to one another, which is an impossibility. Both
methods using JCT.
No fault in my logic. But a big fault in Chris's understanding of what
I am saying. So drop that Graph crap. And Moebius could never have done
these methods because the JCT did not exist in his time.
Your problem Chris is that your mind is closed to the idea that anyone
will ever produce a simple one page proof. And that is what I have
done. So open your mind and take another look. Or talk to some of your
friends in mathematics that will explain to you what I have outlined
above.
a_plu...@hotmail.com
argument clear, and so I attempt another explanation:
Short one page proof of 4 Color Mapping:
Construct a continent that has 5 countries and the continent is
surrounded by blue colored oceans. Can you have those 5 countries all
adjacent to one another and yet
simultaneously all owning a beachfront property to the ocean? The
answer is no because the 5th country will have to surround the other 4
countries in order to satisfy the condition of all 5 countries being
adjacent. Thus the ocean blue color can be used on some of those
landlocked countries. Now the case for 7 or 8 or 9 or 10 country
continent surrounded by blue color ocean becomes another cell where 4
colors are necessary and sufficient. Now the case of a 4 country
continent becomes necessary and sufficient that 4 colors does the job
keeping in mind the ocean is blue. So for continents of 5 or more
countries surrounded by blue ocean becomes "cells" of landocked
countries where 4 colors are necessary and sufficient. The Jordan Curve
Theorem enters in that each country is a closed figure of inside and
outside to uphold the concept of adjacency. Now suppose there exists a
5 country continent that required 5 different colors keeping in mind
the ocean is blue. Contradiction to the Jordan Curve theorem because
then you would have 5 countries all adjacent to one another and yet all
5 countries owning beachfront ocean property.
It is a contradiction because the countries would have to be open and
not closed curves.
Proginoskes
a_plu...@hotmail.com wrote:
> Chris Heckman wrote:
> Yes, but it is wrong. This "proof" was discovered by Moebius in 1840
> and has been rediscovered hundreds of times by amateurs like yourself.
> And it is always refuted in the same way: If a graph contains K5, then
> it's not 4-colorable, but the reverse is not necessarily true, and this
> is what the proof hangs on. Furthermore, if it were a valid proof, then
> Appel and Haken would never have bothered with their computer proof.
You haven't thought through what I said at the end of the paragraph, so
here it is again:
If Moebius's proof were valid, Appel and Haken would never have
bothered with their computer proof. They would not have needed to; it
would have been unnecessarily complicating the problem. And Robinson,
Sanders, Seymour, and Thomas would never have started on their proof,
either. Kempe wouldn't have tried to prove it, either. Neither would
Tait or Heesch.
If Moebius had indeed found a short proof that was correct, none of the
names above would mean anything.
> A.P. writes:
>
> I am curious now, did Moebius use the Jordan Curve Theorem, and I guess
> not since it was discovered when? 1890s???
>
> So how can mine be a Moebius replica?
It's not a photocopy; the details are different. But both proofs depend
on the fallacy that if a map needs to be colored with 5 different
colors, then it must have five regions, where each is adjacent to the
other 4. (Your ocean is the "5th region".)
> And another fault of your argument Chris is that you constantly inject
> Graph theory into every 4 Color Mapping reply. Whereas everyone else
> never uses Graph theory in these discussions, you depend on them. So
> you are not onboard and is in a different sea from the rest of us.
That is because if you use the region-coloring version of the problem,
you have to word it carefully. If you allow two regions that share a
vertex to be neighbors, there is no 4CT; if you say that the regions
don't have to be "homeomorphic to a unit disk", then there is no 4CT
(as proven by Hud Hudson a few years ago).
But the graph theory version has no possible misinterpretation.
(Robertson, Sanders, Seymour, and Thomas actually colored the _edges_
of their graphs, but they use an old theorem that says if you can
3-color the edges of a planar GRAPH, then you can 4-color the vertices
of a GRAPH.
> And I disagree with your Colorado Denver living analogy for it has
> nothing to do with my proof argument. Let me restate my proof argument
> instead of landlocked and surrounding ocean for a 4 country continent.
> Let me outline the proof with the concept of Adjacency. Is it possible
> to have 5 countries who are all adjacent to one another? No! So 4
> colors are necessary and sufficient. [...]
You did it _again_. You're confusing "if P, then Q" with "if Q, then
P". You are assuming that
This is basic logic here you're not getting, something which you should
get, as it's as fundamental of an idea as reducto ad absurdum. (And
it's not _my_ misunderstanding of logic, unless you want to include
every mathematician going back to the ancient Greeks.) And it has
nothing to do with graph theory, either.
The structure of your argument (ignoring the details) boils down to the
following: You have "P implies Q", and then prove P, and thus conclude
Q. Here, P is the statement "It is impossible to have 5 countries all
adjacent to each other", and Q is the statement of the 4 Color Theorem,
and the fancy term for this reasoning is Modus Ponens.
Do you agree with the paragraph above, about the structure of your
proof?
Yes ___ No ___
Now, the flaw in the argument is that "P implies Q" is not necessarily
true; it is "Q implies P" that is true. And if you know that the
statements "Q implies P" and P are both true, you can not conclude
anything about whether Q is true. This is because "False implies True"
and "True implies True" are both true.
--- Christopher Heckman
P.S. Sometimes I wonder whether you really were a math major; certainly
you should not have passed a real analysis course with such sloppy
logic skills. Other times, I wonder whether you really do see why I'm
right, but you keep on needling me for entertainment.
Proginoskes
a_plu...@hotmail.com wrote:
> I feel my last reply to Chris Heckman, still did not make my proof
> argument clear, and so I attempt another explanation:
>
> Short one page proof of 4 Color Mapping:
>
> Construct a continent that has 5 countries and the continent is
> surrounded by blue colored oceans.
Good so far.
> Can you have those 5 countries all
> adjacent to one another and yet
> simultaneously all owning a beachfront property to the ocean?
And I agree the answer to this question is NO. But ...
Answering 'no' to this question is NECESSARY for the 4CT to be true,
but it's not SUFFICIENT. If you were constructing a reducto ad absurdum
argument, and the answer to this question is 'yes', then you would have
disproved the 4CT, and the argument would be valid.
Your argument is like asking whether the equation x^2 = 4 has any real
solutions, and then pointing out that 1^2 is not 4. This second fact
does NOT prove that there are no real solutions to the original
equation. You have eliminated ONE possibility, but not EVERY
possibility. You should ponder the previous sentence; it explains the
source of your error. (After some thought) Well, maybe it's not the
main issue here, but it's still worth pondering.
(Also, if you change the number of countries from 5 to 4, the answer is
still no.)
> The answer is no because the 5th country will have to surround the other 4
> countries in order to satisfy the condition of all 5 countries being
> adjacent.
Well, the answer is actually no for other reasons, which I won't argue
about. Look up Moebius's proof.
> Thus the ocean blue color can be used on some of those
> landlocked countries.
True.
> Now the case for 7 or 8 or 9 or 10 country
> continent surrounded by blue color ocean becomes another cell where 4
> colors are necessary and sufficient.
Actually, the argument is more complicated than this. People have used
brute force to prove that every map with at most N countries is
4-colorable, before computers, where N is (I think) somewhere between
20 and 30.
(After some more thought) It's that phrase "becomes another cell" that
has me stuck on this sentence. I think what you mean here is that you
want to unite a bunch of land-locked countries, which will take you
back to the situation where you only have 5 countries in total. It's a
variation on Fermat's "method of descent" ...
Or maybe what you want to do is to approach the map in "layers", where
you consider all the countries which all have beachfront property, then
you remove those, consider the countries which now have beachfront
property, remove those, etc., until you end up with at most 4
countries.
If this is the case, the argument carries over to Graph Theory without
any real trouble, and it's a lot like what Ibrahim Cahit is doing with
Spiral Colorings, except that Cahit's procedure doesn't "merge"
countries. (That paper is still in the "undecided" pile.)
Since it's tired and I'm late, maybe I should stop here. Maybe an
example would help here, like the lower 48 states of the USA
(pretending Canada and Mexico are ocean), or maybe Europe, where there
aren't as many regions.
--- Christopher Heckman
a_plu...@hotmail.com
And I agree the answer to this question is NO. But ...
Answering 'no' to this question is NECESSARY for the 4CT to be true,
but it's not SUFFICIENT. If you were constructing a reducto ad absurdum
argument, and the answer to this question is 'yes', then you would have
disproved the 4CT, and the argument would be valid.
A.P. writes:
Okay Chris, I think I grasp your necessary and sufficient calls. And my
proof outline is still unclear and murky. However, after taking a break
I realized how to fix it. So I am going to post it here and start a new
thread on proof of the 4 Color Mapping.
I think I made a post recently saying words to the effect that a
triangle is the smallest Jordan Curve that is closed. One would say a
circle like figure is the smallest Jordan Curve that is closed. But I
need some sort of definition of these concepts. If you take 1 or 2 line
segments you cannot arrange them into a Jordan Closed Curve. If you
take 3 line segments you can arrange them into a Jordan Closed Curve.
So is there a definition or concept name for this idea that 3 line
segments form the smallest closed Jordan Curve?
One page proof of 4 Color Mapping Problem:
You are given one country and let us say it is a long strip shape now
place a 2nd strip adjacent to the first strip and now take a 3rd strip
and arrange it so that it is adjacent to the 1st strip and to the 2nd
strip where you are able to bend the 3rd strip so that all three strips
are adjacent. Now take and draw a 4th figure so that it is adjacent to
strips 1, 2, 3. It can be done. I like to think of nested spiral arms
with the 4th one odd shaped in order to force it to be adjacent to that
of 1,2,3.
Okay so far, but when we inspect our construction we notice a pecular
thing. We notice that in order for the 4 countries (strips) to be
adjacent that one of them becomes landlocked. And if the ocean were
surrounding these four countries and colored blue, we can use that blue
color on the landlocked country.
Now here is the necessary and sufficient part of the proof that 4
colors suffice and are necessary. Of the 4 countries, one of them
becomes landlocked and landlocked means inside the other 3 countries.
So the 3 countries act as a Jordan closed curve.
Now we can bring into the proof the Jordan Curve Theorem because 3
countries have inside themselves the 4th country. So why is it in
mathematics that 4 adjacent countries will always landlock one of those
four? The answer is that the smallest Jordan Closed Curve is formed by
a triangle or 3 line segments. So the JCT proves that if you have 4
adjacent countries then one will be landlocked and thus take the color
of the surrounding ocean blue color.
What Chris was objecting about my proof was that it was not necessary
and sufficient. Because I did not elaborate on how the Jordan Curve
Theorem forces 4 adjacent countries to landlock one of those countries
and thus 4 colors are necessary and sufficient. If you can take 4
countries and arrange them in such a manner that they are all adjacent
and for which none of them is landlocked then 4 colors will not suffice
because the ocean water blue cannot be used on one of those 4 countries
and that you will be required to have 4 new colors none of which are
blue.
What I failed to elaborate on is how the Jordan Curve theorem forces
one of the 4 countries to be landlocked and it does so because the
smallest closed curve is formed by 3 countries with the 4 inside the
other 3, that is a triangle is the smallest Jordan Closed curve.
Landlocked is equivalent to Jordan Closed Curve.
Proginoskes
a_plu...@hotmail.com wrote:
> Chris Heckman wrote:
> And I agree the answer to this question is NO. But ...
>
> Answering 'no' to this question is NECESSARY for the 4CT to be true,
> but it's not SUFFICIENT. If you were constructing a reducto ad absurdum
> argument, and the answer to this question is 'yes', then you would have
> disproved the 4CT, and the argument would be valid.
>
> A.P. writes:
>
> Okay Chris, I think I grasp your necessary and sufficient calls. And my
> proof outline is still unclear and murky. However, after taking a break
> I realized how to fix it. So I am going to post it here and start a new
Posting your proof helped a lot, since I was under the impression that
you were proceeding along the same lines as Moebius. I was still under
that impression when I wrote the part of the post that you quoted
above.
Something that would help a lot would be to take a map like the USA,
and show explicitly how your proof colors the states with 4 colors. (A
standard way of getting an algorithm is to assume that your map is a
counterexample, and work through your proof with this particular map,
and see what happens.)
--- Christopher Heckman
b92...@yahoo.com
That is ridiculous! What are you really trying to say?