A Simple Four Color Theorem Proof Etc.

Abstract:

The four color map theorem states that no more than four colors are needed to color the regions of any map so that no two adjacent regions have the same color. More specifically, the theorem states that no more than four colors are required to color the vertices of any planar graph where any two adjacent vertices don't have the same color. This paper shows how the theorem arises from a broken correlation between the number of colors and the number of vertices (dots), and, how repairing this broken correlation leads to maps requiring more than four colors. However, if three dimensions are not invoked on a on planer graph, it can be shown that no more than four colors are needed to color it.

Premise: All maps, regardless of their shapes and sizes, can be represented by planer dot-and-line diagrams. Such drawings make analysis easier and more universal. Each dot (vertex) represents a country and each line represents a border. The rules are any two dots connected by a line cannot be the same color and lines cannot intersect.

Suppose there is a single country. We can represent it with one dot and one color:

Take two randomly-positioned dots. Draw a line (border) between them. These two dots and the line can be mapped to a straight horizontal figure. This will allow us to make apples-to-apples comparisons between different random maps. Also note that we now have two dots and two colors.

Next we have three randomly-positioned dots mutually-connected by three lines. Here again we can map this to a horizontal figure. We make sure the lines do not intersect since they represent borders on a two-dimensional (2D) map. There is now three dots and three colors.

We repeat the previous process with four dots and four colors:

There appears to be a pattern: one dot requires one color; two dots require two colors: three mutually-connected dots require three colors--and four dots, mutually-connected, require at least four colors. If we connect five dots, will we need at least five colors? According to the diagrams below, we can't mutually connect all five dots without intersecting lines:

Since the purple dot and the blue dot are not connected, they can be the same color:

The pattern we observed earlier is broken. Five dots don't require a minimum of five colors as expected--they only require four. Perhaps six dots require at least five colors:

Nope! Maybe seven or eight dots require at least five colors:

Wrong again. But take note of a new pattern: when we add a new dot beyond four, and do our best to mutually connect all dots so their colors cannot be reused, at least two dots are not connected. As a result, they can be the same color, and, the total number of required colors never exceeds four no matter how many dots we have, no matter how big and complex the map. However, this is only true if a 2D map never invokes a third dimension (3D), i.e, there is only north, south, east, west, and no elevation or up and down. At the diagram below, we invoke 3D by allowing the orange line to cross over the black line. Try to imagine the orange line lifting off the page to cross over.

Now all five dots are mutally-connected. Five dots now require a minimum of five colors. In 3D, the ratio of colors (c) to dots (d) is always one. In 2D, this ratio breaks down when there's more than four dots:

Below is an example of eight dots, mutually-connected in 3D, requiring at least eight colors:

If 3D is invoked on a 2D plane, lines are allowed to cross. This potentially allows up to an infinite number of mutual connections between an infinite number of dots, and such a map requires up to an infinite number of colors.

If we examine other dimensions (D) of space, we make the following observations: if D = 0, there is always one dot and one color. If D = 1, there is only one type of map: a straight line. It is fairly obvious that only two colors are needed for this map, no matter how big it is.

When D = 2, and lines are not allowed to cross, it appears that no more than four mutual connections can be made between dots, and no more than four colors are ever needed no matter how big or complex the map. As mentioned earlier, a 3D map could require up to an infinite number of colors. Further, we should note that 0D only accommodates 0D maps (a single dot), 1D accommodates 0D and 1D maps, 2D accommodates 0D, 1D, 2D maps, and finally, our 3D space can accommodate all maps from 0D to 3D. Below is a graph that illustrates our findings:

Take a look at the 2D column. It indicates that 1-color, 2-color, 3-color, and 4-color maps are possible. But are four colors really the maximum needed for any possible 2D map that needs more than three colors? It would take a great deal of computing power to test an infinite number of different types of maps that can exist in 2D space. The data table below lists what we know and a question mark where there is uncertainty:

The data table suggests a pattern where we can predict that the question mark should equal four, i.e., no more than four colors are required to color any 2D map. But is there really a pattern or are the numbers listed coincidental? Can we show that the numbers are logically and mathematically connected? The answer is yes:

Dimensions zero through three each have a maximum number of colors necessary to color maps in their respective spaces. The pattern between them is a tan function (see equation 3). So we can logically conclude that no more than four colors are needed to color any 2D map? Yes, as long as 3D is not invoked on a 2D map. If 3D is invoked, here's what is possible:

Imagine an exoplanet approximately 50 light-years from earth. It orbits a yellow star within the Goldilocks zone. It has a vast ocean. In the middle of that ocean is an island continent with five countries: purple, red, yellow, green, and brown.

In the middle of the island is a mountain. The red and purple countries share a border at the mountain top. The green country created a tunnel through the mountain to connect with and share a border with the yellow country. Each country shares a common border with the remaining four. Not counting the ocean, at least five colors are needed to color this 2D map.

References:

1. Four Color Map Theorem. Wikipedia

2. Weisstein, EW. 2002. Four Color Theorem. Wolfram MathWorld

3. Najera, Jesus. 10/22/2019, The Four-Color Theorem. Cantor's Paradise

4. Planer Graph. javatpoint.com.

Proof that Various Infinities Have a Finite Value

In the late 19th century, German mathematician Georg Cantor demonstrated that there are a variety of infinities of various sizes. Equations 1 and 2 below compare two different arbitrary infinities:

The infinity at equation 2 is n times bigger than the infinity at equation 1.

Srinivasa Ramanujan, an Indian mathematician (1887–1920), was cleverly able to extract finite values from various divergent infinite series. If having various infinities isn't weird enough, imagine various infinities having a finite value! Normally, the process of extracting a finite value from an infinity requires tackling that infinity with a unique set of steps. Thus it is not clear if all infinities have a finite value. Perhaps there are some that do and some that don't. Can mathematics provide any clues? Let's begin with an arbitrary function f that diverges to infinity:

What makes this particular infinity unique is not that variable N tends to infinity, but that N is multiplied by a coefficient alpha. This infinity is alpha times the size of a benchmark infinity that is simply N with an infinite limit. Below is another infinity (function s) with a coefficient of beta, where beta may not equal alpha:

The relationship between f and s is as follows:

Using some Ramanujanian algebra, we can solve s:

We can now solve f by making a substitution:

If coefficient k is finite and x doesn't equal 1, the solution to any arbitrary infinite function is finite. Otherwise it is infinite.

References:

1. Matson, John. 07/19/2007. Strange but True: Infinity Comes in Different Sizes. Scientific American

2. Dodds, Mark. 09/02/2018. The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12? Cantor's Paradise

How to Have Unlimited Orthogonal Space and Time Dimensions within 4D Spacetime

ABSTRACT:

By means of a thought experiment and a mathematical proof, it can be shown that unlimited space and time dimensions are possible, and, that only three space dimensions are mutually perpendicular.

Imagine you are throwing a party. You have invited n number of guests. You want to know the following: the starting location and time of each guest and the time each guest arrives at your party. You know the location of the party and you know what time the guests are supposed to arrive. All of this information consists of 2n+1 bits of time and n+1 bits of location. Assuming each location has coordinates x, y and z, the total bits of space information is 3n + 3.

Each bit of information is statistically independent, i.e., orthogonal to all the rest. We can think of any two bits as having a 90 degree separation. In the case of coordinates x, y, and z, such separation can be easily drawn on graph paper. In all other cases, such separation may be purely abstract and unimaginable. In any case, we can argue that 3n+3 space dimensions and 2n+1 time dimensions are necessary. To have all the information you want, you need 5n+4 dimensions. If n = 100 guests, you only need 504 spacetime dimensions!

It's fairly obvious that the above thought experiment only involves three space dimensions that are mutually perpendicular. But is this universally true? Is there, say, a mathematical proof? The following equation suggests there's no upper limit to how many space dimensions you can have:

It seems like the only constraint re: the number of space dimensions is an empirical one. Let's see if we can find a mathematical one. Let's begin with the following premises:

1. w, x, y and z are unit vectors.

2. w is an arbitrary extra space dimension. What is true for w is true for any extra space dimension.

3. A unit vector is consistent with 1D space and points in only one direction.

4. if two unit vectors (x, y) are perpendicular, they define a plane that is consistent with 2D space. Thus plane xy is not perpendicular to plane xy.

Following these premises we have:

At steps 2 through 4 we assume that all four unit vectors are mutually perpendicular. At 5 we assume w is perpendicular to plane xy. At 6 we assume z is also perpendicular to plane xy. At 7 we conclude that either w is parallel to z or if w is perpendicular to z, then plane xy must be perpendicular to plane xy--which violates premise 4. So w is not an extra dimension. According to premise 2, what is true for w is true for any alleged extra space dimension. Thus, we can further conclude there are only three space dimensions that are mutually perpendicular. If such a conclusion is valid, we should be able to falsify the following 7D cross product table:

From this table we can gather that the unit vector e1 is the solution to three cross-products involving 6 dimensions (see equation 8 below). Premise 3 stipulates that a unit vector only points in one direction. It is also obvious that e2e3, e4e5, e6e7 make three planes persuant to premise 4. Vector e1 can't point in just one direction if it is normal to all three planes, unless all three planes are subsets of the same plane. The inevitable conclusion is not all of these dimensions are mutually perpendicular.

Now, let's take a look at the so-called extra dimensions 4 through 7. At each of the equations 9 through 12 below, the unit vectors circled in red contribute to planes with normal vectors pointing in different directions. Thus they can't all have the same normal vector or cross-product solution.

Therefore, it is safe to say that dimensions 4 through 7 do not behave like mutually perpendicular dimensions.

Circling back to our hypercube at equation 1, we can conclude that there is no upper limit to how many dimensions the cube can have, but only three are mutually perpendicular. The rest may or may not be orthogonal in the sense that they are statistically independent.

References:

1. Seven-dimensional Cross Product. Wikipedia

2. Octonian. Wikipedia

Resolving the Liar's Paradox: "This Statement is False"

"I am a liar," is the original liar's paradoxical statement, but we're going to focus on one of its variations: "This statement is false." If true, then it is false. If false, it must be true. To resolve this paradox, we begin with the following premises:

1. If a statement is true or false, it is one or the other and not both, and, we can determine whether it is true or false.

2. If a statement is true and false, it is a challenge to give it a truth value.

Now let's define some statements: Statement A = "This statement has five words." Statements B and C = "This statement is false." Statement D = "This statement has four words."

Given the foregoing premises, we can determine the truth value of each statement using an AND-gate truth table and by following these steps:

1. Read the statement and determine if it is true or false. If this can't be done, assume it is true (or false if you prefer). Mark the left side of the truth table T for true or F for false.

2. Then ask, "If I assert the statement is true (false), does it become false (true)? If not, repeat your previous mark. If so, add the opposite mark.

If these steps are followed, the truth table for statements A, B, C, D should look like this:

For the true statement A, there is no contradiction or paradox so the two marks are TT. Statement B equals statement C. At B we start out assuming the statement is true, and, at C we begin assuming it is false. That leads to scores TF and FT, respectively. Statement D is determined to be false and remains false, so we have FF. At the far right column are the final truth values. The true statement is true; the false statement is false, and, the paradoxical statement is false. This last result is consistent with the law of non-contradiction. Statements that are contradictory or lead to a contradiction are not credible and should be considered false, notwithstanding any claim to the contrary.

References:

1. Mano, M. Morris and Charles R. Kime. Logic and Computer Design Fundamentals, Third Edition. Prentice-Hall, 2004. p. 73.

2. Epimenides paradox has "All Cretans are liars." Titus 1:12

3. Jan E.M. Houben (1995). "Bhartrhari's solution to the Liar and some other paradoxes". Journal of Indian Philosophy. 23 (4): 381–401.

4. Hájek, P.; Paris, J.; Shepherdson, J. (Mar 2000). "The Liar Paradox and Fuzzy Logic". The Journal of Symbolic Logic. 61 (1): 339–346.

5. Mills, Eugene (1998). "A simple solution to the Liar". Philosophical Studies. 89 (2/3): 197–212.

Debunking Extra Space Dimensions and Minimum Distance

ABSTRACT:

By means of two thought experiments and some mathematics this paper shows that extra space dimensions are untenable. This paper also shows that the minimum distance is many orders of magnitude shorter than the Planck length.

Imagine a 2D universe on an x-y plane (see diagram below). Imagine a normal vector intersecting this plane at point p. 2D-guy inhabits this universe. He can't see the vector that intersects point p. He can only detect point p, so he has no reason to believe the normal vector exists. Now, to avoid point p, he goes around it (see red arrows).

He knows it's possible to draw an imaginary line through point p that can serve as an axis. He also notices when he goes around point p he's not encircling the x-axis or the y-axis--the two dimensions of his space. Thus, he infers that the imaginary axis he's going around does not belong to his 2D universe. He realizes he has discovered a new dimension!

Now, what happens if we apply 2D-guy's process to 3D space? Will we discover a fourth dimension? Let's try it. First we must scale everything up one dimension: The universe becomes 3D; the normal vector becomes a normal plane; Point p becomes line L. Let's assume there's a fourth dimension w, and let's define the normal plane as wx. Plane wx intersects our universe at line L which runs along the x-axis. We should not be able to detect the w-axis nor the bulk of the wx plane. We illustrate this with broken lines at the diagram below:

To avoid line L, we circle around it (see red circular path). We know we can draw an imaginary plane through line L. We know that x is one dimension of the plane. We know the axis we are circling (to avoid line L) is the plane's other dimension. We note we are not going around the x-axis nor the z-axis. That leaves the w-axis, but notice that the w-axis is indistinguishable from the y-axis. Therefore, our assumption that w is a new dimension and is undectable beyond line L is false. Unlike 2D-guy, we have not discovered a new dimension. However, we learned from 2D-guy that if a new dimension exists, it should be possible to do a rotation around an axis that does not exist in our universe. Until someone demonstrates such a rotation, we can conclude, for now, that the highest dimension of space is 3D.

But what if there are extra dimensions that are very small and curled up? If that's the case we should be able to enter alternate universes and those from alternate universes should be able to enter ours. Let me demonstrate what I mean. Imagine a line and pretend it is 3D space. Extending from it is a small extra curled-up dimension:

Let's introduce an arbitrary red object that is way too big to enter the tiny curled-up dimension:

Because the red object is too big to fit, it is assumed there is no way for the big red object to enter or detect the existence of the curled-up dimension. But didn't Euclid say something about a line existing between any two points? (In this case the line would be 3D.)

There's no reason why the big red object can't follow the path of this new line (3D space)and wind up in an alternate universe adjacent to ours:

As you can see, the big red object still can't enter the small, curled-up dimension, but the curled dimension facilitates access to alternate universes. The fact that big objects don't disappear from our universe and don't seemingly emerge from nowhere is strong evidence that microscopic curled-up dimensions don't exist. But wait! Quantum particles pop into existence and vanish all the time. It is hypothetically believed they enter a curled-up dimension (vanish), then leave that dimension and re-enter our universe. However, there's an alternate hypothesis: particles are really particle-waves. Waves experience constructive and destructive interference. When there's an excitation of a field, a particle pops into existence. That excitation could be or is equivalent to constructive interference. When there's destructive interference, energy vanishes--leaving the impression that the particle has disappeared.

The foregoing arguments seem to kill any notion that there are more than three space dimensions, but what about 4D spacetime? Or, what about the 6D object that can be found in Las Vegas? Let's address the 6D object first.

The 6D object I'm referring to is the die. The die has six orthogonal sides. Each side is statistically independent. We can change the value of a side without impacting the value of the other sides. If we change, say, the one to a seven, the other sides will still be two, three, four, five, and six. The most important point we can take away from the die is it is possible to have more than three orthogonal dimensions within 3D space! The die is a 6D object but it is also a 3D cube.

Spacetime, on the other hand, involves three dimensions of space and one dimension of time. If time is multiplied by a velocity, it has units of distance and is treated as a fourth space dimension. But is it really? Let's see what the math has to say:

Equation 1 represents a photon propagating through dimensions x, y, and z over a period of time t. It covers a distance of ct or r. For the sake of keeping the math simple, at equation 2 we rotate the path r so it is along the x-axis. Equation 3 reveals that space and time are not statistically independent, i.e., orthogonal to each other. The the value of time t depends on how far the photon propagates along x, and the value of x depends on how much time t lapses. This is the consequence of converting t into distance units by multiplying it by velocity c. So ct is not a true space dimension that is orthogonal to x. However, time t without c is a very useful statistically-independent parameter. For example, coordinates x, y, z tell you where to be for your dentist appointment and time t tells you when. A change in location does not have to change the time of the appointment, nor does a change in time have to change the location. So what can be done to make ct orthogonal to x? How about multiplying ct and x by factors of g? (See equation 6.) A change in x still causes a change in t, but g-sub-tt can be adjusted so the term stays constant. By the same token, the other term stays constant if g-sub-xx is adjusted when a change in t changes x.

So can we now credibly argue that (g-sub-tt)ct is a genuine fourth space dimension? Well, no 3D space dimension (x,y,z) has to be a function of (depend on) the others. We can, for example, eliminate y and z and still have x. But we can't eliminate a photon's path (x, y and/or z) and still have ct--the distance along a non-existent path. And, if there's no ct, then there's no (g-sub-tt)ct. Therefore, (g-sub-tt)ct is a pseudo-dimension at best.

So far, it seems we've only debunked a fourth dimension of space. What about dimensions five through infinity? Well, how we label a dimension is arbitrary. Any extra dimension can be labeled the fourth dimension. Thus, all arguments we have made against dimension four apply to any extra space dimension.

Now let's turn our attention to the concept of the shortest distance. The popular choice is the Planck length. In fact some theorists quantize space with Planck-size cubes or Planck-size tetrahedrons or Planck-size strings:

In the above diagram, the cube and tetrahedron have sides that are each one Planck length. However, the red diagonal lines reveal shorter lengths all the way down to a single point. These shorter lengths are absolutely necessary to create the shapes desired. Without a zero-length point, for example, there can be no corners for cubes and tetrahedrons. Additionally, there can be no strings in any string theory, since a string is a 1D object. A 1D object implies a zero cross-section or single point. A minimum-distance-greater-than-zero requirement would be a nightmare for M-theorists, since all D-branes would have to be 10 dimensions (including strings!). To have less than 10 dimensions requires zero distance for one or more dimensions. So it can be argued that the minimum distance is really zero, at least on paper. What about the physical world?

Equation 7 tells us that the shortest wavelength is determined by the highest energy. When the universe was a singularity, how short was the singularity's wavelength? If we only account for the energy in the known universe, that wavelength would be approximately a Planck length of a Plank length of a Planck length! Not exactly zero, but far less than a Planck length. Add energy beyond our known universe, and the distance is even shorter.

From a philosophical standpoint, the very concept of length implies a 1D object in the same manner the concept of area implies a 2D object. To measure length requires that we ignore all but one dimension, i.e., we set all but one dimension to zero. So zero distance is necessary, at least in the mind's eye. Since the mind's eye lives in this universe, we can infer that the minimum distance in this universe is zero.

In conclusion, any extra space dimension would allow rotations around an imaginary axis that is not part of 3D space. It would also allow any object access to an alternate universe. The shortest distance is many orders of magnitude shorter than the Planck length, and the Planck length may only be a lower limit of what we can successfully measure.

References:

1. Greene, Brian. 2003. The Elegant Universe. W. W. Norton

2. Irwin, Klee. 04/23/2017. The Tetrahedron. Quantum Gravity Research.

3. Sutter, Paul. 02/23/2022. Loop Quantum Gravity: Does Space-time Come in Tiny Chunks? Space.com