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4-Color Theorem: Suppose regions which share a border of some length must have different colors. Then any map of regions on a plane or a sphere can be colored in such a way that only four colors are needed.

Neil Robertson et all recount the history of the this theorem, reporting that in 1852, while mapping the counties of England, Francis Guthrie realized that he needed only four colors to color the map (Figure 1). Intrigued by the anomaly, Guthrie inquired of his brother concerning this peculiarity who in turn presented the enigma to his professor, Augustus De Morgan. Since then mathematicians around the world have struggled with the 4-Color Problem as the conundrum came to be called (Robertson et all).

John McCarthy points out that in 1976 Wolfgang Haken and Kenneth Appel, mathematicians at the University of Illinois, proved Guthrie's conjecture correct by mapping out 1936 map configurations with the aid of computers (McCarthy). Thanks to Haken and Appel the 4-Color Problem became the 4-Color Theorem.

In addition to the 4-Color Theorem which applies to planes and spheres, mathematicians have since determined the maximum number of colors needed to color the surface of a torus with one hole. 7-Color Theorem: Suppose regions which share a border of some length must have different colors. Then any map of regions on a torus with one hole can be colored in such a way that only seven colors are needed.

Fig. 2

Creating a Torus

1. Begin with a rectangle.

2. Roll it into a cylinder.

3. Bend the cylinder so
that the bases coincide.

Mathematicians figured the number of colors necessary to color any map of regions on a plane or sphere with the 4-Color Theorem and on a torus with the 7-Color Theorem, but as far as I have been able to research, never went beyond these theorems to calculate the number of colors needed to color the surface area of other 3-dimensional figures. This is not surprising since Guthrie first came across the 4-Color Problem while working on maps which deal with planes and spheres.

I have expanded this area of geometry and have discovered the formula by which one can calculate the number of colors required to color the surface area of any 3-dimensional figure depending on the number of holes in the figure.

I also propose that the terribly complex proof that Haken and Appel use in their computer calculations to prove the 4-Color Theorem can be further reduced to a simpler proof which does not require the aid of a computer.

However, one could consider this part of my research as still hypothetical. Furthermore, whether or not this proof of the 4-Color Theorem is correct or not does not affect the formula for determining the number of colors needed to color the surface area of any 3-dimensional figure, which I know is correct. Therefore, if the reader is already well versed in the 4-Color Theorem and 7-Color Theorem, one may wish to skip to Coloring the Surface Area of Any 3-dimensional Figure. This section well may be considered an introduction to why the 4-Color Theorem is true for dummies. I must apologize for the self-created terms used here in my proof. This is due to the fact that no one has ever bothered to research this area of geometry.

The basis of the 4-Color Theorem is that 5 or more regions cannot share a border of some length, each bordering all of the four other regions. Here I will show simply that this is a fact. No other proof is necessary for the 4-Color Theorem because a map would have to have a region border all four other regions in order to disprove the theorem.

The Corner Effect: The point at which 4 or more regions meet does not allow all the regions to share a border of some length, each region with all others as shown here. I have termed this the corner effect (Figure 3).

The Going-Around Effect: Because of the corner effect, regions which do border each other in the location of the corner effect would have to reach around adjacent regions to border those regions they would otherwise not (Figure 4).

The Cut-Off Effect: Because of the going-around effect, regions are trapped by those regions reaching around to border non-adjacent regions. Thus further regions, even ones that encircle the rest of the map cannot share a border of some length with the trapped or cut-off regions (Figure 5).

This may seem overly elementary, but here is the key: because of the cut-off effect, any map of regions bordering the edge of this map would border it with only 3 colors or less. Were this not so, and a map requiring 4 colors to border this map would in and of itself require 5 colors.

One could argue that this explanation of the 4-Color Theorem is circulatory and thus not a true proof of the theorem. However, as I mentioned before, I am not concerned so much with the proof of the 4-Color Theorem as I am with the expansion of the existing theorems.

One could use another way to view the cut-off effect by using a simple arc and node system. 5 nodes represent 5 regions and arcs represent their borders of length. It is impossible to connect all 5 nodes so that each shares an arc with all others without overlapping the arcs. For example, in Figure 6, nodes 2 and 4 do not connect.

There is a way to state an ultimately simple proof of the 4-Color Theorem. The vast amount of chains of regions that mathematicians have used in the past to prove the 4-Color Theorem are in reality vestigial because of the simple fact in order to disprove the theorem, somewhere in the chain of regions one must have 5 regions that border one another so that a fifth color is necessary. The 4-Color Theorem is true simply because no more than four regions can all share a border of some length on a plane or sphere.

The surface area of a sphere is no different than a plane in its properties regarding coloration because the corner effect is still in effect. That is, even using the circular property of the sphere to reach around adjacent regions does not allow more than 4 regions to share a border of some length each with another (Figures 7 and 8).

Mathematicians, as far as I have been able to find, never expanded the 4-Color Theorem past planes and spheres, but I realized that the theorem actually applies to any 3-dimensional figure without holes. That is, that only 4 colors are needed to color any map on the surface area of figures such as cubes, tetrahedrons, and so forth. Thus, the 4-Color Theorem should read: Suppose regions which share a border of some length must have different colors. Then any map of regions on a plane or surface area of a 3-dimensional without holes can be colored in such a way that only four colors are needed.

Of course, as explained before, the torus requires 7 colors because of the hole in the figure. This is simply because the hole eliminates the corner effect so that, unlike on a sphere, the non-adjacent regions, such as those at the top and bottom of a map or pattern, can border each other while regions on the left and right sides of the pattern can border each other also (Figure 9).

One must not confuse a hole with a depression. A true hole allows passage from one side of a figure to another side. For example, a teacup is a torus because of the hole formed by its handle. However, one would not consider the bowl shape depression of the cup a hole for the purposes of coloration (Figure 10).

If a plane and 3-dimensional figures without holes required at most 4 colors, and a torus with one hole requires 7 colors, how many colors are needed to color the surface area of a toroid with two holes, or three, or three-hundred, or more? This question is at the heart of my research. I first started experimenting with simple, two fold toroids (Figure 11).

Then I came across the toroid in Figure 12 in which two single fold toroids combine to form a new figure. Unlike the toroid in Figure 11, the two single fold toroids intersect at two points instead of one; and unlike Figure 11, this object does not have a simple system of holes.

How many holes does the toroid in Figure 12 for the purposes of coloration? I determined the number of holes in the figure by manipulating it similar to the process shown in Figure 10 (Figure 13). Figure 13 shows that the toroid in Figure 12 has 3 holes for the purpose of coloring its surface area.

Following this same process, through experimentation I arrived at a formula one could use to determine the number of holes in any figure for coloration. For example, determining the number of holes that the object in Figure 14 has may seem difficult at first.

Instead of having to go through the lengthy process in Figure 15 to determine the number of holes in any given object, I found that by subtracting the number of hole systems from the number of entry ways in the 3-dimensional object, one could determine the number of holes.

A hole system is the intersection of one or more holes within a figure. For example, the object in Figure 14 has one hole system because all the holes in the object intersect. The object if Figure 16 has three hole systems because there are three separate complexes of holes that do not intersect. A hole that does not intersect with other holes is its own system.

An entry way is the point at which a hole appears at the outer surface of an object. For instance, the object in Figure 16 has 9 entry ways.

Rather than go through the complex process of simplifying the object in Figure 16 to determine the number of holes in has, one could simply use the formula I discovered by working with such models.

Hole Formula
For determining the number of true holes in any 3-dimensional figure for the purpose of coloring the surface area:

where h is the number of holes,
e is the number of entry ways,
and s is the number of hole systems.

Using this formula, one could more easily determine the number of holes in any 3-dimensional figure. For example, the object in Figure 12 has 4 entry ways and 1 hole system, since all hole intersect. 4 minus 1 equals 3, therefore the object has 3 holes as already shown in Figure 13. The object at the beginning of Figure 15 has 3 entry ways and 1 hole system, giving it 2 holes. The object at the end of Figure 15 also has 2 holes because it has 4 entry ways and 2 hole systems. In essence, they are the same object.

One could use this hole formula to determine the number of holes in more complex figures such as the one in Figure 16. With 9 entry ways and 3 hole systems, the figure has 6 holes. Although these models are very simple, I have worked with more complex figures to assure that the hole formula is correct.

Finally, once one has determined the number of holes in a 3-dimensional figure, one can use the coloration formula to determine the maximum number of colors needed to color any map or pattern on the surface area of the given object. I labored for years trying to determine this formula only to realize the simple fact that with each hole, another 3 colors are needed.

where C is the number of colors needed,
and h is the number of holes in the 3-dimensional figure.

One can try out these formulas on such figures as the one at the top of this document. Assuming each hole in the figure has one visable entry way, how many holes does the figure have? How many colors would one need to color its surface area?

Unlike Haken and Appel, because of lack of facilities, time, and funding, I have not the means to prove that these formulas are true. I can only say that they are true and so simple that I do not see why no one has ever thought of them, as far as I have been able to research.

One possible reason why no one ever bothered to expand the 4 and 7-Color Theorems, as far as I have been able to find, is the lack of a need. Outside of antiquated mapping techniques and geometric curiosity, there has been little or no application of such math, until now.

Beyond Swiss cheese, I have in the past speculated that the coloration formula could be used to study star nebula or complex chains of cyclo-hydrocarbons, though I have no idea exactly why anyone would want to map the surface area of either. Later, I realized that realized that the formula could have more useful applications such as assisting with the Human Genome Project, DNA being one long and immensely complex toroid, as well as mapping out theoretical figures in physics such as those that Brian Greene talks about in his book, The Elegant Universe. Speaking of string theory, Greene states, "A typical Calabi-Yau shape contains holes that are analogous to those found at the center of a phonograph record, or a doughnut, or a 'multidoughnut.'" (216).

With the advent of the Internet, the coloration formula could have valuable applications in computer security. For instance, the company, D & G Sciences has developed Leonardo, an "encryption paradigm based on the 4-color map theorem." D & G Sciences markets Leonardo as a system to encrypt confidential messages for safe transmission over the Internet. If a company can design a marketable computer security program using just 4-Color Theorem, the simplest part of my research, what could one do with an infinite amount of color theorems?

Appel, Kenneth and Woflgang Haken. "Every planar map is four colorable." Illinois Journal of Mathematics, vol. 21, 1977, pp. 429-567.

Coloration Formula