In 2005, Georges Gonthier and Benjamin Werner developed a formal proof. This was an improvement, because it allowed to use theorem-proving software, for the first time. The software used is called Coq.
The four color theorem is the first big mathematical problem that was proved with the help of a computer. Because the proof cannot be done by a human, some mathematicians did not recognize it as correct. To verify the proof, it is necessary to rely on a correctly working software and hardware to validate the proof. Because the proof was done by computer, it is also not very elegant.
The four color theorem has been notorious for attracting a large number of false proofs and disproofs in its long history. At first, "The New York Times" refused to report on the Appel–Haken proof. The newspaper did this as a matter of policy; it feared that the proof would be shown false like the ones before it .
Some proofs took a long time, till they could be falsified: For Kempe's and Tait's proofs falsifying them took over a decade.
The simplest counterexamples generally try to create one region which touches all the others. This forces the remaining regions to be colored with only three colors. Because the four color theorem is true, this is always possible; however, because the person drawing the map is focused on the one large region, they do not notice that the remaining regions can in fact be colored with three colors.
This trick can be generalized: If the colors of some regions in a map are chosen beforehand, it becomes impossible to color the remaining regions in such a way that in total, only four colors are used. Someone verifying the counterexample may not think that it may be needed to change the color of these regions. This will make the counterexample look valid, even though it is not.
Perhaps one effect underlying this common misconception is the fact that the color restriction is not transitive: a region only has to be colored differently from regions it touches directly, not regions touching regions that it touches. If this were the restriction, planar graphs would require arbitrarily large numbers of colors.
Other false disproofs violate the assumptions of the theorem in unexpected ways, such as using a region that has multiple disconnected parts, or not allowing regions of the same color from touching at a point.
In real life, many countries have exclaves or colonies. Since they belong to the country, they need to be colored with the same color as the parent country. This means that usually, more than four colors are needed to color such a map. When mathematicians talk about the graph associated with the problem, they say that is not planar. Even though it is easy to check if a graph is planar, finding the fewest number of colors needed to color it is very difficult. It is NP-complete, which is one of the most difficult problems that exist. The fewest number of colors needed to color a graph is known as its "chromatic number". Many of the problems that happen when trying to solve the four color theorem are related to discrete mathematics. For this reason, methods from algebraic topology are often used.
The four color theorem requires the "map" to be on a flat surface, what mathematicians call a plane. In 1890, Percy John Heawood created what is called Heawood conjecture today: It asks the same question as the four color theorem, but for any topological object. As an example, a torus can be colored with at most seven colors. The Heawood conjecture gives a formula that works for all such objects, except the Klein bottle.
Theorem
A theorem is a proven idea in mathematics. Theorems are proved using logic and other theorems that have already been proved. A minor theorem that one must prove to prove a major theorem is called a lemma. Theorems are made of two parts: hypotheses and conclusions.
Theorems use deduction, in contrast to theories which are empirical.
Some theorems are trivial, since they directly follow from the propositions. Other theorems are called "deep", because their proof is long and difficult. Sometimes, such proofs involve other areas of mathematics or show connections between different areas. A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem, and there are many other examples of simple yet deep theorems in number theory and combinatorics (among other areas).
There are other theorems for which a proof is known, but cannot be easily written down. Among the best examples are the four color theorem and the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search which is then verified by a computer program. At first, many mathematicians did not accept this form of proof, but it has become more widely accepted over the years. The mathematician Doron Zeilberger has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved. Many mathematical theorems can be reduced to simpler computations, including polynomial identities, trigonometric identities and hypergeometric identities.
Triceratops
Triceratops was a huge herbivorous ceratopsid dinosaur from the late Cretaceous. Its name came from having three horns on its head. They were mainly found in North America. As adults, they grew up to 30 feet long by 9 feet tall (), and probably weighed around 5,400 kg (12,000 lb).
"Triceratops" was a low browser with a bony beak in front of its jaws. The jaws had close-set grinding teeth. Its defence had to stand up to attacks from taller theropods, hence the bony shield which covered its neck. Holes made by teeth have been found on the bony frill behind the horns, and on the sacrum (the part of the spine above the pelvis).
Many fossils of "Triceratops" have been collected since the genus was first described in 1889. There is at least one complete individual skeleton. Paleontologist John Scannella observed: "It is hard to walk out into the Hell Creek Formation and not stumble upon a "Triceratops" weathering out of a hillside". Forty-seven complete or partial skulls were discovered in just that area during the decade 2000–2010. Specimens showing life stages from hatchling to adult have been found.
Individual "Triceratops" were about 7.9 to 9.0 m (26.0–29.5 ft) in length, 2.9 to 3.0 m (9.5–9.8 ft) in height, and 6.1–12.0 tonnes (13,000–26,000 lb) in weight.
The most distinctive feature is their large skull, among the largest of all land animals. The largest known skull (specimen BYU 12183) is estimated to have been in length when complete, and could reach almost a third of the length of the entire animal. It bore a single horn on the snout, above the nostrils, and a pair of horns approximately 1 m (3 ft) long, with one above each eye. Most other ceratopsids had large holes (fenestrae) in their frills, while those of "Triceratops" were noticeably solid.
"Triceratops" species were sturdy, with strong limbs and short three-hoofed hands and four-hoofed feet.
The posture of these dinosaurs has long been the subject of some debate. Originally, it was believed that the front legs of the animal had to be sprawling at angles from the thorax, in order to better bear the weight of the head. This stance can be seen in paintings by Charles Knight and Rudolph Zallinger. However, evidence of trackways, and reconstructions of skeletons show that "Triceratops" and other ceratopsids had an upright stance during normal locomotion, with the elbows flexed and slightly bowed out. The stance was intermediate between fully upright and fully sprawling, similar the modern rhinoceros.
Although "Triceratops" are commonly portrayed as herd animals, there is little evidence that they lived in herds.