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Hollandie's Tiling Lemma is an obscure theorem in combinatorial geometry first discovered by Dutch mathematician Zseiger Hollandie (1938-1991) and is the second of Hollandie's Seven Fundamental Lemmas of Combinatorics. It has had far-reaching impacts on many fields, including architecture, biology and both the combinatorial and geometry fields of mathematics.

Hollandie's Tiling Lemma states that for any finite set of squares whose area sum to 2n squared, there exists a subset of said squares that can completely cover a n by n square for any real number n. For example: In any finite set of squares whose area sum to 16 cm2, there exists a subset (not necessarily proper) of the squares such that they cover a square of sides 2 cm by 2 cm.

Hollandie's Tiling Lemma forms a key part of the subsequent Zseiger-Oreilly collection of lemmas and conjectures on key combinatorial geometry identities. The theorem is named after the mathematician thought to have first found a conclusive proof for it, Zseiger Hollandie, and has been used recursively as a plot device in popular culture.

History
Zseiger Hollandie was believed to have discovered the proof for Hollandie's Tiling Lemma in the autumn of 1951, during his short stint as a research assistant in the Central Cyberneurology Institute of Eastern America, located in Boston. It was there that he addressed a letter to childhood friend Conrad Ecelliorin, whom he would later work with to develop many more important theorems in algebra and number theory.

The contents of the letter, while never publicly disclosed, were believed to be roughly along the lines of the following:

Dearest Conrad,

''While at the Institute, I believe I have stumbled upon a most startling of results, I have barely proven it for a day but my mind already marvels at the myriad of possibilities that this holds for all of the field of mathematics. The result is of a most elegant kind, that the collection of squares with a sum of area 4n, is always capable of completely covering a square of area n. The proof for it is most difficult to convey in words, and yet the idea is simple and can be conveyed with the most basic of diagrams, attached in the following sheets of paper are the steps required to prove the above result.''

Conrad Ecelliorin was unable to decipher Hollandie's proof, so he sent it to the Dutch Royal Academy of Mathematics for verification, though Hollandie's original proof remains unknown, it has been verified by numerous experts worldwide to be true, and now the simplified version of his proof is commonly taught within the math curriculum of most tertiary institutes worldwide and on the Mathematique En Grandoise Academie (Grand Academy of Mathematics) located within the central European Academy of Advanced Sciences.

It first appeared in common circulation in Grigor Fulurae's book, Great Ideas of the 21st Century, listed as "possibly one of the most influential theorems of this generation".

Proof
As mentioned above, the original proof by Hollandie is not known to the public, his letters to Ecelliorin collected and classified by the Dutch Royal Academy of Mathematics upon the beginnings of the Trans-Atlantic Conflict beginning in early 1954. However, a multitude of proofs using a great variety of methods have been developed by other mathematicians, the most simplified and well-known version of which having been created by Conrad Ecelliorin himself in 1957.

Generalizations
As of this current date, no generalisations of Hollandie's Tiling Lemma exist. In 1968, Harold Frump from the Michigan State Mathematical Institute claimed to have found a method generalising the problem to a specific variety of polygons that he had personally named, "Frump-gon"s. Hollandie was quick to point out a counter-example to Frump's proof, and the resulting scandal and public backlash placed the education standard of the Eastern American Union into question, possibly resulting in exacerbation of tensions between the West European Trade Corporation and the Eastern American Union.

There have been equivalent statements of Hollandie's Tiling Lemma in numerous subjects like biology and architecture, in what is essentially a restatement of the original theorem with the key terms changed to the appropriate lexicon of the target field.

In biology
Hollandie's Tiling Lemma has been used in biology to prove numerous results, including the extension of the Cube-Square Law, known as the Cube-Square Permeation Limit, referring to the optimal limit at which any substance can be distributed throughout two given mediums given the rate of propagation within the two mediums and the percentage composition of the two mediums, this information has found great usage not only in biological engineering to create supercreatures, animals that reach their maximal biological size, it has also found medical application in terms of improved medication with faster reaction times.

Scientists at the Central Cyberneurology Institute of East America claim to be able to utilize Hollandie's Tiling Lemma to create an optimized algorithm for their prototype brain-mapping quantum computer, Indigo.

In architecture
As the name suggests, Holllandie's Tiling Lemma is used in architecture for the design and optimization of tiling of buildings. While the results of said theorem may seen to have little to no practical effect on the real world, QuanCore Pte. Ltd., the building firm in charge of the construction of buildings like the Cybernetics wing of the Lunar Academy of Advanced Sciences and the Administrative Tower of New Mesa City, Southern Egypt, claimed to have saved over 350,000 USD in marble tile wastage thanks to the application of Hollandie's Tiling Lemma.

Second-year architecture students are required to have understanding of at least tertiary level mathematics, which involves the first four of Hollandie's Seven Fundamental Lemmas of Combinatorics, with at least a distinction-level grade in the second and fourth lemma, the second lemma being Hollandie's Tiling Lemma.

In popular culture
Following Frump's failure to generalize Hollandie's Tiling Lemma, numerous works were made mocking his attempt to generalize said theorem, typically referring to the generalization of Hollandie's Tiling Lemma to be an impossible task, describing the "Frump-gon generalization" to be a lazy man's way out, yet ultimately wrong. An example would be Jacob Greighem's novel, The A.I. Who Could Cry, in which the protagonists challenges a supposedly omnipotent supercomputer to generalize Hollandie's Tiling Lemma. After two weeks of no results, the supercomputer offers the Frump-gon generalization, to which the protagonist points out a counterexample, causing the supercomputer to explosively short-circuit. Other works referencing a similar theme include The Grindlewarden Effect, Hearken 890, Professor Greg., and the award-winning biographical movie, Hollandifrump: A Tale of Two Mathematicians.

It is worth noting that the Frump-gon generalization is not always portrayed negatively in popular culture. In the fictional novel, The Wyvern's Code, the Frump-gon generalization is revealed to be a strawman of the true Frump-gon generalization, and that Frump was assassinated shortly after creating the original generalisation to be replaced by a sentient android in order to protect Hollandie's reputation so that he could eventually achieve world domination.

The mystery surrounding the original proof of Hollandie's Tiling Lemma as written in the letters to Ecelliorin has also spawned a number of movies and novels exploring numerous conspiracy theories about what those letters may have contained. The most notable of these movies would be Broken Arrows, a nuclear thriller where it is revealed that Hollandie was secretly working as a spy in the Eastern American Union in the buildup to the Trans-Atlantic Conflict, and that his letters contained military secrets compromising the defense of the East American Union. Critics praised the movie as both stunning for its special effects, and the sense of mystery it leaves behind. Greensburg Times film critic, Tom Mackaylin commented, "The movie leaves you feeling satisfied, but also manages to make the audience wonder, "what if?" all the way to the cinema exit."