Equidissection
[4] They are considered interesting because the results are counterintuitive at first, and for a geometry problem with such a simple definition, the theory requires some surprisingly sophisticated algebraic tools.
Some authors restrict their attention to simplicial dissections, especially in the secondary literature, since they are easier to work with.
[9] Affine transformations of the plane are useful for studying equidissections, including translations, uniform and non-uniform scaling, reflections, rotations, shears, and other similarities and linear maps.
This means that one is free to apply any affine transformation to a polygon that might give it a more manageable form.
[10] The fact that affine transformations preserve equidissections also means that certain results can be easily generalized.
The latter follows mutatis mutandis from the proof for the octahedron in [2] Let T(a) be a trapezoid where a is the ratio of parallel side lengths.
[16] More generally, all convex polygons with rational coordinates can be equidissected,[17] although not all of them are principal; see the above example of a kite with a vertex at (3/2, 3/2).
[20] It is conjectured that a similar condition involving stable polynomials may determine whether or not the spectrum is empty for algebraic numbers a of all degrees.
Aigner & Ziegler (2010) remark of Monsky's theorem, "one could have guessed that surely the answer must have been known for a long time (if not to the Greeks).
"[22] But the study of equidissections did not begin until 1965, when Fred Richman was preparing a master's degree exam at New Mexico State University.
Richman wanted to include a question on geometry in the exam, and he noticed that it was difficult to find (what is now called) an odd equidissection of a square.
When nobody else submitted a solution, the proof was published in Mathematics Magazine (Thomas 1968), three years after it was written.
Monsky (1970) then built on Thomas' argument to prove that there are no odd equidissections of a square, without any rationality assumptions.
A clever coloring of the plane then implies that in all dissections of the square, at least one triangle has an area with what amounts to an even denominator, and therefore all equidissections must be even.
The essence of the argument is found already in Thomas (1968), but Monsky (1970) was the first to use a 2-adic valuation to cover dissections with arbitrary coordinates.
Generalization to regular polygons arrived in 1985, during a geometry seminar run by G. D. Chakerian at UC Davis.
[6] Sherman Stein suggested dissections of the square and the cube: "a topic that Chakerian grudgingly admitted was geometric.
[2] Kasimatis & Stein (1990) began the study of the spectra of two particular generalizations of squares: trapezoids and kites.
Several papers have been authored at Hebei Normal University, chiefly by Professor Ding Ren and his students Du Yatao and Su Zhanjun.
[28] Attempting to generalize the results for regular n-gons for even n, Stein (1989) conjectured that no centrally symmetric polygon has an odd equidissection, and he proved the n = 6 and n = 8 cases.
A decade later, Stein made what he describes as "a surprising breakthrough", conjecturing that no polyomino has an odd equidissection.
The topic of equidissections has recently been popularized by treatments in The Mathematical Intelligencer (Stein 2004), a volume of the Carus Mathematical Monographs (Stein & Szabó 2008), and the fourth edition of Proofs from THE BOOK (Aigner & Ziegler 2010).
Then M(n) is 0 for even n and greater than 0 for odd n. Mansow (2003) gave the asymptotic upper bound M(n) = O(1/n2) (see Big O notation).
[29] Schulze (2011) improves the bound to M(n) = O(1/n3) with a better dissection, and he proves that there exist values of a for which M(a, n) decreases arbitrarily quickly.
Labbé, Rote & Ziegler (2018) obtain a superpolynomial upper bound, derived from an explicit construction that uses the Thue–Morse sequence.
