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Equidissection

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In <a href="Geometry" title="Geometry">geometry</a>, an equidissection is a <a href="Partition_of_a_set" title="Partition of a set">partition</a> of a <a href="Polygon" title="Polygon">polygon</a> into <a href="Triangle" title="Triangle">triangles</a> of equal <a href="Area" title="Area">area</a>. The study of equidissections began in the late 1960s with <a href="Monsky's_theorem" title="Monsky's theorem">Monsky's theorem</a>, which states that a <a href="Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a> cannot be equidissected into an odd number of triangles.<a href="#cite_note-FOOTNOTEMonsky1970-1">[1]</a> In fact, <a href="Almost_all" title="Almost all">most</a> polygons cannot be equidissected at all.<a href="#cite_note-FOOTNOTEKasimatisStein1990-2">[2]</a>

Much of the literature is aimed at generalizing Monsky's theorem to broader classes of polygons. The general question is: Which polygons can be equidissected into how many pieces? Particular attention has been given to <a href="Trapezoid" title="Trapezoid">trapezoids</a>, <a href="Kite_(geometry)" title="Kite (geometry)">kites</a>, <a href="Regular_polygon" title="Regular polygon">regular polygons</a>, <a href="Zonogon" title="Zonogon">centrally symmetric polygons</a>, <a href="Polyomino" title="Polyomino">polyominos</a>, and <a href="Hypercube" title="Hypercube">hypercubes</a>.<a href="#cite_note-FOOTNOTEStein2004-3">[3]</a>

Equidissections do not have many direct applications.<a href="#cite_note-FOOTNOTESteinSzabó2008108–109-4">[4]</a> 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. Many of the results rely upon extending <a href="P-adic_valuation" title="P-adic valuation">p-adic valuations</a> to the <a href="Real_number" title="Real number">real numbers</a> and extending <a href="Sperner's_lemma" title="Sperner's lemma">Sperner's lemma</a> to more general <a href="Colored_graph" class="mw-redirect" title="Colored graph">colored graphs</a>.<a href="#cite_note-FOOTNOTEStein200417-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definitions">Definitions</h3></div>

A dissection of a polygon P is a <a href="Finite_set" title="Finite set">finite set</a> of triangles that do not overlap and whose union is all of P. A dissection into n triangles is called an n-dissection, and it is classified as an even dissection or an odd dissection according to whether n is <a href="Parity_(mathematics)" title="Parity (mathematics)">even or odd</a>.<a href="#cite_note-FOOTNOTEStein200417-5">[5]</a>

An equidissection is a dissection in which every triangle has the same area. For a polygon P, the set of all n for which an n-equidissection of P exists is called the spectrum of P and denoted S(P). A general theoretical goal is to compute the spectrum of a given polygon.<a href="#cite_note-FOOTNOTESteinSzabó2008120-6">[6]</a>

A dissection is called <a href="Simplicial_complex" title="Simplicial complex">simplicial</a> if the triangles meet only along common edges. Some authors restrict their attention to simplicial dissections, especially in the secondary literature, since they are easier to work with. For example, the usual statement of Sperner's lemma applies only to simplicial dissections. Often simplicial dissections are called <a href="Polygon_triangulation" title="Polygon triangulation">triangulations</a>, although the vertices of the triangles are not restricted to the vertices or edges of the polygon. Simplicial equidissections are therefore also called equal-area triangulations.<a href="#cite_note-FOOTNOTESchulze2011-7">[7]</a>

The terms can be extended to higher-dimensional <a href="Polytope" title="Polytope">polytopes</a>: an equidissection is set of <a href="Simplex" title="Simplex">simplexes</a> having the same n-volume.<a href="#cite_note-FOOTNOTEMead1979302-8">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="Preliminaries">Preliminaries</h3></div>

It is easy to find an n-equidissection of a triangle for all n. As a result, if a polygon has an m-equidissection, then it also has an mn-equidissection for all n. In fact, often a polygon's spectrum consists precisely of the multiples of some number m; in this case, both the spectrum and the polygon are called principal and the spectrum is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle m\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle m\rangle }</annotation>

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</math><img src="./c3fd17b4da296dfb3b179940f91f0e2b02ae23fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.85ex; height:2.843ex;" alt="{\displaystyle \langle m\rangle }" loading="lazy">.<a href="#cite_note-FOOTNOTEKasimatisStein1990-2">[2]</a> For example, the spectrum of a triangle is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 1\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle 1\rangle }</annotation>

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</math><img src="./9885b032f65684d9cd6c5802a174ad9cf8648fef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \langle 1\rangle }" loading="lazy">. A simple example of a non-principal polygon is the quadrilateral with vertices (0, 0), (1, 0), (0, 1), (3/2, 3/2); its spectrum includes 2 and 3 but not 1.<a href="#cite_note-FOOTNOTESteinSzabó2008126-9">[9]</a>

<a href="Affine_transformation" title="Affine transformation">Affine transformations</a> of the plane are useful for studying equidissections, including <a href="Translation_(geometry)" title="Translation (geometry)">translations</a>, uniform and non-uniform <a href="Scaling_(geometry)" title="Scaling (geometry)">scaling</a>, <a href="Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a>, <a href="Rotation" title="Rotation">rotations</a>, <a href="Shear_mapping" title="Shear mapping">shears</a>, and other <a href="Similarity_(geometry)" title="Similarity (geometry)">similarities</a> and <a href="Linear_map" title="Linear map">linear maps</a>. Since an affine transformation preserves straight lines and ratios of areas, it sends equidissections to equidissections. This means that one is free to apply any affine transformation to a polygon that might give it a more manageable form. For example, it is common to choose coordinates such that three of the vertices of a polygon are (0, 1), (0, 0), and (1, 0).<a href="#cite_note-FOOTNOTESteinSzabó2008121,_128,_131-10">[10]</a>

The fact that affine transformations preserve equidissections also means that certain results can be easily generalized. All results stated for a regular polygon also hold for <a href="Affine-regular_polygon" title="Affine-regular polygon">affine-regular polygons</a>; in particular, results concerning the <a href="Unit_square" title="Unit square">unit square</a> also apply to other parallelograms, including <a href="Rectangle" title="Rectangle">rectangles</a> and <a href="Rhombus" title="Rhombus">rhombuses</a>. All results stated for polygons with <a href="Integer" title="Integer">integer</a> coordinates also apply to polygons with <a href="Rational_number" title="Rational number">rational</a> coordinates, or polygons whose vertices fall on any other <a href="Lattice_(group)" title="Lattice (group)">lattice</a>.<a href="#cite_note-FOOTNOTEStein200412–20-11">[11]</a>

<div class="mw-heading mw-heading3"><h3 id="Best_results">Best results</h3></div>

<a href="Monsky's_theorem" title="Monsky's theorem">Monsky's theorem</a> states that a square has no odd equidissections, so its spectrum is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 2\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle 2\rangle }</annotation>

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</math><img src="./992716a16cf8643ff23b13988e7f83583766531b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \langle 2\rangle }" loading="lazy">.<a href="#cite_note-FOOTNOTEMonsky1970-1">[1]</a> More generally, it is known that <a href="Centrally_symmetric" class="mw-redirect" title="Centrally symmetric">centrally symmetric</a> polygons and <a href="Polyomino" title="Polyomino">polyominos</a> have no odd equidissections.<a href="#cite_note-12">[12]</a> A conjecture by <a href="Sherman_K._Stein" title="Sherman K. Stein">Sherman K. Stein</a> proposes that no special polygon has an odd equidissection, where a special polygon is one whose <a href="Equivalence_class" title="Equivalence class">equivalence classes</a> of <a href="Parallel_(geometry)" title="Parallel (geometry)">parallel</a> edges each sum to the <a href="Zero_vector" class="mw-redirect" title="Zero vector">zero vector</a>. Squares, <a href="Zonogon" title="Zonogon">centrally symmetric polygons</a>, <a href="Polyomino" title="Polyomino">polyominos</a>, and <a href="Polyhex_(mathematics)" title="Polyhex (mathematics)">polyhexes</a> are all special polygons.<a href="#cite_note-FOOTNOTEStein200420-13">[13]</a>

For n > 4, the spectrum of a regular n-gon is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle n\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle n\rangle }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \langle n!\rangle }</annotation>

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</math><img src="./ac63fffaff193cf331059d73166ef8453e5da97a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.851ex; height:2.843ex;" alt="{\displaystyle \langle n!\rangle }" loading="lazy">, where n! is the <a href="Factorial" title="Factorial">factorial</a> of n.<a href="#cite_note-FOOTNOTEMead1979-15">[15]</a> and the spectrum of an n-dimensional <a href="Cross-polytope" title="Cross-polytope">cross-polytope</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 2^{n-1}\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle 2^{n-1}\rangle }</annotation>

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</math><img src="./34c25531d1cfa5b30e4f6f7588bf2fbd818f3405.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.291ex; height:3.176ex;" alt="{\displaystyle \langle 2^{n-1}\rangle }" loading="lazy">. The latter follows <a href="Mutatis_mutandis" title="Mutatis mutandis">mutatis mutandis</a> from the proof for the octahedron in <a href="#cite_note-FOOTNOTEKasimatisStein1990-2">[2]</a>

Let T(a) be a <a href="Trapezoid" title="Trapezoid">trapezoid</a> where a is the ratio of parallel side lengths. If a is a <a href="Rational_number" title="Rational number">rational number</a>, then T(a) is principal. In fact, if r/s is a fraction in lowest terms, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(T(r/s))=\langle r+s\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle S(T(r/s))=\langle r+s\rangle }</annotation>

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</math><img src="./fb332b292056aa87bea3a0e533dcfe59ec64bfb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.943ex; height:2.843ex;" alt="{\displaystyle S(T(r/s))=\langle r+s\rangle }" loading="lazy">.<a href="#cite_note-FOOTNOTESteinSzabó2008122-16">[16]</a> More generally, all <a href="Convex_polygon" title="Convex polygon">convex polygons</a> with rational coordinates can be equidissected,<a href="#cite_note-FOOTNOTESuDing2003-17">[17]</a> although not all of them are principal; see the above example of a kite with a vertex at (3/2, 3/2).

At the other extreme, if a is a <a href="Transcendental_number" title="Transcendental number">transcendental number</a>, then T(a) has no equidissection. More generally, no polygon whose vertex coordinates are <a href="Algebraically_independent" class="mw-redirect" title="Algebraically independent">algebraically independent</a> has an equidissection.<a href="#cite_note-18">[18]</a> This means that <a href="Almost_all" title="Almost all">almost all</a> polygons with more than three sides cannot be equidissected. Although most polygons cannot be cut into equal-area triangles, all polygons can be cut into equal-area quadrilaterals.<a href="#cite_note-FOOTNOTEHalesStraus198242-19">[19]</a>

If a is an <a href="Algebraic_number" title="Algebraic number">algebraic</a> <a href="Irrational_number" title="Irrational number">irrational number</a>, then T(a) is a trickier case. If a is algebraic of <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> 2 or 3 (<a href="Quadratic_irrational" class="mw-redirect" title="Quadratic irrational">quadratic</a> or cubic), and its <a href="Conjugate_element_(field_theory)" title="Conjugate element (field theory)">conjugates</a> all have positive <a href="Real_part" class="mw-redirect" title="Real part">real parts</a>, then S(T(a)) contains all sufficiently large n such that n/(1 + a) is an <a href="Algebraic_integer" title="Algebraic integer">algebraic integer</a>.<a href="#cite_note-FOOTNOTEJepsenMonsky2008-20">[20]</a> It is conjectured that a similar condition involving <a href="Stable_polynomial" title="Stable polynomial">stable polynomials</a> may determine whether or not the spectrum is empty for algebraic numbers a of all degrees.<a href="#cite_note-21">[21]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

The idea of an equidissection seems like the kind of elementary geometric concept that should be quite old. <a href="#CITEREFAignerZiegler2010">Aigner & Ziegler (2010)</a> 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)."<a href="#cite_note-FOOTNOTEAignerZiegler2010131-22">[22]</a> But the study of equidissections did not begin until 1965, when Fred Richman was preparing a <a href="Master's_degree" title="Master's degree">master's degree</a> exam at <a href="New_Mexico_State_University" title="New Mexico State University">New Mexico State University</a>.

<div class="mw-heading mw-heading3"><h3 id="Monsky's_theorem">Monsky's theorem</h3></div>

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. Richman proved to himself that it was impossible for 3 or 5, that the existence of an n-equidissection implies the existence of an (n + 2)-dissection, and that certain quadrilaterals arbitrarily close to being squares have odd equidissections.<a href="#cite_note-FOOTNOTEThomas1968187-23">[23]</a> However, he did not solve the general problem of odd equidissections of squares, and he left it off the exam. Richman's friend John Thomas became interested in the problem; in his recollection,

<dl><dd>"Everyone to whom the problem was put (myself included) said something like 'that is not my area but the question surely must have been considered and the answer is probably well known.' Some thought they had seen it, but could not remember where. I was interested because it reminded me of <a href="Sperner's_Lemma" class="mw-redirect" title="Sperner's Lemma">Sperner's Lemma</a> in <a href="Topology" title="Topology">topology</a>, which has a clever odd-even proof."<a href="#cite_note-FOOTNOTESteinSzabó2008107-24">[24]</a></dd></dl>

Thomas proved that an odd equidissection was impossible if the coordinates of the vertices are rational numbers with odd denominators. He submitted this proof to <a href="Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a>, but it was put on hold:

<dl><dd>"The referee's reaction was predictable. He thought the problem might be fairly easy (although he could not solve it) and was possibly well-known (although he could find no reference to it)."<a href="#cite_note-FOOTNOTESteinSzabó2008108-25">[25]</a></dd></dl>

The question was instead given as an Advanced Problem in the <a href="American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a> (<a href="#CITEREFRichmanThomas1967">Richman & Thomas 1967</a>). When nobody else submitted a solution, the proof was published in Mathematics Magazine (<a href="#CITEREFThomas1968">Thomas 1968</a>), three years after it was written. <a href="#CITEREFMonsky1970">Monsky (1970)</a> then built on Thomas' argument to prove that there are no odd equidissections of a square, without any rationality assumptions.<a href="#cite_note-FOOTNOTESteinSzabó2008108-25">[25]</a>

Monsky's proof relies on two pillars: a <a href="Combinatorics" title="Combinatorics">combinatorial</a> result that generalizes Sperner's lemma and an <a href="Algebra" title="Algebra">algebraic</a> result, the existence of a <a href="P-adic_valuation" title="P-adic valuation">2-adic valuation</a> on the real numbers. A clever <a href="Graph_coloring" title="Graph coloring">coloring</a> 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 <a href="#CITEREFThomas1968">Thomas (1968)</a>, but <a href="#CITEREFMonsky1970">Monsky (1970)</a> was the first to use a 2-adic valuation to cover dissections with arbitrary coordinates.<a href="#cite_note-26">[26]</a>

<div class="mw-heading mw-heading3"><h3 id="Generalizations">Generalizations</h3></div>

The first generalization of Monsky's theorem was <a href="#CITEREFMead1979">Mead (1979)</a>, who proved that the spectrum of an n-dimensional cube is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle n!\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle n!\rangle }</annotation>

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Generalization to regular polygons arrived in 1985, during a geometry seminar run by G. D. Chakerian at <a href="University_of_California%2C_Davis" title="University of California, Davis">UC Davis</a>. <a href="Elaine_Kasimatis" title="Elaine Kasimatis">Elaine Kasimatis</a>, a graduate student, "was looking for some algebraic topic she could slip into" the seminar.<a href="#cite_note-FOOTNOTESteinSzabó2008120-6">[6]</a> Sherman Stein suggested dissections of the square and the cube: "a topic that Chakerian grudgingly admitted was geometric."<a href="#cite_note-FOOTNOTESteinSzabó2008120-6">[6]</a> After her talk, Stein asked about regular pentagons. Kasimatis answered with <a href="#CITEREFKasimatis1989">Kasimatis (1989)</a>, proving that for n > 5, the spectrum of a regular n-gon is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle n\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle n\rangle }</annotation>

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</math><img src="./591b81beaaf66f9e21fe885d4a9521c0390bee67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.204ex; height:2.843ex;" alt="{\displaystyle \langle n\rangle }" loading="lazy">. Her proof builds on Monsky's proof, extending the p-adic valuation to the complex numbers for each prime <a href="Divisor" title="Divisor">divisor</a> of n and applying some elementary results from the theory of <a href="Cyclotomic_field" title="Cyclotomic field">cyclotomic fields</a>. It is also the first proof to explicitly use an affine transformation to set up a convenient coordinate system.<a href="#cite_note-FOOTNOTEStein200418-27">[27]</a> <a href="#CITEREFKasimatisStein1990">Kasimatis & Stein (1990)</a> then framed the problem of finding the spectrum of a general polygon, introducing the terms spectrum and principal.<a href="#cite_note-FOOTNOTESteinSzabó2008120-6">[6]</a> They proved that almost all polygons lack equidissections, and that not all polygons are principal.<a href="#cite_note-FOOTNOTEKasimatisStein1990-2">[2]</a>

<a href="#CITEREFKasimatisStein1990">Kasimatis & Stein (1990)</a> began the study of the spectra of two particular generalizations of squares: trapezoids and kites. Trapezoids have been further studied by <a href="#CITEREFJepsen1996">Jepsen (1996)</a>, <a href="#CITEREFMonsky1996">Monsky (1996)</a>, and <a href="#CITEREFJepsenMonsky2008">Jepsen & Monsky (2008)</a>. Kites have been further studied by <a href="#CITEREFJepsenSedberryHoyer2009">Jepsen, Sedberry & Hoyer (2009)</a>. General quadrilaterals have been studied in <a href="#CITEREFSuDing2003">Su & Ding (2003)</a>. Several papers have been authored at <a href="Hebei_Normal_University" title="Hebei Normal University">Hebei Normal University</a>, chiefly by Professor Ding Ren and his students Du Yatao and Su Zhanjun.<a href="#cite_note-28">[28]</a>

Attempting to generalize the results for regular n-gons for even n, <a href="#CITEREFStein1989">Stein (1989)</a> conjectured that no centrally symmetric polygon has an odd equidissection, and he proved the n = 6 and n = 8 cases. The full conjecture was proved by <a href="#CITEREFMonsky1990">Monsky (1990)</a>. A decade later, Stein made what he describes as "a surprising breakthrough", conjecturing that no polyomino has an odd equidissection. He proved the result of a polyomino with an odd number of squares in <a href="#CITEREFStein1999">Stein (1999)</a>. The full conjecture was proved when <a href="#CITEREFPraton2002">Praton (2002)</a> treated the even case.

The topic of equidissections has recently been popularized by treatments in <a href="The_Mathematical_Intelligencer" title="The Mathematical Intelligencer">The Mathematical Intelligencer</a> (<a href="#CITEREFStein2004">Stein 2004</a>), a volume of the <a href="Carus_Mathematical_Monographs" title="Carus Mathematical Monographs">Carus Mathematical Monographs</a> (<a href="#CITEREFSteinSzabó2008">Stein & Szabó 2008</a>), and the fourth edition of <a href="Proofs_from_THE_BOOK" title="Proofs from THE BOOK">Proofs from THE BOOK</a> (<a href="#CITEREFAignerZiegler2010">Aigner & Ziegler 2010</a>).

<div class="mw-heading mw-heading2"><h2 id="Related_problems">Related problems</h2></div>

<a href="#CITEREFSakaiNaraUrrutia2005">Sakai, Nara & Urrutia (2005)</a> consider a variation of the problem: Given a convex polygon K, how much of its area can be covered by n non-overlapping triangles of equal area inside K? The ratio of the area of the best possible coverage to the area of K is denoted tn(K). If K has an n-equidissection, then tn(K) = 1; otherwise it is less than 1. The authors show that for a quadrilateral K, tn(K) ≥ 4n/(4n + 1), with t2(K) = 8/9 if and only if K is affinely congruent to the trapezoid T(2/3). For a pentagon, t2(K) ≥ 2/3, t3(K) ≥ 3/4, and tn(K) ≥ 2n/(2n + 1) for n ≥ 5.

<a href="G%C3%BCnter_M._Ziegler" title="Günter M. Ziegler">Günter M. Ziegler</a> asked the converse problem in 2003: Given a dissection of the whole of a polygon into n triangles, how close can the triangle areas be to equal? In particular, what is the smallest possible difference between the areas of the smallest and largest triangles? Let the smallest difference be M(n) for a square and M(a, n) for the trapezoid T(a). Then M(n) is 0 for even n and greater than 0 for odd n. <a href="#CITEREFMansow2003">Mansow (2003)</a> gave the asymptotic upper bound M(n) = O(1/n2) (see <a href="Big_O_notation" title="Big O notation">Big O notation</a>).<a href="#cite_note-FOOTNOTESchulze20112-29">[29]</a> <a href="#CITEREFSchulze2011">Schulze (2011)</a> 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. <a href="#CITEREFLabbéRoteZiegler2018">Labbé, Rote & Ziegler (2018)</a> obtain a superpolynomial upper bound, derived from an explicit construction that uses the <a href="Thue%E2%80%93Morse_sequence" title="Thue–Morse sequence">Thue–Morse sequence</a>.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-FOOTNOTEMonsky1970-1">^ <a href="#cite_ref-FOOTNOTEMonsky1970_1-0">a</a> <a href="#cite_ref-FOOTNOTEMonsky1970_1-1">b</a> <a href="#CITEREFMonsky1970">Monsky 1970</a>.

</li>

<li id="cite_note-FOOTNOTEKasimatisStein1990-2">^ <a href="#cite_ref-FOOTNOTEKasimatisStein1990_2-0">a</a> <a href="#cite_ref-FOOTNOTEKasimatisStein1990_2-1">b</a> <a href="#cite_ref-FOOTNOTEKasimatisStein1990_2-2">c</a> <a href="#cite_ref-FOOTNOTEKasimatisStein1990_2-3">d</a> <a href="#CITEREFKasimatisStein1990">Kasimatis & Stein 1990</a>.

</li>

<li id="cite_note-FOOTNOTEStein2004-3"><a href="#cite_ref-FOOTNOTEStein2004_3-0">^</a> <a href="#CITEREFStein2004">Stein 2004</a>.

</li>

<li id="cite_note-FOOTNOTESteinSzabó2008108–109-4"><a href="#cite_ref-FOOTNOTESteinSzabó2008108–109_4-0">^</a> <a href="#CITEREFSteinSzabó2008">Stein & Szabó 2008</a>, pp. 108–109.

</li>

<li id="cite_note-FOOTNOTEStein200417-5">^ <a href="#cite_ref-FOOTNOTEStein200417_5-0">a</a> <a href="#cite_ref-FOOTNOTEStein200417_5-1">b</a> <a href="#CITEREFStein2004">Stein 2004</a>, p. 17.

</li>

<li id="cite_note-FOOTNOTESteinSzabó2008120-6">^ <a href="#cite_ref-FOOTNOTESteinSzabó2008120_6-0">a</a> <a href="#cite_ref-FOOTNOTESteinSzabó2008120_6-1">b</a> <a href="#cite_ref-FOOTNOTESteinSzabó2008120_6-2">c</a> <a href="#cite_ref-FOOTNOTESteinSzabó2008120_6-3">d</a> <a href="#CITEREFSteinSzabó2008">Stein & Szabó 2008</a>, p. 120.

</li>

<li id="cite_note-FOOTNOTESchulze2011-7"><a href="#cite_ref-FOOTNOTESchulze2011_7-0">^</a> <a href="#CITEREFSchulze2011">Schulze 2011</a>.

</li>

<li id="cite_note-FOOTNOTEMead1979302-8"><a href="#cite_ref-FOOTNOTEMead1979302_8-0">^</a> <a href="#CITEREFMead1979">Mead 1979</a>, p. 302.

</li>

<li id="cite_note-FOOTNOTESteinSzabó2008126-9"><a href="#cite_ref-FOOTNOTESteinSzabó2008126_9-0">^</a> <a href="#CITEREFSteinSzabó2008">Stein & Szabó 2008</a>, p. 126.

</li>

<li id="cite_note-FOOTNOTESteinSzabó2008121,_128,_131-10"><a href="#cite_ref-FOOTNOTESteinSzabó2008121,_128,_131_10-0">^</a> <a href="#CITEREFSteinSzabó2008">Stein & Szabó 2008</a>, pp. 121, 128, 131.

</li>

<li id="cite_note-FOOTNOTEStein200412–20-11"><a href="#cite_ref-FOOTNOTEStein200412–20_11-0">^</a> <a href="#CITEREFStein2004">Stein 2004</a>, pp. 12–20.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFMonsky1990">Monsky 1990</a>; <a href="#CITEREFPraton2002">Praton 2002</a>

</li>

<li id="cite_note-FOOTNOTEStein200420-13"><a href="#cite_ref-FOOTNOTEStein200420_13-0">^</a> <a href="#CITEREFStein2004">Stein 2004</a>, p. 20.

</li>

<li id="cite_note-FOOTNOTEKasimatis1989-14"><a href="#cite_ref-FOOTNOTEKasimatis1989_14-0">^</a> <a href="#CITEREFKasimatis1989">Kasimatis 1989</a>.

</li>

<li id="cite_note-FOOTNOTEMead1979-15"><a href="#cite_ref-FOOTNOTEMead1979_15-0">^</a> <a href="#CITEREFMead1979">Mead 1979</a>.

</li>

<li id="cite_note-FOOTNOTESteinSzabó2008122-16"><a href="#cite_ref-FOOTNOTESteinSzabó2008122_16-0">^</a> <a href="#CITEREFSteinSzabó2008">Stein & Szabó 2008</a>, p. 122.

</li>

<li id="cite_note-FOOTNOTESuDing2003-17"><a href="#cite_ref-FOOTNOTESuDing2003_17-0">^</a> <a href="#CITEREFSuDing2003">Su & Ding 2003</a>.

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> See <a href="#CITEREFSuDing2003">Su & Ding (2003)</a> for more precise statements of this principle.

</li>

<li id="cite_note-FOOTNOTEHalesStraus198242-19"><a href="#cite_ref-FOOTNOTEHalesStraus198242_19-0">^</a> <a href="#CITEREFHalesStraus1982">Hales & Straus 1982</a>, p. 42.

</li>

<li id="cite_note-FOOTNOTEJepsenMonsky2008-20"><a href="#cite_ref-FOOTNOTEJepsenMonsky2008_20-0">^</a> <a href="#CITEREFJepsenMonsky2008">Jepsen & Monsky 2008</a>.

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFStein2004">Stein 2004</a>, p. 21; <a href="#CITEREFJepsenMonsky2008">Jepsen & Monsky 2008</a>, p. 3

</li>

<li id="cite_note-FOOTNOTEAignerZiegler2010131-22"><a href="#cite_ref-FOOTNOTEAignerZiegler2010131_22-0">^</a> <a href="#CITEREFAignerZiegler2010">Aigner & Ziegler 2010</a>, p. 131.

</li>

<li id="cite_note-FOOTNOTEThomas1968187-23"><a href="#cite_ref-FOOTNOTEThomas1968187_23-0">^</a> <a href="#CITEREFThomas1968">Thomas 1968</a>, p. 187.

</li>

<li id="cite_note-FOOTNOTESteinSzabó2008107-24"><a href="#cite_ref-FOOTNOTESteinSzabó2008107_24-0">^</a> <a href="#CITEREFSteinSzabó2008">Stein & Szabó 2008</a>, p. 107.

</li>

<li id="cite_note-FOOTNOTESteinSzabó2008108-25">^ <a href="#cite_ref-FOOTNOTESteinSzabó2008108_25-0">a</a> <a href="#cite_ref-FOOTNOTESteinSzabó2008108_25-1">b</a> <a href="#CITEREFSteinSzabó2008">Stein & Szabó 2008</a>, p. 108.

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFMonsky1970">Monsky 1970</a>, p. 251; <a href="#CITEREFBekkerNetsvetaev1998">Bekker & Netsvetaev 1998</a>, p. 3492

</li>

<li id="cite_note-FOOTNOTEStein200418-27"><a href="#cite_ref-FOOTNOTEStein200418_27-0">^</a> <a href="#CITEREFStein2004">Stein 2004</a>, p. 18.

</li>

</li>

<li id="cite_note-FOOTNOTESchulze20112-29"><a href="#cite_ref-FOOTNOTESchulze20112_29-0">^</a> <a href="#CITEREFSchulze2011">Schulze 2011</a>, p. 2.

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

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<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110524005026/http://people.fas.harvard.edu/~amathew/HMMT.pdf">Sperner’s Lemma, Brouwer’s Fixed-Point Theorem, And The Subdivision Of Squares Into Triangles</a> - Notes by Akhil Mathew</li>

<li><a rel="nofollow" class="external text" href="http://page.math.tu-berlin.de/~felsner/Lehre/TilingSlides/schmitt_100130_1.pdf">Über die Zerlegung eines Quadrats in Dreiecke gleicher Fläche</a> - Notes by Moritz W. Schmitt (German language)</li>

<li><a rel="nofollow" class="external text" href="http://morkofffkin.narod.ru/diff/tiling.ps">Tiling Polygons by Triangles of Equal Area</a> - Notes by AlexGhitza</li></ul></div><div><div class="zim-footer">

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