How nearly equal an odd number of triangles can be

No odd number of equal triangles proved one of the strangest theorems in elementary geometry. A square can be cut into two triangles of equal area, or four, or any even number, by cutting it into strips and each strip along a diagonal. It cannot be cut into three triangles of equal area, or five, or any odd number, however the triangles are shaped and arranged. Paul Monsky’s proof of 1970 colours every point of the plane by how divisible its coordinates are by two and shows that some triangle must have an area that is not $1/n$ for odd $n$ — not because it is too large or too small by any visible amount, but for reasons of arithmetic.

That leaves a question the theorem says nothing about. If the areas cannot all be $1/n$, how close can they come? The theorem rules out a spread of nought between the largest and smallest triangle; it does not say whether the spread must be a quarter, or a hundredth, or a millionth. This essay answers by search, for three, five, seven and nine triangles, and finds both how fast the spread can shrink and a closed form for the best five-triangle dissection it found.

Three triangles: a half and two quarters #

With three triangles there is almost no freedom. Three triangles have only nine corners between them, and four of those must be the square’s own corners. The best the search found, in hundreds of starts, is the dissection on the left of the figure: one triangle of area a half resting on the top side with its apex on the bottom, and two triangles of a quarter beside it, a spread of a quarter.

The obvious route to equal thirds visibly fails. A triangle of area a third resting on a whole side of the square must have its apex two thirds of the way across. If the apex is inside the square, what is left is a pentagon, which cannot be cut into fewer than three triangles. If the apex is on one of the neighbouring sides, what is left is a quadrilateral, and either of its diagonals cuts it into triangles of a half and a sixth. A square is too small a polygon for three triangles to share evenly, and the search’s quarter is the price.

Five triangles, and the golden ratio #

With five triangles the freedom is real. A dissection can have vertices on the sides of the square as well as its corners, and vertices inside it, and the number of each is tied to the number of triangles: with $i$ vertices inside the square there must be $7 - 2i$ on its boundary, corners included. Within each arrangement of which vertices form which triangles, the vertices can slide — a boundary vertex along its side, an interior one anywhere — and each area changes as they do.

The search finds the best by brute force. It picks vertices at random, joins them by the Delaunay triangulation to fix an arrangement, and if the arrangement has five triangles, it moves the vertices one at a time, keeping any move that shrinks the spread and refusing any that turns a triangle inside out. Then it starts again, hundreds of times.

The best it found has a spread of 0.022543: three triangles of area 0.190983 and two of 0.213525. Those numbers are recognisable. The small area is $(3 - \sqrt5)/4$, which is $1/(2\varphi^2)$ where $\varphi$ is the golden ratio, the large one is $(3\sqrt5 - 5)/8$, and the spread is

$$\frac{5\sqrt5 - 11}{8} = 0.0225425\ldots,$$

agreeing with the search to five significant figures. The interior vertex sits 0.3820 from the left side of the square — $1/\varphi^2$ again. The golden ratio appears in a problem about a square and areas, for the reason it usually appears: the optimum balances proportions that must reproduce themselves, here three equal small areas against two equal large ones, and the equations that balance them come down to the quadratic whose root is $\varphi$. Its continued fraction is all ones, which is a different expression of the same self-reproduction.

Whether this is the best possible five-triangle dissection the search cannot say; it is the best of the arrangements it reached. But the closed form is strong evidence that it is a genuine optimum of its arrangement rather than an accident of where the search stopped, since a search that stops early does not land on algebraic numbers.

The spread as one vertex moves #

What stops the spread from reaching nought at the optimum is visible by moving one vertex.

Each triangle’s area is a linear function of any one vertex’s position, so as the vertex moves, every area changes in a straight line, and the spread — the largest minus the smallest — is made of straight pieces. At the best position the curves come to a corner, where the identity of the largest or the smallest triangle changes. The corner is at 0.0225, not at nought. Moving one vertex enlarges some triangles and shrinks others in fixed proportions, and no position of it balances all five; the best position is where the triangles it cannot balance are as close as they can be made.

That is the mechanism of every optimum the search finds, and the next figure shows its signature in the areas themselves.

At the best five, two areas sit exactly at the largest value and three exactly at the smallest; at the best seven, three at the largest, three at the smallest and one between. That clustering is what an optimum of a largest-minus-smallest objective looks like: if only one triangle were largest, some small move would shrink it without making anything else as large, and the spread would fall. Equality is impossible, so the best the areas can do is form two tight groups, one just above $1/n$ and one just below.

How fast the spread falls #

The best spreads found are a quarter for three triangles, 0.0225 for five, 0.00307 for seven and 0.000146 for nine. Each step of two triangles cuts the spread by a factor of 11, then 7, then 21: on a logarithmic scale the points fall roughly along a line, which is what exponential decrease looks like. For even $n$, by contrast, the spread is exactly nought. Nothing in the picture explains why the two kinds of $n$ behave so differently, because the difference is not geometric. It is Monsky’s 2-adic colouring, and the geometric figures show only its shadow: the odd cases can get as close to equality as anyone likes and never arrive.

The values at seven and nine are less certain than at five. Different runs of the search found 0.003071 and 0.003077 for seven, and for nine the best of several runs differed by a factor of six. Published searches — Jean-Philippe Labbé, Günter Rote and Günter Ziegler have carried out the most thorough ones — have done at least as well as these, and the true minimum for each $n$ is known only where such a search has been pushed to the end and the result checked by exact algebra. A search can show that a spread is achievable; it cannot show that nothing smaller is.

Close in one sense, far in another #

The reason a spread can be tiny but never nought is that Monsky’s obstruction is not measured in the way the spread is. His proof colours every point of the plane with one of three colours according to the 2-adic sizes of its coordinates — how many times 2 divides them, extended to all real numbers by a valuation that agrees with that count on fractions. The colouring is arranged so that the square’s corners carry all three colours in a pattern that forces, by Sperner’s lemma, a triangle in any dissection whose three corners are coloured differently. The area of such a triangle is 2-adically large: its 2-adic absolute value is at least 2. And $1/n$, for odd $n$, has 2-adic absolute value exactly 1. So that triangle’s area is not $1/n$.

The 2-adic absolute value measures something unrelated to ordinary size. In it, a number is small when it is divisible by a high power of 2, so 1024 is tiny and $1/1024$ is huge, and the series $1 + 2 + 4 + 8 + \cdots$ converges, to minus one. Two real numbers can be extremely close in the ordinary sense and far apart 2-adically, and the near-equal dissections exploit exactly that. The rainbow triangle’s area is within two hundredths of $1/5$ in the five-triangle dissection above, and within a ten-thousandth of $1/9$ in the nine; it is always 2-adically far from $1/n$, because the colouring guarantees it, and it can be as near as anyone likes in the ordinary sense, because the colouring does not care.

That is also why lower bounds on the spread are so weak. To turn “2-adically different” into “differs by at least so much”, an argument has to use the fact that the areas are built from finitely many coordinates by a bounded number of multiplications and additions, so that a nonzero difference cannot be smaller than a bound depending on how complicated those coordinates are. Such bounds exist, and they are astronomically small, because a sufficiently complicated arrangement of vertices can make a nonzero quantity extremely small. The searches find spreads that are small, but nowhere near that small.

Squares, trapezoids and regular polygons #

Monsky’s theorem is about the square, and the question of which polygons can be cut into $m$ triangles of equal area — equidissections — has different answers for different shapes. Any triangle can be cut into $m$ equal triangles for every $m$, by dividing one side into $m$ equal parts. A parallelogram behaves like the square: even numbers only. Elaine Kasimatis showed in 1989 that a regular polygon with five or more sides can be cut into $m$ triangles of equal area only when $m$ is a multiple of its number of sides; a regular pentagon, for example, can be cut into 5, 10 or 15 equal triangles, but not 6 or 7. For trapezoids, the answer depends on the ratio of the parallel sides, and for some ratios no equidissection exists at all.

Each of these is proved the same way, with a valuation chosen to suit the polygon’s coordinates and a Sperner-type count, and each leaves the same quantitative question open: for the forbidden numbers of triangles, how close to equal can the areas come? The square is the case that has been searched most thoroughly, and even there, as the figures show, the answer is known only by example.

What a search can and cannot prove #

Of 780 starts that produced five triangles, ten reached the best value. Most settled at arrangements whose own best is several times worse, some more than ten times worse, because the search moves vertices within an arrangement and cannot change which vertices form which triangles. Each arrangement has its own optimum, and the global optimum is the best of those. A search reaches it only if it starts in the right arrangement, which is why the search restarts hundreds of times and why the number of arrangements, which grows quickly with $n$, is what makes nine triangles so much harder than five.

The search has a second blind spot, subtler than the first. It chooses arrangements by drawing random vertices and joining them by their Delaunay triangulation, the triangulation in which no vertex lies inside any triangle’s circumcircle. Once an arrangement is chosen, its vertices can move anywhere that keeps every triangle the right way round, and they often end far from Delaunay. But an arrangement can only be chosen if it is the Delaunay triangulation of some starting positions, and not every way of joining vertices into triangles is. An arrangement that is never Delaunay — whose triangles can be realised by some positions but never by positions with empty circumcircles — is invisible to this search however long it runs. Searches designed to be exhaustive enumerate the arrangements combinatorially instead, and then optimise each one, which is the approach the published searches take.

That limitation is the same one met in the count of pieces a dissection needs: searches produce dissections with few pieces, and nothing proves that fewer are impossible. For near-equal triangles the upper bounds — dissections actually found — come from search and construction, and lower bounds come from arithmetic. An argument in the style of Monsky’s shows that some triangle’s area differs from $1/n$, and with more work it bounds how small the difference can be; but such bounds are extremely small, far smaller than anything the searches find. Between the arithmetic floor and the searched ceiling lies a gap of many orders of magnitude, and the true behaviour of the best spread as $n$ grows is not known.

Why even numbers are so easy #

The contrast with even $n$ deserves its own sentence. Four triangles of equal area: cut the square along both diagonals. Six: cut it into three equal strips and each strip along a diagonal. Any even number works the same way, and the construction is so simple that the theorem for odd numbers can seem perverse. Equal area is enough for cutting polygons into each other — any two polygons of equal area can be cut into the same pieces — and yet equal area cannot be shared out equally among an odd number of triangles.

The areas of the best odd dissections always straddle $1/n$ in a balanced way, because they must add up to the whole square. In the five-triangle optimum the three small triangles each fall short of $1/5$ by 0.00902 and the two large ones each exceed it by 0.01353, and three times the first is exactly two times the second, 0.0271: what the small triangles lack, the large ones carry. A spread of nought would need both deficits to vanish at once, and the arithmetic says the deficits can be pushed down together but never both to nought. In an even dissection the same bookkeeping balances at zero, with every triangle exactly $1/n$.

The two facts are about different things. The Bolyai–Gerwien theorem allows any pieces, and cuts them freely; Monsky’s theorem restricts the pieces to triangles and asks that their number be fixed. Equally a rectangle made only of squares can be cut into squares exactly when its sides are commensurable — another case where a restriction on the shape of the pieces turns a geometric question into an arithmetic one. In each of these the arithmetic decides existence, and the geometry decides only how close a near miss can come.

Still open: the rate, and the arrangements #

The rate at which the best spread for $n$ odd triangles goes to nought is not known. Constructions by Bernd Schulze and by Labbé, Rote and Ziegler give upper bounds that decrease rapidly, and the searches suggest that the best spreads decrease faster still; the lower bounds that come from arithmetic are vastly smaller, and closing the gap would require either constructions that approach equality as fast as the arithmetic permits or arithmetic arguments that explain why they cannot.

Nor is there a description of which arrangements of vertices give the best spreads. At five triangles the best arrangement found has one interior vertex and a closed form involving $\sqrt5$; at seven and nine the search reports numbers, and whether they too are algebraic numbers of small degree, with a pattern behind them, is not known. A structural answer — a family of arrangements, one for each odd $n$, whose spreads can be computed exactly — would turn the searched ceiling into a theorem, and none has been found. Even the smallest open case is instructive: for seven triangles the best arrangement found has three triangles at one area, three at another and one between, and whether its spread is an algebraic number as simple as the five-triangle $(5\sqrt5 - 11)/8$ is a question the search leaves open.