IBM Research | Ponder This | December 2003 solutions
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December 2003

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(Part 1):

If |AB|=|DC| then P is a parallelogram, which we already solved. Assume |AB| > |DC| (AB is longer than DC). Let h denote the height, the distance between the parallel lines AB and DC. Continue sides AD and BC until they meet at point E, at height g=h*(|AB|)/(|AB|-|DC|) above AB. Draw a line parallel to AB, at distance √{(g-h)*g} = g*√{|DC|/|AB|} from E. This line partitions P into two trapezoids, similar to each other, one larger than the other by a factor √{|AB|/|DC|}.

(Part 2):

Let P be a convex quadrilateral whose angles are linearly independent over the rationals; that is, no nontrivial integer linear combination of the four angles gives 0. For concreteness, suppose their measures (in degrees) are

α1=180/√(5), α-2=180/√(7), α3=180/√(11), α4=360-α123.

Suppose P admits a "similar dissection" into n tiles, each having k sides. Let the angles of the tile be β1,...,βk, where each βi is strictly between 0 and 360, and none is 180. (A k-gon with an angle of 180 is really a k-1-gon.)

The dissected polygon has (say) a total of m+r+4 vertices: m interior vertices where the angles of the various tiles sum to 360; r vertices where the angles of the various tiles sum to 180 (these vertices can be on a side of P or on a side of another tile, which would contribute the other 180 degrees); and the four vertices A,B,C,D of the original polygon. Since each tile contributes 180(k-2) degrees, we have

360*m+180*r+(α1234)=180*(k-2)*n, 2*m+r+2=(k-2)*n.

We have m+r+4 equations, each expressing an angle (αi, 360 or 180) as a nonnegative integer combination of βj. When βj appears in one of these equations, we say that βj is a "component" of the appropriate angle (αi, 360 or 180).

Suppose first that one of these m interior vertices meets only two tiles, so that two angles sum to 360: βab=360, with βab. So βa>180, and since P is convex, βa is not a component of any αi, nor of any of the r instances of 180. So βa appears n times in our m+r+4 equations, each time as a component of 360. In each of these appearances, if it is not exactly βab=360, but rather βa+(∑ cj βj), we will find another of our equations that contains βb but not βa, and exchange the (∑ cj βj) with the βb, to produce an equation of the form βab=360. Our equations no longer correspond to the dissection, but that's all right.

If we eliminate the n equations of the form βab=360, and eliminate the two angles βa and βb from our tiles, we come up with a set of (m-n)+r+4 equations relating the k-2 remaining tile-angles. It is as if we had started with (k-2)-gons, m-n vertices of 360, r vertices of 180, and the four angles αi.

Continue until there is no equation of the form βij=360.

So we can assume that each of the m equations for 360 involves at least three tile angles (counting multiplicity), and each of the r equations for 180 involves at least two. Among the m+r+4 equations, the number of tile angles (including multiplicity) is then

n*k ≥ 3*m + 2*r + 4.

Combine with the earlier equation

n*(k-2) = 2*m + r + 2,

(subtracting 3 times the equation from 2 times the inequality)

to obtain

n*(6-k) ≥ r+2.

So k is at most 5.

Because no linear relation exists among αi, the four equations expressing αi as sums of βj must involve at least four distinct βj. So k must be at least 4.

If k=4, then from α1234=360=β1234, we must have αii, i=1,2,3,4. Then r must be 0 (no combination of βi gives 180). Our transformations (to get rid of βab=360) did not change r, nor did they decrease the number of tile angles composing αi, so that in the original dissection we must have had r=0 (no tile vertices along the sides of the large polygon) and each angle αi was just one tile angle. So the dissection was a single tile.

Suppose k=5. If αii for each i=1,2,3,4, then from ∑ αi=360 and ∑ βi=540 we would have β5=180, impossible. Not all βj are components of αi, since otherwise ∑ αi=360 would be at least ∑ βi=540. So exactly four of the βj are components of αi, and we can write 360=∑ αi=c11+...+c44, with all cj positive integers. One of them, say c1, is at least 2.

Take one of the occurrences of angle β5, either as one of the m equations expressing 360, or as one of the r equations expressing 180; in the latter case, multiply by 2 to get 360 on the right hand side. We get an equation giving 360 as a nonnegative integer linear combination of β1,...,β5: say 360=d11+...+d55, with d5>0. So we have three equations on the βi:

c11 + ... + c44 = 360, d11 + ... + d55 = 360, β1 + ... + β5 = 540, with ci and dj nonnegative integers and c1 ≥ 2, c2 ≥ 1, c3 ≥ 1, c4 ≥ 1, d5 ≥ 1.

Compute (3*d5-2) times the first equation, plus 2 times the second equation, minus (2*d5) times the third equation. The right-hand sides cancel, as do the β5. We are left with the equation f11 + .. + f44 = 0, with fi integers.

But since (α1,...,α4) are linearly independent over the integers, (β1,...,β4) are also linearly independent, so that fi=0.

In particular,

0 = f1 = (3*d5-2)*c1 + 2*d1 - 2*d5.

The only solution satisfying d5 ≥ 1, c1 ≥ 2, d1 ≥ 0 is d5=1, c1=2, d1=0.

So we learn that d5=0.

Now looking at fi, i=2,3,4:

0 = fi = (3*d5-2)*ci + 2*di - 2*d5 0 = c*i + 2*di - 2.

The only integer solution with ci ≥ 1 and di ≥ 0 is ci=2, di=0.

Then β5=360, again impossible.

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