The Shortest Planar Arc of Width 1

Ani Adhikari Department of Statistics Sequoia Hall Stanford University Stanford, CA 94305

and

Jim Pitman Department of Statistics University of California Berkeley, CA 94720

Technical Report No. 113 September 1987

Department of Statistics University of California Berkeley, California 1.1

THE SHORTEST PLANAR ARC OF WIDTH 1

ANI ADHIKARI AND JIM PITMAN Departrnent of Statistics, Sequoia Hall, Stanford University, Stanford, CA 94305 and Department of Statistics, University of California, Berkeley, CA 94720

1. Introduction. A floor is ruled with parallel lines spaced unit distance apart. You are given a piece of wire of length I which you are free to bend but not stretch. Can you bend the wire so that if dropped on the floor the bent piece of wire is certain to cross at least one of the lines, no matter how it falls? In this article we find the least l such that this can be done, and show how the wire should be bent.

More formally, let a: [0, 1] -4 R2 be a continuous rectifiable arc, of length 1(a). For 0.9.< r, define the width of a between parallels at angle 9 by

w0(a) = distance between supporting parallel lines at an angle 9 to the x-axis.

FIG. 1. The width between parallels at angle 0

And define the width of a by 1.2

w (a) = inf wQ(x)

Our problem is to identify an arc of minimal length among all arcs of width at least 1. A cir- cular arc of diameter 1 has length it= 3.14* h.Tree sides of the unit square reduces the length to 3. Two sides of an equilateral triangle of altitude I further reduces the length to 4/t3= 2.309401 . The minimal length tums out only slightly less, namely

2.27829 * - *, and is achieved by bending the wire into a shape which we call the calliper, shown in Fig. 2.

FIG. 2. The calliper.

In Section 2 we identify the calliper of length 2.27829 - as the shortest convex arc of width 1. In Sections 3 and 4 we show that the calliper is in fact shortest among all arcs, con- vex and non-convex. For the most part, our arguments are geometric, in the spirit of Kazar- inoff (1961), Yaglom and Boltyanski (1961) and Niven (1982).

Our problem is a deterministic variation of Buffon's famous needle problem: in case you do not bend the wire at all, but leave it as a straight needle of length 1, what is the chance that the needle crosses at least one line if dropped at random on the floor? As observed by Bertrand in the last century, and argued in detail by Barbier (1860), under natural assumptions of random- ness the expected number of crossings of the grid by a needle bent into a planar curve is a constant c times the length 1, regardless of the shape of the curve. A closed convex curve of constant width 1 must cross the lines exactly twice, no matter how it falls. Since a circle of diameter I has constant width 1 and length it, it follows that 1.3

(i) the constant is c = 2/xt,

(ii) every closed convex curve of constant width 1 has the same length it, and (iii) for I . 1 the probability that a randomly dropped straight needle crosses the lines is (2/i)l.

For other questions about randomly dropped curves, and further references, see DeTemple and Robertson (1980).

The problem solved in this article is a special case of a stochastic geometry problem which we do not know how to solve for all l. Given a wire of length l, how should it be bent to max- imize the probability that it crosses at least one line when tossed at random on the ruled floor? Here we just find the least length L such that this maximum probability is one. For 0 < I < 1 the best strategy is easily shown to be leaving the wire straight, with crossing probability (2/7i)1. But for 1 <1 < L we do not know how to bend the wire to maximize the crossing probability. Some remarks on this and other problems in the same vein are mentioned in the final section.

Notation. Since rotation, translation, and reflection of an arc in the plane affect neither its length nor its width, we will often use these operations to reduce calculations to arcs with some convenient orientation. A generic arc a will often be denoted instead by A-B to indi-

A cate that it starts at A and ends at B. To -indicate the arc is convex we write A B instead of A-B. For points A and B in the plane we use the following notation: AB for the straight line through A and B, AB for the segment of this line between A and B, I AB I for the length of this line segment.

Constructions of arcs may be indicated by notation such as this: if A B is some arc under consideration, then A BA is the closed arc obtained by following A-B from A to B, then returning to A along a line segment. In such constructions the precise paramaterization of the arc will be irrelevant, though the order in which points are visited may be important 2.1

2. The Shortest Convex Arc of Width One. Suppose now that the arc A B is convex,

A meaning that the closed arc A BA is the boundary of a convex subset of the plane:

A B A B A FIG. 3. Some convex arcs.

A A convex arc A B must lie on one side of the line AB. Without loss of generality, we take AB to be horizontal, and suppose A B lies above AB. We propose to find the shortest such

A arc of width one. In case A B extends to the left of A or to the right of B, a shorter arc with greater width is obtained by dropping a perpendicular to AB:

A 3 FIG. 4. Reduction to arcs over an interval.

So we may assume that d = AB 2 1, and confine our search to arcs A B lying in the strip above AB, such as the leftmost arc in Fig. 3. In this case the width condition can be reformu- lated more simply as follows, with reference to the horizontal segment A'B' at unit height above AB, and open discs Disc(A) and Disc(B) centered at A and B, with radii 1, as shown in the next figure:

Al . ;; _-- Ds(B)"C

I,A

A JB

FIG. 5. An arc of width greatera an 1. LEMMA. For a convex arc A B above AB with l AB > 1 width (A B) > 1 if and only if the following three conditions all hold:

A (i) A B intersects A'B' at some point F

A (ii) A F does not intersect Disc (B)

A (iii) F B does not intersect Disc (A)

Proof of Sufficiency. Suppose the three conditions hold. Width at least 1 in the horizontal and vertical directions is clear. For the width between down-sloping parallels, consider two paral- lels distance 1 apart, one through A, the other above it tangent to Disc (A ). The upper paral- lel must hit the arc between F and B by (i) and (iii). Similarly, width at least 1 between up- sloping parallels is implied by (i) and (ii).

Proof of Necessity. If (i) fails, the vertical width is less than 1. The only remaining case is if

A (i) holds but either (ii) or (iii) fails. If say (ii) fails, there exist points C and D on A B as in the following diagram such that C and D lie on Circle (B) bounding Disc (B), and apart from these points arc C D lies inside the open Disc (B).

defined in tems of CD and isc (B ). Th upe-uprigl'e fABprle oC A'.A

By convexity of A B, the arc lies inside the shaded region in the diagram, with boundary d d e f n c . ppersupportinglineofA 2.3

must therefore intersect Disc (B ), giving a width less than 1.

According to the lemma, for fixed A and B at distance d . 1, the shortest convex arc A B of width 1 lying above AB is the shortest path from A to B via some F on A'B', not entering Disc (B ) on the way from A to F, and not entering Disc (A ) on the way from F to B.

In case d > 2/43, this shortest path is obvious. For F the midpoint of A'B' the straight lines AF and FB do not meet the discs, so AFB is the shortest, with length V1Id. For

d =2h13, the length is 2.30940... ..

FIG. 7. Two sides of an equilateral triangle: d =P3

If d < 2/43, the constraint of the discs is felt. For any F on A'B', it is easy to see that the shortest way to get from A to F without entering Disc (B) is to go along AF, if this can be done without entering Disc (B); else to go from A to Circle (B ) along a tangent, stay on the circle awhile, and leave for F along another tangent. And the shortest way to return from F to B without entering Disc (A) can be described similarly, with Circle (A) replacing Circle (B). A ' .;.F

A.:. FIG. 8. The shortest path from A to B via F satisfying (i), (ii), (iii).

To find the shortest convex arc from A to B sadsfying (i)-(iii), reflect the arc in A'B' after it first touches A'B' at F. The problem reduces to finding the shortest arc from A to B , never 2.4 entering Disc (B ) or Disc (A ), as in Fig. 9. A4

A

.....

A B A B

- FIG. 9. Paths from A to B avoiding the discs.

It is easy to see that the minimal arc must be straight as it passes through F. This can only happen if F is the midpoint of A'B', so that the minimal arc is symmetric.

By symmetry, the length of this shortest arc is 21(d) where 1(d) is the length of the arc between A and F, d = AB I, and 1 d 2/43. By the elementary geometry of Fig. 10,

I (d) = _ 1 [x1/2 arctan \71Y- 2arctan (d/2)] + d/2

A' F

(A C) = -A-&- i

ip A^ (DF) = AL/2 HK( D

m IIIA - oz & A

FIG. 10. Calculation of I (d). 2.5

To find the d which minimizes this length, differentiate I with respect to d:

I'(d) = [ _ _ 112 1+d2,4]

f+_ 1 [d2-4 1 d 2 1d2+4 J Set this equal to 0 to see that the minimum is achieved when z = d2 satisfies

4(z - 1)(4+z)2- z(z-4)2 = 0

Solving the cubic yields the minimizing d = 1.04359 * and I (d ) = 2.27829 . The shortest convex arc of width 1 so obtained is shown in the center of the next figure. We call this arc the calliper.

Also shown are the arcs considered above for other values of d near d*. For d 2 d* the argument above shows these are the shortest convex arcs of width 1 with endpoints distant d apart. But for d . d we are unable to draw this conclusion due to the initial step of drop- ping perpendiculars to reduqe to the case of arcs over the interval between the ends. All we can say is that these arcs are shortest among arcs of width 1 that stay in the strip over the interval.

......

SL=f sL=1025 ^ J^= g O+.,. a ~~~~I. 06S- , c_ 22X0.

FIG. 11. The calliper for various values of d. 3.1

3. Results for Non-convex Arcs. The aim of this section and the next is to show that the calliper defined in Section 2 is in fact a shortest arc of width 1 among all arcs, not just among convex ones as shown already. This seems to be a lot harder than you might expect. Intui- tively, it is hard to believe that you could do any better by allowing the arc to cross through its convex hull. But a difficulty can be seen from the fact that there are arcs such as those shown below for which there is no shorter arc with the same convex hull. ~~~~~~~~~~~~~~~~".~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0

FIG. 12. Arcs of minimal length with given convex hull.

It must somehow be argued that such arcs are still far from the shortest for the given width. A first step in this direction is provided by the following inequality:

PROPOSITION 1. Let I be the length of an arc, d the distance between its ends, and w its width. Then I 2 NI4w2 + d2.

Proof. By definition of w, in every direction there are two parallel lines distant w apart, each touched by the arc. Consider two such parallels, taken parallel also to the line between the endpoints in case d > 0, as in the left panel below. Reflecting a portion of the arc as in the right panel produces an arc of the same length I joining points at a distance of l4w2 + d2 by Pythagoras.

______J.~ -.----r

w oS _ _t * * _ v I~~~~~O

_.

FIG. 13. Construction for the proof of Proposition 1. 3.2

REMARKS. For width w = 1 all we get immediately is 1 . 2, using d . 0. But this is very crude. For d . 2/43 the above bound is attained by two line segments, as remarked already in Section 2. For 0 < d < 2/,I3 it is not hard to show that the bound cannot be attained. In the case of a closed convex curve, when d = 0, the least possible length is xt, attained by any curve of constant width 1. This result is given for closed convex curves in Exercise 7-18 of Yaglom and Boltyanskii (1961). A quick proof in this case can be given by consideration of the expected number of crossings in the Buffon setting described in the introduction. This result can be extended to non-convex closed curves by an easy variation of arguments in this section. But we do not know the best inequality for 0 < d < 2/d.

To get a handle on possibly non-convex arcs, we work with the class CM of all arcs starting at a fixed point A of length at most 3, consisting of at most M line segments, where M is a fixed integer. It is easy to show that length and width are a continuous functions for a topol- ogy which makes this class of arcs compact. This class therefore contains a shortest arc of width at least 1. Pick such a minimal arc and call it a . We propose to study the properties * of a , as a preliminary to getting lower bounds on its length which do not involve M, the bound on the number of segments. Clearly, any such lower bound is also a lower bound on the length of any arc whatsoever with width one.

LEMMA 2. The sequence of endpoints of segments of a minimal arc a* is a permutation of the set of corners of the convex hull of a

Proof. Let H be the set of corners of the convex hull of a . The proof is in two steps. Step (i). Each element of H is visited exactly once by a*. Step (ii). Each endpoint of a segment of a is an element of H.

Proof of Step (i). Clearly every corner of the convex hull is an endpoint of some line segment of a*, so visited at least once by a . Suppose some element h e H is visited more than * once by a . Then an arc with one less segment formed by cutting the corner h at one of its 3.3 appearances is shorter than (X , belongs to CM, and has the same convex hull as cX . Since two arcs with the same convex hull have the same width, this contradicts the minimality of ax

Proof of Step (ii). Each segment endpoint is in H because if (x had a segment endpoint not in H, a shorter arc with fewer segments and the same convex hull could be made by cutting the corner just as in step (i).

LEMMA 3. A minimal arc aC* has no self-intersections.

Proof Lemma 2 implies that if a* does cross itself, it must do so stictly inside its convex hull. Suppose there is such a crossing. Then there are points D, E, F, G along the arc, visited in that order, such that segment DE of the arc intersects segment FG. Thus A-B = A_DFFG'B where A'D or G B might be trivial. Now DE and FG are the diagonals of a quadrilateral, of which DF and EG are two opposite sides. So by adding two applications of the triangle inequality

IDFI + IEGI < IDEI + IFGI

Consider the new arc A-`DFEG-B where FE is EF reversed. The convex hull of the new arc is the same as that of a , but by the above inequality the new arc is shorter, contradicting the minimality of a*. F F

A 8 A a FIG. 14. Construction for the proof of Lemma 3. 3.4

According to the above, a minimal arc oc must visit each corner of its convex hull exactly once without ever intersecting itself, by a succession of line segments between corners of the convex hull. For want of a better term let us call such an arc standard. The initial segment of a standard arc must be an edge of its convex hull, to avoid self-intersection. The simplest standard arcs are convex arcs forming all but one side of a convex polygon. We say such an arc has one gap. Then there are arcs of standard form with two or more gaps:

A A A FIG. 15. Standard arcs with 1, 2 and 3 gaps.

In general, an arc of standard form must comprise all segments of its convex hull except for a finite number of these segments, which we call gaps.

* LEMMA 4. A minimal arc a of width at least I has endpoints at distance d (a*) 2 1.

Proof. Let a have endpoints A at the origin and B on the positive x -axis in x ,y coordinates. Let CAD be the corner of the convex hull of a at A, EBF the corner at B. Because of the standard form of a , one of the segments AC and AD must be a segment of a*, while the other must be a gap on the convex hull. Similarly for BE and BF. We will show that angles CAD and EBF are both at most right angles.

AA, Iz, c. F'

~

FIG. 16. Construction for proof of Lemma 4. 3.5

Once this is proved, it is clear that ca must lie within the strip bounded by the lines x =0 and x =d(ac). Since the width of cc in the horizontal direction is at least 1, d(ac) must be at least 1.

Suppose angle CAD is obtuse.

A DpF FIG. 17. Construction in case angle CAD is obtuse.

Assume to be definite that AC is the first segment of a*, so AD is a gap. Let A' be the foot of the perpendicular dropped from C to AD. Let CB denote the portion of a& starting at C and ending at B. Consider the new arc a =A 'CB This arc a is clearly shorter than a But its convex hull contains that of a, so its width is at least 1. This contradicts the minimal- ityof a.

Thus angle CAD is at most a right angle. A similar argument shows angle DBE is at most a right angle, completing the proof of the Lemma.

Now by Proposition 1, no matter what the bound on the number of segments, for a minimal arc a* of width at least 1,

l(a*) 2 l¶54+d2(a*) > = 2.236 ... Approximating any arc by arcs with a finite number of straight segments, it follows that every arc of width at least 1 has length at least '15. But more work is required to get up to the length 2.278.. of the calliper of Section 2. This is the subject of the next section. 4.1

4. Convexity of minimal arcs. It will be shown in this section that for any positive integer M, a minimal arc aXiof width at least I with at most M segments must be convex. It follows that the calliper defined in Section 2 is in fact a shortest arc among all arcs with width at least 1. According to the previous section a is standard. That is to say, it must visit each corner of its convex hull exactly once, without ever intersecting itself, by a succession of line segments between corners of the convex hull. In general, a standard arc must pass along each edge of its convex hull except for a finite number of edges which we call gaps. Assum- ing that the arc starts clockwise from A, it must wind clockwise around the hull until it meets a gap. This might be the end, in which case the arc is convex with just one gap. Otherwise, if the arc has n gaps it must cross back and forth across the interior of the hull by say n- 1 straight segments, winding successively anticlockwise and clockwise around the hull between these crossings.

FIG. 18. A standard arc with 5 gaps.

Associated with each gap of a standard arc is a convex portion of the arc which connects the ends of the gap. Call this portion of arc the arch across the gap. A standard arc thus consists of a succession of arches across gaps, with adjacent arches overlapping by a segment crossing the hull. Define dh aldtude of an arch to be the width of the arch between parallels to its gap.

7? %7%%1-%-%- >

FIG. 19. The arches of the arc in Fig. 18 and their altitudes ai 4.2

A key step in our argument is provided by the following Lemma.

LEMMA 6. If a standard arc has width w, at least one of its arches has altitude at least w.

Proof. For a convex arc with one gap this is trivial. Consider then an arc with two arches of altitudes a 1 and a2:

A FIG. 20. An arc with 2 arches.

The first arch is taken to be above the first gap, which is assumed horizontal. If a1 2 w there is nothing to prove. So suppose a I < w. Then to achieve width w between horizontals the end of the second gap at B must lie above the horizontal at level a 1. The second gap must therefore have positive slope. By convexity, the parallel to this gap supporting the arc from below must touch the arc at a point P on the second arch. Consequently, this arch has altitude a2 2 w.

This argument extends to a standard arc with n arches as follows. Keep track of the order in which the arches are traversed. Suppose that the supporting parallel opposite a gap touches the arc at a point of the arch over that gap, or on some later arch. Then if that point of contact is over the arch, the altitude of the arch must be at least w. And in case that point of contact is not on the arch over the gap, but later, the supporting parallel opposite the next gap touches the arc on or after the next arch. This is shown by the same argument as above in case n = 2. This serves as an inductive step which forces some arch to have altitude at least w, because there are only a finite number n of arches. 4.3

LEMMA 7. A minimal arc a with at most M segments is convex.

Proof. For convenience, rotate and reflect a so that the gap corresponding to the tallest arch is horizontal. By Lemma 6, this arch has altitude at least 1. Let C1 and C2 denote the left and right endpoints of the gap.

Suppose C2 is not an endpoint of the arc. Then there is a last point of the arc on the convex hull before C2, call it L1, which is the end of another gap, say L 1L2. It is clear that L 1L2 cannot be parallel to CIC2 since if it were the length of a would be at least 3. Thus L1L2 must meet C1C2 at some point E, say, to the right of C2, as in the figure below. This point E must also be to the right of a perpendicular dropped vertically from L1, since otherwise this perpendicular would offer a shorter path with bigger convex hull.

A _

FIG. 21. Case (i).

Case (i). ILIL21 < IL C21

In this case we could replace the portion L FB of a by the path L1L£RS, as in Fig. 21 where R is the rightmost point of a , and S is the foot of a perpendicular dropped from R to

C 1C2. But since IRS C2L22 this gives a shorter arc with the same convex hull. 4.4 Case (ii). ILL21 . IL1C21 Now L2 must lie on or outside the circle with center L1 and radius IL C2 , as in Fig. 22. Let L 1L2 intersect the circle at D, and let the tangent to the circle at D intersect C 1C2 at F.

\ D 4- ,/4L A,/ N-. CZ E FIG. 22. Case (ii).

By the geometry of the diagram

IDF I < IFL21 < IC2L21< l(C2 L2) s IDE 1. The last inequality is due to the minimality of a*, since otherwise we could replace the portion L l L2 of a& by the two straight line segments LD and DE to get a shorter arc with a bigger convex hull. Hence I DF I < DE I. That is,

angle C2ELI = angle FED <450.

Let T denote a point on the arch from C1 to C2 farthest away from the gap, so the perpendic- ular distance between T and CIC2 is the altitude of the arch. Reflect T about CIC2, and do the same with L2. Let L'2 be the reflection of L2, and T' the reflection of T.

Then

1(rC2) + l (C2 L2) I TC21 + C2L2I1 = ITC21 + IC2L'21 . distance of Tfrom TE "~~~~ 4.5

; L45e

I - L rE~~~~~~~ 'U~~~~T FIG. 23. The reflection argument. This distance is least when angle C2EL2 is 45°, that is, angle TET'=90. And in this case the distance is at least 4F. So

1(a) > I(Cj rC -B) . i+4 > 2.4.

But by Lemma 1, l (a*) must be less than 2.309 . This contradicts the minimality of a, so C2 must be an end of the arc.

Equally, C1 must be the other end of the arc. That is to say, a*is a convex arc with a single arch. 5.1

5. Exercises and Open Problems.

Uniqueness of the shortest arc of width one. A weakness of the indirect argument of the last section is that it does not seem to settle the uniqueness question. While the arguments of section 3 show that the calliper is unique among convex arcs, and some variation of the argu- ment of the last section should ensure uniqueness among arcs that cross their convex hull only a finite number of times, what about arcs that cross an infinite number of times? It is hard to imagine such an arc of width one of the same length as the calliper, but we do not have a proof that no such arc exists.

Maximizing the probability of crossing. Consider again the problem, posed in the introduc- tion, of maximizing the probability that a dropped piece of wire of length I will cross parallel lines distant 1 apart. In case the arc is not planar there may be some ambiguity about what constitutes a crossing. But however this ambiguity is resolved, it is obviously best to keep the wire bent in a planar arc. If the arc has width w9 between parallels at angle 0 to the x-axis, then by conditioning on the angle 0 at which the arc falls relative to the parallels, the chance of crossing is

P (cross) = (1/)fmin(we, 1)dO 0

The problem is to maximize this integral over all arcs of length 1. Call the maximal value of P (cross) so obtained P *(1). The existence of an arc of length I attaining this bound can be shown by the usual kind of compactness argument. For I . 1, the optimal arc is straight- And for I > 2.27829 *- *, the length of the calliper, P *(I) is 1. But for I between 1 and the length of the calliper, we do not know how to find an optimal arc, whether it is unique, or what is the value of P*(). 5.2

Upper bounds on P (1). Because the expected number of crossings is (2/it)l, no matter what the arc, Markov's inequality gives

P *(I) < (2/n)l

The graph of the right hand side as a function of I is the straight line through the origin in Figure 25. For I < 1 this Markov bound is attained. And for I greater than the length of the calliper, P *(1) is 1. But we do not know any better upper bounds on P *(I) than 1 and the Markov bound.

Lower bounds on P*(l)

1. Leaving the arc straight. Lower bounds on P (1) are of course easier to obtain, by compu- tation of P (cross) for particular arcs of length 1. For a straight arc of length 1 . 1 the cross- ing probability is well known and easily calculated:

Pstraighut(1) = (2/1)( 1(1 - sinf) + (3) where 1 = arccos(1/l).

The graph of Pstraight (I) is the lowest curve in Figure 24.

2. Bending the arc in the middle. The next easiest case is an arc of two straight segments bent in the middle. Let d be the distance between the ends, h = 1/2, the length of each segment, as in Figure 23.

FIG. 23. A straight arc bent in the middle.

In case d < 1 and h < 1, when the triangle defined by the arc must cross the lines either zero or two times, the crossing probability is simply

(2/1)(perimeter of triangle)/2 = (1 + d)/ir, 5.3

since the expected number of crossings is (2/1r)(perimeter of triangle). So for fixed 1 . 1 < 2 it is best to make d 2 1. According to our calculations, which we invite the reader to check as an exercise, for l between 0 and 2/43 it is best to leave the arc straight. For I > 2/4s3, the arc should be bent in the middle so that the distance d between its ends equals 2/'I3. Let P*middle (1) denote the maximum crossing probability for an arc of length I bent in the middle. Then

. Pmiddle(1) = Pstraight(') for 0 1 5 2/-%I3 =l+ 3 for 2/43 < l 5 2 1~ 3 -=l+ __4Lf-~42_- - arcosIlh i for 2 < l < 4/'I, 1c 3 ICU = lfor l 2 4/Fl.

* The function P Mi"C (1) is the middle curve in Figure 24. By the formula above,

Pmid* e(2.27829 . . . ) = .99794*v. So P (l) is very close to P*mid&,(1) for I near the length of the calliper.

1.0

0.0

0 L.0 ,9 ,0 0.0 1.0 2.0 3.0 length ofarc

FIG. 24. Crossing probabilities for various arcs 5.4

3. Bending the arc into two straight segments.

We conjecture that P *middle (1) is in fact the maximum crossing probability over all arcs of length I bent, not necessarily in the middle, into two straight segments. We can calculate the crossing probability of any arc bent into two straight pieces. But the formula for the probabil- ity is forbidding, and we have not gone through the calculus necessary to verify our conjecture. To see what the formula is like, consider the triangle of Fig. 25.

FIG. 25. Triangle defined by into two straight segments. The crossing probability is the integral

fmin(wo, 1)dO, divided by 7t. As can be seen from the diagram, this integral breaks into three parts, each representing the contribution from an interval of angles 0 for which the width is attained by a particular side of the triangle. The length of this interval is the angle opposite the side. Here is the contribution from the interval of angles 0 from 0 to i in Figure 25 for which side b attains the width, call it Int(f). The contributions of angles a and fy are similar, with cyclic changes in sides and angles.

Int(5) = Jmin(b sin(0+ a) , 1 ) dO. 0

= b (cosy+cosac) if b < 1. 5.5

For b . 1, the integral is more complicated, but reduces after some calculation, as follows. Let (D+ = (X/2-a) + arcos l/b, O_ = (r/2-a) - arcos l/b For a function f of 0, let If (0)] Y denote f (y) -f (x). Then,

Int(p) = [-bcos(a+O)]O- + 2arcos(1/b) + [-bcos(ac+O)I if 0< (_ < D+ < p = (D++ [-bcos(ac+0)], if (D_ < 0< (D <1 = [-bcos(a+O)]o- + (3-(D.) if O < (_ < 13< D+ = f3 < 0< <(+ Add this expression and two similar expressions for Int (a) and Int (y), and divide by ic to get the crossing probability for an arc composed of two straight segments. The reader is invited to check this analysis, and verify our conjecture that bending in the middle always maximizes this crossing probability for given length.

Let Pmiddle (1, d) denote the chance that an arc of length I bent in the middle into two sege- ments, with endpoints distance d apart, crosses at least once. It is interesting to note that in the case I < 2, d 2 1, it is possible to bend the arc at points other than the center and still have chance Pmiddle (I, d) of crossing. Suppose the arc is bent so that the distance between the two endpoints is still d, but the two arms have different lengths. Then provided both arms have length less than one, the chance that the arc crosses at least once is P,d&,e(I, d). To see- this, consider the triangular closed arc. This crosses either zero, two, or four times. Since the distance between the ends is fixed, the perimeter of the triangle is the same no matter where the arc is bent. So the expected number of crossings is the same, no matter where the arc is bent. And if both arms have length less than 1, then four crossings are made if and only if the base of the triangle crosses twice. So the probability of crossing four times is the same no matter where the arc is bent, provided both arms have length less than 1. This implies the chance of at least two crossings remains the same. Thus if our conjecture above is correct, then in the case 1 < 2 there are infinitely many ways to bend the arc in two to maximize the probability of crossing. 5.6

4. Bending into 3 or more segments.

This will of course lead to better lower bounds on P (I ). But after our calculations above for two segments we do not see how to organize the computations sensibly, let alone evaluate P *(l) as a limit. All we know for sure is that this maximal probability as a function of I lies somewhere in between the top two curves in Figure 24. Once again, the reader is invited to help.

A variation of the problem. A nice conjecture about a variation of the problem was sug- gested to us by David Goldschmidt. Suppose you are allowed to cut the wire and glue the pieces back together to form a connected union of arcs, call it a gadget. How now can you improve the crossing probability, and what is the shortest length fromn which you can make a gadget of width 1? The conjecture for the shortest gadget of width 1 is three straight arms connecting the corners of an equilateral triangle of altitude 1 to its center. Since this gadget is so much simpler to define than the calliper, perhaps there is a simple proof.

The problem higher dimensions. What is the shortest arc of width one in d dimensions? This problem seems very much harder in three or more dimensions. The three dimensional problem can be presented as follows. Suppose there is a one inch gap between the parallel plane sides of your stove and your kdtchen cupboard. What is the shortest length of a piece of wire that can be bent in such a way that it cannot fall into the crack? Presumably the solution is shaped something like three connected sides of a regular tetrahedron of altitude one inch, but we have no idea of what the exact shape must be. References.

1. M.E. Barbier, Note sur le problEme de l'aiguille et jeu du joint couvert, J. Mathem. Pures et Appl., (2) 5 (1860) 272-286.

2. Duane W. DeTemple and Jack M. Robertson, Constructing Buffon curves from their distributions, Amer. Math. Monthly, 87 (1980) 779-784.

3. N.D. Kazarinoff, Geometric Inequalities, Random House 1961.

4. I. Niven, Maxima and minima without calculus, Dolciani Math. Expos. #6, Math. Assoc. Amer., 1982.

5. I.M. Yaglom and V.G. Boltyanskii, Convex Figures, Holt, Rinehart and Wilson, 1961. TECHNICAL REPORTS Statistics Departnent University of Califomia, Berkeley 1. BREIMAN, L. and FREEDMAN, D. (Nov. 1981, revised Feb. 1982). How many variables should be entered in a regression equation? Jour. Amer. tatist. Aso. March 1983, 78, No. 381, 131-136. 2. BRILLINGER, D. R. (Jan. 1982). Some contrasting examples of the time and frequency domain approaches to time series analysis. Time S s Methods in Hydrosciences. (A. H. El-Shaarawi and S. R. Esterby, eds.) Elsevier Scientific Publishing Co., Amsterdam, 1982, pp. 1-15. 3. DOKSUM, K. A. (Jan. 1982). On the performance of estimates in proportional hazard and log-linear models. Survival Analysis. (John Crowley and Richard A. Johnson, eds.) IMS Lecture Notes - Monograph Series, (Shanti S. Gupta, series ed.) 1982, 74-84. 4. BICKEL, P. J. and BREIMAN, L. (Feb. 1982). Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test.AnL Prob.. Feb. 1982, 11. No. 1, 185-214. 5. BRILLINGER, D. R. and TUKEY, J. W. (March 1982). Spectrum estimation and system identification relying on a Fourier transform. The Collected Works Lf Ww. Tukey. vol. 2, Wadsworth, 1985, 1001-1141. 6. BERAN, R. (May 1982). Jackknife approximation to bootstrap estimates. Anna Statjst. March 1984, 12 No. 1, 101-118. 7. BICKEL, P. J. and FREEDMAN, D. A. (June 1982). Bootstrapping regression models with many parameters. Lehmarm Festschrift. (P. J. Bickel, K. Doksum and J. L. Hodges, Jr., eds.) Wadsworth Press, Belmont, 1983, 28-48. 8. BICKEL, P. J. and COLLINS, J. (March 1982). Minimizing Fisher information over mixtures of distributions. Saky 1983, 45, Series A, Pt. 1, 1-19. 9. BREIMAN, L. and FRIEDMAN, J. (July 1982). Estimating optimal transformations for multiple regression and correlation. 10. FREEDMAN, D. A. and PETERS, S. (July 1982, revised Aug. 1983). Bootstrapping a regression equation: some empirical results. JASA. 1984, 79, 97-106. 11. EATON, M. L. and FREEDMAN, D. A. (Sept. 1982). A remark on adjusting for covariates in multiple regression. 12. BICKEL, P. J. (April 1982). Minimax estimation of the mean of a mean of a normal distribution subject to doing well at a point. Rent Advances E Statistics, Academic Press, 1983. 14. FREEDMAN, D. A., ROTHENBERG, T. and SUTCH, R. (Oct 1982). A review of a residential energy end use model. 15. BRILLINGER, D. and PREISLER, H. (Nov. 1982). Maximum likelihood estimation in a latent variable problem. Studies in Econometrics. Time Sene. and Multivariate Statistics. (eds. S. Karlin, T. Amemiya, L. A. Goodman). Academic Press, New York, 1983, pp. 31-65. 16. BICKEL, P. J. (Nov. 1982). Robust regression based on infinitesimal neighborhoods. Ann. Sist Dec. 1984, 12, 1349-1368. 17. DRAPER, D. C. (Feb. 1983). Rank-based robust analysis of linear models. I. Exposition and review. 18. DRAPER, D. C. (Feb 1983). Rank-based robust inference in regression models with several observations per cell. 19. FREEDMAN, D. A. and FIENBERG, S. (Feb. 1983, revised April 1983). Statistics and the scientific method, Comments on and reactions to Freedman, A rejoinder to Fienberg's comments. Springer New York 1985 Chori Analysis in Social Research. (W. M. Mason and S. E. Fienberg, eds.). 20. FREEDMAN, D. A. and PETERS, S. C. (March 1983, revised Jan. 1984). Using the bootstrap to evaluate forecasting equations. L .f Foreca ng 1985, Vol. 4, 251-262. 21. FREEDMAN, D. A. and PETERS, S. C. (March 1983, revised Aug. 1983). Bootstrapping an econometric model: some empirical results. IBES 1985, 2, 150-158. - 2 -

22. FREEDMAN, D. A. (March 1983). Structural-equation models: a case study. 23. DAGGElT, R. S. and FREEDMAN, D. (April 1983, revised Sept. 1983). Econometrics and the law: a case study in the proof of antitrust damages. c gf dih Berkeley Conference. in honor of Jerzy Neyman and Jack Kiefer. Vol I pp. 123-172. (L. Le Can, R. Olshen eds.) Wadsworth, 1985. 24. DOKSUM, K. and YANDELL, B. (April 1983). Tests for exponentiality. Handbook of Statistics, (P. R. Krishnaiah and P. K. Sen, eds.) 4, 1984. 25. FREEDMAN, D. A. (May 1983). Comments on a paper by Markus. 26. FREEDMAN, D. (Oct. 1983, revised March 1984). On bootstrapping two-stage least-squares estimates in stationary linear models. AlDL Statist.. 1984, 12, 827-842. 27. DOKSUM, K. A. (Dec. 1983). An extension of partial likelihood methods for proportional hazard models to general transformation models. Arm, Sist. 1987, 15, 325-345. 28. BICKEL, P. J., GOETZE, F. and VAN ZWET, W. R. (Jan. 1984). A simple analysis of third order efficiency of estimates. Prc f -the Neyman-Kiefer Conference. (L. Le Can, ed.) Wadsworth, 1985. 29. BICKEL, P. J. and FREEDMAN, D. A. Asymptotic normality and the bootstrap in stratified sampling. Aim. Statist. 12 470-482. 30. FREEDMAN, D. A. (Jan. 1984). The mean vs. the median: a case study in 4-R Act litigation. JB.ES.1985 Vol 3 pp. 1-13. 31. STONE, C. J. (Feb. 1984). An asymptotically optimal window selection rule for kerel density estimates. Ann. Statist. Dec. 1984, 12, 1285-1297. 32. BREIMAN, L. (May 1984). Nail finders, edifices, and Oz. 33. STONE, C. J. (Oct. 1984). Additive regression and other nonparametric models. Azm. Statis 1985, 13, 689-705. 34. STONE, C. J. (June 1984). An asymptotically optimal histogram selection rule. Proc. of the Berkeley Conf. in Honor of Jerzy Neyman and Jack Kiefer (L. Le Cam and R. A. Olshen, eds.), II, 513-520. 35. FREEDMAN, D. A. and NAVIDI, W. C. (Sept. 1984, revised Jan. 1985). Regression models for adjusting the 1980 Census. Statistical Science. Feb 1986, Vol. 1, No. 1, 3-39. 36. FREEDMAN, D. A. (Sept. 1984, revised Nov. 1984). De Finetti's theorem in continuous time. 37. DIACONIS, P. and FREEDMAN, D. (Oct. 1984). An elementary proof of Stirling's formula. Amer. M.ath Monthly. Feb 1986, Vol. 93, No. 2, 123-125. 38. LE CAM, L. (Nov. 1984). Sur l'approximation de familles de mesures par des familles Gaussiennes. Ann. Inst. khnri Poincare. 1985, 21, 225-287. 39. DIACONIS, P. and FREEDMAN, D. A. (Nov. 1984). A note on weak star uniformities. 40. BRE1MAN, L. and IHAKA. R. (Dec. 1984). Nonlinear discriminant analysis via SCALING and ACE. 41. STONE, C. J. (Jan. 1985). The dimensionality reduction principle for generalized additive models. 42. LE CAM, L (Jan. 1985). On the normal approximation for sums of independent variables. 43. BICKEL, P. J. and YAHAV, J. A. (1985). On estimating the number of unseen species: how many executions were there? 44. BRILLINGER, D. R. (1985). The natural variability of vital rates and associated statistics. Biometric to appear. 45. BRILLINGER, D. R. (1985). Fourier inference: some methods for the analysis of array and nonGaussian series data Water Resources Bulletin, 1985, 21, 743-756. 46. BREIMAN, L. and STONE, C. J. (1985). Broad spectrum estimates and confidence intervals for tail quantiles. - 3 -

47. DABROWSKA, D. M. and DOKSUM, K. A. (1985, revised March 1987). Partial likelihood in transfonnation models with censored data. 48. HAYCOCK, K. A. and BRILLINGER, D. R. (November 1985). LIBDRB: A subroutine library for elementary time series analysis. 49. BRILLINGER, D. R. (October 1985). Fitting cosines: some procedures and some physical examples. Festschrift, 1986. D. Reidel. 50. BRILLINGER, D. R. (November 1985). What do seismology and neurophysiology have in common? - Statistics! Compte Rendus Math. Rp Acad. Si Canada. January, 1986. 51. COX, D. D. and O'SULLIVAN, F. (October 1985). Analysis of penalized likelihood-type estimators with application to generalized smoothing in Sobolev Spaces. 52. O'SULLIVAN, F. (November 1985). A practical perspective on ill-posed inverse problems: A review with some new developments. To appear in Joumal of Statistical Science. 53. LE CAM, L. and YANG, G. L. (November 1985, revised March 1987). On the preservation of local asymptotic normality under information loss. 54. BLACKWELL, D. (November 1985). Approximate normality of large products. 55. FREEDMAN, D. A. (December 1985, revised Dec. 1986). As others see us: A case study in path analysis. Prepared for the Joumal of Educational Statistics. 56. LE CAM, L. and YANG, G. L. (January 1986). Distinguished Statistics, Loss of information and a theorem of Robert B. Davies. 57. LE CAM, L. (February 1986). On the Bernstein - von Mises theorem. 58. O'SULLIVAN, F. (January 1986). Estimation of Densities and Hazards by the Method of Penalized likelihood. 59. ALDOUS, D. and DIACONIS, P. (February 1986). Strong Uniform Times and Finite Random Walks. 60. ALDOUS, D. (March 1986). On the Markov Chain simulation Method for Uniform Combinatonral Distributions and Simulated Annealing. 61. CHENG, C-S. (April 1986). An Optimization Problem with Applications to Optimal Design Theory. 62. CHENG, C-S., MAJUMDAR, D., STUFKEN, J. & TURE, T. E. (May 1986, revised Jan 1987). Optimal step type design for comparing test treatments with a control. 63. CHENG, C-S. (May 1986, revised Jan. 1987). An Application of the Kiefer-Wolfowitz Equivalence Theorem. 64. O'SULLIVAN, F. (May 1986). Nonparametric Estimation in the Cox Proportional Hazards Model. 65. ALDOUS, D. (JUNE 1986). Finite-Time Implications of Relaxation Times for Stochastically Monotone Processes. 66. PITMAN, J. (JULY 1986, revised November 1986). Stationary Excursions. 67. DABROWSKA, D. and DOKSUM, K. (July 1986, revised November 1986). Estimates and confidence intervals for median and mean life in the proportional hazard model with censored data. 68. LE CAM, L. and YANG, G.L. (July 1986). Distinguished Statistics, Loss of information and a theoren of Robert B. Davies (Fourth edition). 69. STONE, C.J. (July 1986). Asymptotic properties of logspline density estimation. 71. BICKEL, P.J. and YAHAV, J.A. (July 1986). Richardson Extrapolation and the Bootstrap. 72. LEHMANN, E.L. (July 1986). Statistics - an overview. 73. STONE, C.J. (August 1986). A nonparametric framework for statistical modelling. - 4 -

74. BIANE, PH. and YOR, M. (August 1986). A relation between Lkvy's stochastic area formula, Legendre polynomials, and some continued fractions of Gauss. 75. LEHMANN, E.L. (August 1986, revised July 1987). Comparing Location Experiments. 76. O'SULLIVAN, F. (September 1986). Relative risk estimation. 77. O'SULLIVAN, F. (September 1986). Deconvolution of episodic hormone data. 78. PrTMAN, J. & YOR, M. (September 1987). Further asymptotic laws of planar Brownian motion. 79. FREEDMAN, DA. & ZEISEL, H. (November 1986). From mouse to man: The quantitative assessment of cancer risks. 80. BRILLINGER, D.R. (October 1986). Maximum likelihood analysis of spike trains of interacting nerve cells. 81. DABROWSKA, D.M. (November 1986). Nonparametric regression with censored survival time data. 82. DOKSUM, K.J. and LO, A.Y. (November 1986). Consistent and robust Bayes Procedures for Location based on Partial Information. 83. DABROWSKA, D.M., DOKSUM, KA. and MIURA, R. (November 1986). Rank estimates in a class of semiparametric two-sample models. 84. BRILLINGER, D. (Decenber 1986). Some statistical methods for random process data from seismology and neurophysiology. 85. DIACONIS, P. and FREEDMAN, D. (December 1986). A dozen de Finetti-style results in search of a theory. 86. DABROWSKA, D.M. (January 1987). Uniforn consistency of nearest neighbour and kemel conditional Kaplan - Meier estimates. 87. FREEDMAN, DA., NAVIDI, W. and PETERS, S.C. (February 1987). On the impact of variable selection in fitting regression equations. 88. ALDOUS, D. (February 1987, revised April 1987). Hashing with linear probing, undernon-uniform probabilities. 89. DABROWSKA, D.M. and DOKSUM, KA. (March 1987). Estimating and testing in a two sample generalized odds rate model. 90. DABROWSKA, D.M. (March 1987). Rank tests for matched pair experiments with censored data. 91. DIACONIS, P and FREEDMAN, D.A. (March 1987). A finite version of de Finetti's theorem for exponential families, with uniform asymptotic estimates. 92. DABROWSKA, D.M. (April 1987). Kaplan-Meier estimate on the plane. 92a. ALDOUS, D. (April 1987). The Harnonic mean formula for probabilities of Unions: Applications to sparse random graphs. 93. DABROWSKA, D.M. (June 1987). Nonparametric quantile regression with censored data. 94. DONOHO, D.L & STARK, PB. (June 1987). Uncertainty principles and signal recovery. 95. RIZARDI, F. (Aug 1987). Two-Sample t-tests where one population SD is known. 96. BRILLINGER, D.R. (June 1987). Some examples of the statistical analysis of seismological data. To appear in Proceedings, Centennial Anniversary Symposiwn, Seismographic Stations, University of California, Berkeley. 97. FREEDMAN, DA. and NAVIDI, W. (June 1987). On the multi-stage model for cancer. 98. O'SULLIVAN, F. and WONG, T. (June 1987). Determining a fimction diffusion coefficient in the heat equation. - 5 -

99. O'SULLIVAN, F. (June 1987). Constrained non-linear regularization with application to some system identification problems. 100. LE CAM, L. (July 1987). On the standard asymptotic confidence ellipsoids of Wald. 101. DONOHO, D.L. and LIU, R.C. (July 1987). Pathologies of some minimum distance estimators. 102. BRILLINGER, D.R., DOWNING, K.H. and GLAESER, R.M. (July 1987). Some statistical aspects of low-dose electron imaging of crystals. 103. LE CAM, L. (August 1987). Harald Cramer and sums of independent random variables. 104. DONOHO, A.W., DONOHO, D.L. and GASKO, M. (August 1987). Macspin: Dynamic graphics on a desktop computer. 105. DONOHO, D.L. and LIU, R.C. (August 1987). On minimax estimation of linear fimctionals. 106. DABROWSKA, D.M. (August 1987). Kaplan-Meier estimate on the plane: weak convergence, LIL and the bootstrap. 107. CHENG, C-S. (August 1987). Some orthogonal main-effect plans for asymmetrical factorials. 108. CHENG, C-S. and JACROUX, M. (August 1987). On the construction of trend-free run orders of two-level factorial designs. 109. KLASS, M.J. (August 1987). Maximizing E max S'iES': A prophet inequality for sums of I.I.D. 1 5k!gnk mean zero variates. 110. DONOHO, D.L. and LIU, R.C. (August 1987). The "automatic" robustness of minimum distance functionals. 111. BICKEL, P.J. and GHOSH, J.K. (August 1987). A decomposition for the likelihood ratio statistic and the Bartlett correction - a Bayesian argument. 112. BURDZY, K., PITMAN, J.W. and YOR, M. (September 1987). Some asymptotic laws for crossings and excursions. 113. ADHIKARI, A. and PITMAN, J. (September 1987). The shortest planar arc of width 1. Copies of these Reports plus the most recent additions to the Technical Report series are available from the Statistics Department technical typist in room 379 Evans Hall or may be requested by mail from: Department of Statistics University of Califomia Berkeley, California 94720 Cost: $1 per copy.