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Adornments, Flowers, and Kneser-Poulsen

Bob Connelly

Cornell University

(visiting University of Cambridge)

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Part I

Carpenter’s Rule Questionwith Erik Demaine and Günter Rote

2000

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Carpenter’s rule resultGiven a closed polygon in the plane that does not intersect itself there is a

continuous motion that moves it in such a way that no edges cross and all the edges stay the same length and the final configuration is convex.

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Carpenter’s rule resultGiven a closed polygon in the plane that does not intersect itself

there is a continuous motion that moves it in such a way that no edges cross and all the edges stay the same length and the final configuration is convex.

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Two casesThe opening result applies to the case of an open

chain or a simple closed chain.

Closed CaseOpen case

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Moral of this story

• The polygon opens by expanding. In other words, each pair of vertices of the polygon, that can conceivably get further apart during the motion, do.

• Expanding polygons can’t have self-intersections. The only way to stop is when it becomes convex.

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Methods

• (CDR 2000) Canonical expansion which varies continuously as the data is varied.

• (I. Streinu 2000) Series of one-parameter expansive motions using pseudo-triangulations.

• (J. Canterella, E. Demaine, H. Iben, J. O’brien 2004) Energy driven opening, but not necessarily expansive.

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Part II

Adornmentswith E. Demaine, M. Demaine, S. Fekete, S. Langerman, J.

Mitchell, A. Ribo Mor, G. Rote.

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What about a chain of shapes other than line segments?

Some shapes can lock preventing them from opening, such as the following:

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Is there an interesting class of shapes that do open?

Attach non-overlapping adornments to each line segment. As the chain opens, carry the adornments along. How daring can you be in allowing adornments so they don’t bump into each other? The following is a somewhat timid example.

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Slender Adornments

We propose that each adornment be a region bounded by a curve on each side of the line segment such that the distance along the curve to each end point of the line segment be (weakly) monotone. Call these slender adornments.

Obtuse angle

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Symmetric Slender Adornments

An adornment attached to a line segment is symmetric if it is symmetric about the line.

Symmetric,and slender.

One sided, not symmetric, slender.

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Adornment Result

Theorem (CDDFLMRR): Any closed or open polygonal chain with non-overlapping slender adornments piece-wise smoothly opens without overlapping to a configuration with the core chain convex.

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Adornments might not expandRemark: Even when the underlying chain is

expanding, some points of the adornment may get closer together. The red points expand, but the blue points contract in the Figure.

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Part III

Kneser-Poulsen questionswith K. Bezdek

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Unions of circular disks

Suppose that we have a union of circular disks, and we rearrange the disks so that their centers expand.

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Unions of circular disks

Suppose that we have a union of circular disks, and we rearrange the disks so that their centers expand.

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Kneser-Poulsen Conjecture

(Kneser 1955, Poulsen 1954): The volume/area of the union (or intersection) of a finite collection of disks in Ed does not get smaller (or larger) when they are rearranged such that each pair of centers p does not get smaller.

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HistoryKirszbraun (1934): If a finite number of disks have an empty

intersection, and their centers p are expanded to q, then the new set of disks with centers at the configuration q and the same radii also have an empty intersection. (In any Ed.)

B. Bollobás (1968): K-P for the plane for all radii equal, and q a continuous expansion of p.

M. Gromov (1987): K-P for any dimension d, but only for d+1 disks.

V. Capoyleas and J. Pach (1991): K-P for any dimension d, but only for d+1 disks.

H. Edelsbrunner (1995): Introduces Voronoi regions to aid in the calculation.

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History (continued)

M. Bern and A. Sahai (1998): K-P for the plane for any radii, and q is a continuous expansion of p inspired by Edelsbrunner’s remarks.

B. Csikós (1995): K-P for all dimensions, any radii, but only when q is a continuous expansion of p. (Inspired by Bern and Sahai?)

K. Bezdek and R. Connelly (2001): The K-P conjecture is correct in the plane when the configuration of centers q is any discrete expansion of p. (Inspired by all the above.)

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FlowersGordon and Meyer (1995) and Gromov (1987) suggested a K-P

conjecture for combinations of unions and intersections of balls. For example for the expression,

(B1 B2) U (B3 B4) U (B5 B6) U B7

Centers of the disks are only allowed to move together or apart as a function of their position in the expression. This is shown with solid or dashed lines in the figure. Area monotonicity follows from Csikós’s formula and/or our results.

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Symmetric slender adornments are flowers

Symmetric slender adornments are the (possibly infinite) union Ux [Da(x) Db(x)], where the disks Da(x) and Db(x) are centered at the two end points a and b respectively, and x, on the boundary of each disk, runs over the boundary of the adornment.

ba x

D (x)D (x)

ba

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Intersecting slender adornmentsTwo slender adornments intersect if and only if four

circular disks corresponding to the four end points of the two line segments intersect.

y

x

b

b

c

d

D (y) D (y)

D (x)

D (x)

a

d

a

c

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Adorned chains again

Theorem: If a chain, which has non-overlapping symmetric slender adornments attached, is expanded (discretely) to another chain, the corresponding adornments are still non-overlapping.

Proof: If the expanded chain has overlapping adornments, some 4 circular disks about the end points must intersect, while the original disks did not. This contradicts Kirszbraun’s Theorem.

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Discrete non-symmetric adornments can intersect

If the slender adornment is not symmetric, a discrete expansion can create intersections.

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Continuous expansions

Theorem (CDDFLMRR): If slender, non-overlapping, but not necessarily symmetric, adornments are attached to a chain, and the expansion is continuous, then the adornments on the expanded chain will not overlap.

Proof: For each pair of adornments wait until the symmetric other half can be attached disjointly and apply the discrete result. It is enough to look at the two pairs of intersecting disks on each side.

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Some cases

Symmetrize both adornmentsand apply the symmetrizedtheorem.

Apply symmetrized verstion tothe chain segment and theother symmetrized adornment.

The base chainsare expanding

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Extra flowers

• If symmetric slender adornments do intersect, then under a discrete expansion of plane, the area of the union does not decrease. (C-B applied to flowers.)

• This works in any dimension if the assumed expansion is continuous (Csikós applied to flowers).

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Best possible?

There are examples of adornments to chains that are slightly larger than slender, and the chains cannot open in any way without the adornments colliding.