Adornments, Flowers, and
Kneser-Poulsen
Bob Connelly
Cornell University
(visiting University of Cambridge)
1
Part I
Carpenter’s Rule Question
with Erik Demaine and Günter Rote
2000
2
Carpenter’s rule result
Given 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.
3
Carpenter’s rule result
Given 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.
4
Two cases
The opening result applies to the case of an open
chain or a simple closed chain.
Open case
Closed 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 selfintersections. The only way to stop is when
it becomes convex.
6
Methods
• (CDR 2000) Canonical expansion which
varies continuously as the data is varied.
• (I. Streinu 2000) Series of one-parameter
expansive motions using pseudotriangulations.
• (J. Canterella, E. Demaine, H. Iben, J.
O’brien 2004) Energy driven opening, but
not necessarily expansive.
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Part II
Adornments
with 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.
One sided,
not symmetric,
slender.
Symmetric,
and 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 expand
Remark: 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 questions
with 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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History
Kirszbraun (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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Flowers
Gordon 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.
1
2
6
3
7
5
4
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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.
Da(x)
a
x
D (x)
b
b
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Intersecting slender adornments
Two slender adornments intersect if and only if four
circular disks corresponding to the four end points
of the two line segments intersect.
Dc(y)
c
a
Dd(y)
x
d
y
b
D (x)
b
Da(x)
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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 nonoverlapping.
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, nonoverlapping, 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
The base chains
are expanding
Apply symmetrized verstion to
the chain segment and the
other symmetrized adornment.
Symmetrize both adornments
and apply the symmetrized
theorem.
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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.
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