Polytopes with very stiff Faces

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Polytopes with very stiff Faces

• Jürgen Richter-Gebert
• Technical University Munich
• joint work with
• Alexander Below, ETH Zurich

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Polytopes and their face lattices
Each polytope in R3 that has the same face
lattice as a polytope P is called a realization
of P.
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Polytopes and their face lattices
Different realizations of the cube
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Prescribing faces
Dimension 3 Let P be a polytope containing an
n-gon G. Every realization
of G can be completed to a realization of P.
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Prescribing faces
Dimension 3 Let P be a polytope containing an
n-gon G. Every realization
of G can be completed to a realization of P.
Dimension 4 Let P be a polytope containing an
face G. Not every
realization of G can be completed to a
realization of P.
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The key tool for Fixing ShapesLawrence
extensions

• Start with polytope G,...
• and a point q outside G,...
• choose a new direction,...
• - and a line l trough q,...
• - put two points q, q- on l,...
• - take conv(G, q, q-).

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A Transmitter for Shapes
Thm. In every realization of P the faces F
and F - of P are projectively
equivalent.
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What can be prescribed?
(4,3) There are 4-polytopes, such that the
shape of a 3-face cannot be prescribed. (4,2)
There are 4-polytopes, such that the shape of
a 2-face cannot be prescribed. (d,k)
For dgt4 and 1ltkltd there are d-polytopes, such
that the shape of a k-face cannot be prescribed.
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Variations of the Transmitter
An edge-forgetting polytope
A vertex-forgetting polytope
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Making Faces more stiff
The connected sum operation

• Take two polytopes P and Q.
• Make sure they have a projectively equivalent
  face F.
• Glue (transformed copies of) P and Q along F.
• - The resulting polytope is P F Q.

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Necessarily flat faces
The connected sum operation
But this
should not happen !!
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Necessarily flat faces
The connected sum operation
But this
should not happen !!
Therefore F should be necessarily flat.
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Superimposing Obstructions
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Superimposing Obstructions
Glue two twisted copies of polytope X Along the
prism
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Other Useful Gadgets
glue along prism
Gluing two transmitters produces a
connector with four pyramid as outputs
T1 T2
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A very stiff octagon
Connecting 8 X-Polytopes in a suitable way
produces a polytope with an octagon with these
incidence properties.
Thm. There is a 4-polytope P with 33 vertices
that has an octagonal 2-face G, such that
in every realization of P, G is projectively
equivalent to a regular octagon.
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What we can hope for
For every polygon G with algebraic vertex
coordinates there is a 4-polytope P that has a
2-face G such that In every realization G is
projectively equivalent to G.
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What we can hope for
For every polygon G with algebraic vertex
coordinates there is a 4-polytope P that has a
2-face G such that In every realization G is
projectively equivalent to G.
Realizing a polytope means solving a system
of polynomial equations and inequalities. Every
polytope can be realized with algebraic vertex
coordinates.
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What we can hope for
For every polygon G with algebraic vertex
coordinates there is a 4-polytope P that has a
2-face G such that In every realization G is
projectively equivalent to G.
The face lattice of a polytope is invariant
under (small) projective transformation. G can
only be prescribed up to projective equivalence
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What we can hope for
For every polygon G with algebraic vertex
coordinates there is a 4-polytope P that has a
2-face G such that In every realization G is
projectively equivalent to G.
3-polytopes cannot prescribe anything. Edges are
trivially prescribed.
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What the universality theorem does
Input A system S of polynomial equations and
inequalities in Shor normal form. 1 lt x1 lt x2
lt x3 lt . . . lt xn xi xj xk xi xj xk
Output A 4-polytope P ? R4m that contains a
2-face G ? R4r and a function f R4r? Rn
such that for every realization of P ? R4m
the image f(G) is a solution of S.
Conversely, for every solution s ? R there is a
realization of P with f(G) s.
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Encoding variables in n-gons
In P the slopes of a centrally symmetric
2(n3)-gon, have the values 0, 1, s1, s2, s3, . .
. , sn, ?. Variables of S are encoded by the
cross ratios
xi ?,01,si
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Fixing slopes and 1-chords
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Slopes and 1-chords determine G
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Slopes and 1-chords determine G
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Slopes and 1-chords determine G
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Slopes and 1-chords determine G
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Slopes and 1-chords determine G
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Slopes and 1-chords determine G
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Fixing 1-chords by a polytope
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Fixing concrete coordinates
How to fix algebraic numbers by Shor normal
forms? Standard trick Define algebraic numbers
by their minimal polynomial and
separating rational left and right bounds.
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Fixing concrete coordinates
How to fix algebraic numbers by Shor normal
forms? Standard trick Define algebraic numbers
by their minimal polynomial and
separating rational left and right bounds.
Almost
trivial Example sqrt(2) x11,
xy?y, 0 lt 1 lt y lt x.
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So, what can we do?
For every centrally symmetric polygon G with
algebraic vertex coordinates there is a
4-polytope P that has a 2-face G such that in
every realization G is projectively equivalent
to G.
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So, what can we do?
For every centrally symmetric polygon G with
algebraic vertex coordinates there is a
4-polytope P that has a 2-face G such that in
every realization G is projectively equivalent
to G.
. . . and general polygons?
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Fixing general polygons
For every polygon G with algebraic vertex
coordinates there is a 4-polytope P that has a
2-face G such that in every realization of P the
face G is projectively equivalent to G.
Flatten and double G to get a centrally symmetric
polygon. Use vertex forgetters to recover the
original G.
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Higher Dimensions
For every d-polytope G with algebraic vertex
coordinates there is a (d2)-polytope P that has
a d-face G such that in every realization of P
the face G is projectively equivalent to G.
Proof Fix G by line images on edges, link
these line images to polygons, fix these
polygons, glue everything together.
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Line Images (dim3)
Let e be an edge of a 3-polytope P. Let F1, F2, .
. . , Fn be the faces of P not incident with e.
Let fi imge(Fi) be the intersection of the
line supporting e with a facet Fi. The sequence
(f1, f2, . . . , fn) is the line image of e.
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Fixing Polytopes by Fixing Line Images
Thm. Let P and P be two polytopes with
identical face lattice. If corresponding line
images of P and P are projectively
equivalent, then P and P are projectively
equivalent.
Rem. In fact, fixing a small number (d) of line
images suffices to determine P.
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Coding line images by polygons
Positions of the points of a line image can be
coded by edge-supporting lines of a polygon Ge
adjacent to e.
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Pasting Ge and P
Embed P in R4. Ge incident to e in a new
direction of space. Take the convex hull of P
and Ge. This links the line image of e of to
Ge.
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The Final Theorem
For every d-polytope G with algebraic vertex
coordinates there is a (d2)-polytope P that has
a d-face G such that in every realization of P
the face G is projectively equivalent to G.
Proof Fix G by line images on edges, link
these line images to polygons, fix these
polygons, glue everything together.
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Open Problems

• Can the prescribed faces be facets ?
• Are there non-trivial very stiff polytopes?
• Find good applications of the theorem.

. . . Happy Conference Peter
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. . . is NP-hard
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. . . is NP-hard
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