Isostatic Structures: Using Richard Rado's Matroid Matchings · Isostatic Structures: Using Richard Rado’s Matroid Matchings Henry Crapo, Les Moutons matheux, La Vacquerie Joint

Isostatic Structures:Using Richard Rado’s Matroid Matchings

Henry Crapo, Les Moutons matheux, La VacquerieJoint work with Tiong Seng Tay, Nat. Univ. Singapore,

and Emanuela Ughi, Univ. Perugia

Workshop on Rigidity, Fields Institute, 11-14 October, 2011

Outline

1 Main Points

2 Basics and Context

3 Semi-simplicial Maps

4 Shelling

5 Freely Shellable Maps

6 Partitions of the Vertex Set

7 Finale

Dedication

I would like to dedicate this talk to two persons,both of whom are architects and engineers.

Dedication

To Janos Baracs,

DedicationTo Janos Baracs,

instigator and cofounder ofthe research group Topologie Structurale,

who learned projective geometryfrom his high school math teacher in Budapest,

and who introduced Ivo Rosenberg and myself tothree dimensional space and rigidity

during a workshop for members of theCentre de recherches mathematiquesin January 1973, over 38 years ago,

Dedication

. . . posing, among other problems:

to characterize generically 3-isostatic graphs

to predict special positions of non-rigidityfor generically 3-isostatic graphs,

to specify the correct placements of cross-bracesin grid frameworks.

to analyze the rigidity of tensegrity frameworks.

to analyze the relation between stresses and liftingof plane polyedral frameworks.

to develop a theory of periodic filling of spaceby copies of one or more associated zonohedra.

Dedication

To Richard Gage,

Dedication

To Richard Gage,

founder and leading member ofthe association Architects and Engineers for 911 Truth,

who has brought a new level of intelligent and systematic inquiry,a new level of organization and energetic public engagement,

to the quest for an independent inquiry intothe state crimes of 11/9/2001

and into this decade of their rain of miserable consequences.

Dedication

To Richard Gage,

founder and leading member ofthe association Architects and Engineers for 911 Truth,

who has brought a new level of intelligent and systematic inquiry,a new level of organization and energetic public engagement,

to the quest for an independent inquiry intothe state crimes of 11/9/2001

and into this decade of their rain of miserable consequences.

Dedication

Everything you ever wanted to knowabout the 9/11 conspiracy theory

in under 5 minutes.

http://www.informationclearinghouse.info/article29110.htm

(surely the central rigidity problem of our era)

Dedication

With special thanks to Walter Whiteley and Bob Connelly,Ileana Streinu and Tibor Jordan,

who have so energeticallykept this beautiful subject alive and well,

expanding its horizons,training the researchers of this new generation,

and making it possible for us to be together today.

Main Points

(1) If a graph G has a shellable semi-simplicial mapto the d-simplex Kd+1,then it is generically d-isostatic.

(2) Conjecture: The converse is true.

(3) We offer a strengthened conjecture:Conjecture: A graph is generically d-isostatic if and only ifit has a freely-shellable semi-simplicial map to the d-simplex.

(4) We investigate further restrictions of the class of mapsto maps that are fewer in number and easier to construct:maps whose vertex packets are broken paths.

Main Points





Main Points





Main Points





Generically Isostatic Graphs

A graph G (V ,E ) is generically d-isostatic if and only ifit is edge-minimal among graphs that are rigidin some (and therefore in almost every) positionin real Euclidean or projective space of dimension d .


We shall deal only with generic behavior of graphs as structures,so we will speak simply of “d-isostatic” graphs,dropping the adjective “generic”.


A graph, as bar-and-joint structure, is isostatic:



in Statics, if and only ifall external equilibrium loads are uniquely resolvable in the edges.



in Mechanics, if and only ifit is edge-minimal among graphs with no internal motion.



in Matroid Theory, if and only ifit is a basis for the generic d-rigidity matroid on KV .



in Statics, if and only ifall external equilibrium loads are uniquely resolvable in the edges.

in Mechanics, if and only ifit is edge-minimal among graphs with no internal motion.

in Matroid Theory, if and only ifit is a basis for the generic d-rigidity matroid on KV .

d-Isostatic Graphs

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1-isostatic graphs are trees.

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A 2-isostatic graph.fe

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A 2-isostatic graph.

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A 3-isostatic graph.(maximal planar)

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A 3-isostatic graph.

Figure: d-Isostatic graphs, for d = 1, 2, 3.

Definition

A semi-simplicial map f : G (V ,E )→ Kd+1,

whereKd+1 = K (I , J)

andI = {1, 2, . . . }, J = {12, 13, . . . },

consists of a pair of maps

f0 : V → I , f1 : E → J,

that preserve incidence.

The subset f −10 (i), for any vertex i ∈ I ,

we call the i th vertex packet of f , denoted Vi .

Definition

That is, an edge e = ab whose vertices a and bhave distinct values f0(a) = i , f0(b) = j in Imust be sent by f1 to ij ∈ J.

We call such an edge e = ab an ij-bridge.



Definition

An edge e = ab whose end vertices go to the same vertex,say f0(a) = i = f0(b),must be sent to an edge ij of K incident to i .

We call such an edge e = ab a loop at i toward j .



Definition



Definition



We shall include in the definition of simplicial mapone crucial additional property:

Definition



We shall include in the definition of simplicial mapone crucial additional property:

(P0) Edge independence: The inverse image f −11 (ij),

denoted Tij , of any edge ij of Kis a tree spanning the union Vi ∪ Vj

of its two related vertex packets.

Definition



(the combined statement:)

A semi-simplicial map f : G (V ,E )→ Kd+1(I , J),consists of a pair of maps f0 : V → I , f1 : E → J,that preserve incidence, and . . .

(P0) Edge independence: The inverse image f −11 (ij),

denoted Tij , of any edge ij of Kis a tree spanning the union Vi ∪ Vj

of its two related vertex packets.

Visual Representation of Maps

Semi-simplicial maps have very satisfactory visual representations,using colors taken from a standard edge-coloring of Kd+1

to specify the images of each edge.

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Figure: A d-isostatic graph, with semi-simplicial map.


The tree-decomposition is then easily comprehended.

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Figure: The trees T12 and T34.



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Figure: All together now!.

Path Connectivity

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Figure: Paths between vertices having distinct/identical images.

If a and b have distinct images i , j under f0,then a and b are connected along a unique path in the tree Tij .

Path Connectivity

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Figure: Paths between vertices having distinct/identical images.

If a and b have the same image i under f0,then they are connected along unique pathsin each of the d trees Tij , for j 6= i .

Shelling

A vertex packet can be shelledif there is a sequence of monochromatic cutsthat reduces it to a subgraph with no edges.

=

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Figure: A sequence of monochromatic cuts.

Special Placement

In the special position given by a semi-simplicial map,any external equilibrium test loadapplied at two vertices a, b is uniquely resolvable.

Special Placement


If f (a) 6= f (b), the external load is resolved (and uniquely so)along the path between a and b in the tree Tij ,all those edges being collinear along the line i ∨ j .

Special Placement


If f (a) = f (b) = i , the external loadcan be uniquely represented as a sumof d + 1 equilibrium loads applied to a, b,one in each of the (independent) directions i ∨ j at i .

These individual loads are then uniquely resolvablealong the paths from a to b in the trees Tij

Theorem

A graph G is generically d-isostatic graph if ithas a shellable semi-simplicial map to the d-simplex.

Maps on Dependent Graphs

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2Packet {bfcg} is not shellable

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Packet {abefgh} is but partially shellable

Figure: Non-shellable maps on a 3-dependent graph.

Converse, d = 2

For d = 2: Any non-shellable maphas an obstacle to shellingin the form of a set of 3 or more verticesco-spanned by sub-trees of two trees.

This is a dependent subgraph.

Converse, d = 2

Theorem:A graph G is generically 2-isostatic graph

if and only ifit has a shellable semi-simplicial map to the triangle,

if and only ifall semi-simplicial maps to the triangle are shellable.

Converse, d = 3?

This is far from being the case in dimension 3.

A 3-isostatic graph may havemany non-shellable maps to the tetrahedron.

Converse, d = 3?

Existence of a non-shellable map establishes only thatthere is a subset Q of some vertex packet ithat is spanned by sub-trees of any pairof the three trees Tij for j 6= i .

Converse, d = 3?

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Figure: The packet V1 contains an obstacle to shelling.

These are the only two edge maps with this vertex map.

Converse, d = 3?

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Figure: A change of one vertex image produces a shellable map.

This vertex map has a unique compatible edge map.

Eliminate Obstacles - Eliminate Shelling

Perhaps the best way to deal with obstacles to shellingwill be to look for maps in which obstacles cannot occur,


that is, those for which the vertex packetsinduce independent subgraphs,that is, cycle-free subgraphs, or forests.


These maps are freely shellable:


Simply proceed edge by edge,each single edge being a monochromatic cut!

Eliminate Obstacles - Eliminate ShellingSimply proceed edge by edge,each single edge being a monochromatic cut!

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Figure: A forest as induced subgraph of packet V2.


Conjecture:

A graph G is generically 3-isostatic if and only ifit has a semi-simplicial map to the tetrahedronin which all vertex packets induce subgraphsthat are independent (ie: forests) as subgraphs of G .


There are four interesting classes of such maps:those in which the vertex packets induce:

F forests

T trees

B broken paths

P paths


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Figure: Vertex packets are trees (l), paths (r).


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Figure: Vertex packets are trees (l), paths (r).

Understanding these drawings:

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Figure: These drawings may seem complicated, but are easily analyzed.


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Figure: Trees T12, T34.


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The vertex set can not always be partitioned into paths.

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Figure: A hinged ring of tetrahedra.

3-isostatic graphs do not necessarily havemaps to K4 in which

(P) induced graphs on vertex packets are paths.


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There must be a path of length ≥ 3,not within a single tetrahedron.

The vertex b is isolated with its image 3.There must be a path of length ≥ 4.


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l must be 4, otherwise there is no 2-path from d to l .Then values 3 and 4 are isolated at b and l ,

So only 1 and 2 are available for tetrahedron efgh.

Freely shellable semi-simplicial maps

In practice, freely-shellable maps seem to abound,and seem much easier to find “by hand”

than more general mapsfor which you must check shellability.

Freely shellable semi-simplicial maps

What is more, freely-shellable mapshave relatively few loops that need to be assigned.

Partitions that Produce Freely Shellable Maps

To prove a graph G (V ,E ) is isostatic,it suffices to exhibit a partition π of the vertex set Vhaving three properties Pi (see below).

The main criterion P3 isRichard Rado’s matroid basis matching condition.

Partitions that Produce Freely Shellable Maps

Theorem: Rado’s Basis Matching TheoremGiven any relation R from a set Xto a set S of elements of a matroid M(S),then there is matching in R from Xto a basis for the matroid M(S)if and only ifthe cardinality |X | = rank ρ(S) of the matroid M,and, for every subset A ⊂ X ,the cardinality |A| ≤ ρ(A),the rank of its image R(A) in M(S).

Bibliography on Matroid Matching

Richard Rado,A Theorem on Independence Relations,Quarterly J. of mathematics, Oxford 13 (1942), 83-89.


Joseph P. S. Kung, Gian-Carlo Rota, Catherine H. Yan,Combinatorics: The Rota Way,Cambridge University Press, 2009.


Kazuo Murota,Matrices and Matroids for Systems AnalysisSpringer Verlag,Algorithms and Combinatorics20 (2000),(revised 2010).


And an article which led us to the possibility ofinsisting that vertex packets induce paths:

Roger K. S. Poh,On the Linear Vertex-Arboricity of a Planar GraphJournal of Graph Theory, 14 No. 1 (1990), 73-75.

A Matroid Union

Given a partition of the vertex set of G ,define bridges and loops,

and for each ij construct the matroid minor:restrict to the induced subgraphon the union of the two packets,

and contract by its bridges.

Then take the matroid union over all pairs ij

A Matroid Union

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Figure: The bridges of a map on the icosahedron.

A Matroid Union

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Figure: Restrictions to packet unions V1 ∪ V4 and V2 ∪ V3.

Characterization of Partitions for Freely-Shellable MapsTheorem: A partition π of the vertex set of a graph G (V ,E )is the inverse image partitionof a freely-shellable semi-simplicial mapf : G → Kd+1 if and only ifthe partition π has the following three properties Pi

(P1) The induced subgraph Gi on any part πi

of π is independent (circuit-free).

(P2) For any pair ij , the bridge subgraph G (Vi ∪ Vj ,Bij)is independent.

(P3) The relation R between the set of loops of Gand the set of elements of the matroid union Msatisfies the Rado condition for basis matching:|L| = ρ(M) and

∀A ⊆ E , |A| ≤ ρ(R(A)).






∀A ⊆ E , |A| ≤ ρ(R(A)).






∀A ⊆ E , |A| ≤ ρ(R(A)).






∀A ⊆ E , |A| ≤ ρ(R(A)).

A Partition Not Satisfying the Rado Condition

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Figure: Partition (a)(befhi)(cd)(g) does not satisfy the Rado condition.


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Figure: Partition (a)(befhi)(cd)(g) has 2 compatible loop maps.


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Figure: A non-Rado partition for K6,6 less 6 edges. (edge di !)


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Figure: The Rado relation R for that partition.

A Partition Satisfying the Rado Condition

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Figure: A partition with 32 compatible loop maps.


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Figure: The Rado relation R for that partition.


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Figure: The symmetry of R is perhaps more visible here.


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Figure: Four independent binary choices, . . . .


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Figure: After four independent binary choices, a cycle remains.

The Road Ahead

It remains to prove that any 3-isostatic graphhas a freely-shellable semi-simplicial map to the simplex K4.

The Road Ahead

This has always been the hard part of the problem!

The Road Ahead

What is likely to happen?

The Road Ahead


Either:

There will be a relatively simple proof,I would guess during the next few months, . . .

The Road Ahead


Either:


Or Jackson and Jordan will hit uswith another magnificent counterexample,

like the biplane(an example on the complete graph K56)

that hit the three towers.

The Road Ahead


Either:


Or Jackson and Jordan will hit uswith another magnificent counterexample,

like the biplane(an example on the complete graph K56)

that hit the three towers.

Followed by a rapid retreatfrom an untenable position!

The Road Ahead

Which properties of isostatic graphsmight permit us to prove the conjecture?

The Road Ahead

Which properties of isostatic graphsmight permit us to prove the conjecture?

We lean toward an analogue in d = 3of Tay’s proof for d = 2.

Toward an Analogue of Tay’s Proof for d = 2

We use the (3v − 6)× 6vprojective rigidity matrix R,

and the (3v + 6)× 6v matrix Swhose rows span the orthogonal complementary subspace.


By Hodge star complementation,the determinants of full-size minors of R

are equal to the determinantsof the complementary full-size minors of S

up to a sign ±1 of the bipartition of the column set,and up to a fixed polynomial quantity Q,

called the pure condition or resolving bracket,which is non-zero exactly when the graph is isostatic.


The column matroids of R and of S are dual to one another,and are independent of the graph G in question!


The column matroids of R and of S are dual to one another,and are independent of the graph G in question!

(Q 6= 0 exactly when the rows of R form a basis forthe space of external equilibrium loads

on the set V of vertices of G ,regarded as a single rigid body.)

The Orthogonal Complementary Matrices S and R

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a3 -a2 a1a4 -a2 a1

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b3 -b2 b1b4 -b2 b1

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Figure: Columns grouped by trees Tij .


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000 000 000 000 000 000000 000 000 000 000 000

000 000 000 000 000 000000 000 000 000 000 000

-ac34ac34 -ac24ac24 -ac23ac23 -ac14ac14 -ac13ac13 -ac12ac12-ad34ad34 -ad24ad24 -ad23ad23 -ad14ad14 -ad13ad13 -ad12ad12

-ae34ae34 -ae24ae24 -ae23ae23 -ae14ae14 -ae13ae13 -ae12ae12-bc34bc34 -bc24bc24 -bc23bc23 -bc14bc14 -bc13bc13 -bc12bc12

-bd34bd34 -bd24bd24 -bd23bd23 -bd14bd14 -bd13bd13 -bd12bd12-be34be34 -be24be24 -be23be23 -be14be14 -be13be13 -be12be12

-cd34cd34 -cd24cd24 -cd23cd23 -cd14cd14 -cd13cd13 -cd12cd12-ce34ce34 -ce24ce24 -ce23ce23 -ce14ce14 -ce13ce13 -ce12ce12-de34de34 -de24de24 -de23de23 -de14de14 -de13de13 -de12de12

a3 -a2 a1a4 -a2 a1

a4 -a3 a1

b3 -b2 b1b4 -b2 b1

b4 -b3 b1

c3 -c2 c1c4 -c2 c1

c4 -c3 c1

d3 -d2 d1d4 -d2 d1

d4 -d3 d1

e3 -e2 e1e4 -e2 e1

e4 -e3 e1

00000 00000 0000 00000 0000 000000000 0000 00000 0000 00000 00000000 00000 00000 0000 0000 00000

00000 00000 0000 00000 0000 000000000 0000 00000 0000 00000 00000000 00000 00000 0000 0000 00000

00000 00000 0000 00000 0000 000000000 0000 00000 0000 00000 00000000 00000 00000 0000 0000 00000

00000 00000 0000 00000 0000 000000000 0000 00000 0000 00000 00000000 00000 00000 0000 0000 00000

00000 00000 0000 00000 0000 000000000 0000 00000 0000 00000 0000

0000 00000 00000 0000 0000 00000

11111 00000 00000 00000 00000 0000011111 00000 00000 00000 00000 00000

11111 00000 00000 00000 00000 0000011111 00000 00000 00000 00000 00000

11111 00000 00000 00000 00000 0000011111 00000 00000 00000 00000 00000

Figure: Columns grouped by vertices v .


Any set of columns in R labeled by a single vertex,say by aand by a circuit in K4,such as 12, 23, 34, 14,are dependent.


Any set of columns in R labeled by a single vertex,say by aand by a circuit in K4,such as 12, 23, 34, 14,are dependent.

Any set of columns in R labeled by a edge of K4,say by 12and by all vertices a,are dependent.


e34d34c34b34a34e24d24c24b24a24e23d23c23b23a23e14d14c14b14a14e13d13c13b13a13e12d12c12b12a12

dececdbebdbcaeadac

e134e124e123

d134d124d123

c134c124c123

b134b124b123

a134a124a123

C34C24C23C14C13C12

000000000000000000000000000000000000

000000000000000000000000000000000000000000000000000000

000000000000000000000000000000000000

000000000000000000000000000000000000

-ac34ac34-ac24ac24-ac23ac23-ac14ac14-ac13ac13-ac12ac12-ad34ad34-ad24ad24-ad23ad23-ad14ad14-ad13ad13-ad12ad12

-ae34ae34-ae24ae24-ae23ae23-ae14ae14-ae13ae13-ae12ae12-bc34bc34-bc24bc24-bc23bc23-bc14bc14-bc13bc13-bc12bc12

-bd34bd34-bd24bd24-bd23bd23-bd14bd14-bd13bd13-bd12bd12-be34be34-be24be24-be23be23-be14be14-be13be13-be12be12

-cd34cd34-cd24cd24-cd23cd23-cd14cd14-cd13cd13-cd12cd12-ce34ce34-ce24ce24-ce23ce23-ce14ce14-ce13ce13-ce12ce12-de34de34-de24de24-de23de23-de14de14-de13de13-de12de12

a3 -a2 a1a4 -a2 a1

a4 -a3 a1

b3 -b2 b1b4 -b2 b1

b4 -b3 b1

c3 -c2 c1c4 -c2 c1

c4 -c3 c1

d3 -d2 d1d4 -d2 d1

d4 -d3 d1

e3 -e2 e1e4 -e2 e1

e4 -e3 e1

000000000000000000000000000000000000000000000000000000000000000000000000000000000

000000000000000000000000000000000000000000000000000000000000000000000000000000000

000000000000000000000000000000000000000000000000000000000000000000000000000000000

000000000000000000000000000000000000000000000000000000000000000000000000000000000

000000000000000000000000000000000000000000000000000000

000000000000000000000000000

11111 000000000000000000000000011111 0000000000000000000000000

11111 000000000000000000000000011111 0000000000000000000000000

11111 0000000000000000000000000111110000000000000000000000000

Figure: From a non-zero diagonal to a (rooted) freely shellable map.


e

d

c

b

a

2

1

1

3

2

Figure: The corresponding rooting of a freely shellable map.

An analogue of Henneberg reduction?

Is it possible to reduce any isostatic graphto an isostatic graph on one fewer vertex,by a procedure that, when repeated,leads, step-by-step, to a map?

Grazie

Thank you for your attention.

This paper should be up on the arXiv soon:

Isostatic Structures:Using Richard Rado’s Independent Matchings

Isostatic Structures: Using Richard Rado's Matroid Matchings · Isostatic Structures: Using Richard Rado’s Matroid Matchings Henry Crapo, Les Moutons matheux, La Vacquerie Joint

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