Cyclic Structures and Incidence Theorems Jrgen Richter-Gebert - - PowerPoint PPT Presentation

Cyclic Structures and Incidence Theorems

Jürgen Richter-Gebert Technical University Munich

Jürgen Richter-Gebert Technical University Munich

Stupid Proofs for Interesting (?) Theorems

Incidence Theorems Oriented Matroids (Pseudoline arrangements) Zonotopal Tilings (Penrose like tylings) Stresses/Liftings/Reciprocal Figures Bracket Polynomials (Algebra of projective Geometry) Circle Patterns DDG

Topics we will meet

Running Example

1 3 2 6 5 4 9 8 7

The collinearity of (123), (456), (159), (168), (249), (267), (348), (357) implies the collineartity

• f (7,8,9).

Pappus‘s Theorem

a x b y c z

A Warmup

Area Method

Claim:

area(a,b,c) + area(x,z,y) + area(a,x,y,b) + area(b,y,z,c) + area(a,x,z,c) = 0

A Warmup

Area Method

Claim:

area(a,b,c) + area(x,z,y) + area(a,x,y,b) + area(b,y,z,c) + area(a,x,z,c) = 0

a x b y c z

A Warmup

Area Method

Claim:

area(a,z,b,w) + area(a,y,d,z) + area(b,z,d,x) + area(a,w,c,y) + area(b,x,c,w) + area(c,x,d,y) = 0

Works for any Manifold

w z y x d c b a

A Warmup

Area Method

Claim:

Decomposing the cube

area(a,z,b,w) + area(a,y,d,z) + area(b,z,d,x) + area(a,w,c,y) + area(b,x,c,w) + area(c,x,d,y) = 0

A Warmup

Area Method

Decomposing the cube

==> Discrete Königs Nets Bobenko, Suris

A Warmup

Area Method

Reciprocal Diagrams

A Warmup

Area Method

Reciprocal Diagrams

A Warmup

Area Method

Reciprocal Diagrams

You never have a problem with the last edge

Realizability / Strechability

Arrangements

• f Pseudolines
• Topological lines in RP
• Two cross exactly once

Is there an equivalent line arrangement?

2

Realizability / Strechability

Is there an equivalent line arrangement?

Arrangements

• f Pseudolines
• Topological lines in RP
• Two cross exactly once

2

Excursion: How to draw pseudoline arrangements?

1st Method:

• fix boundary points
• make a Tutte embedding
• draw smooth splines

Conjecture

Every such pseudoline is „a function graph“ ==> no curls allowed!

Realizability / Strechability

Is there an equivalent line arrangement?

Arrangements

• f Pseudolines
• Topological lines in RP
• Two cross exactly once

2

Excursion: How to draw pseudoline arrangements?

1st Method:

• fix boundary points
• make a Tutte embedding
• draw smooth spines

Conjecture

Every such pseudoline is „a function graph“ ==> no curls allowed!

Realizability / Strechability

Is there an equivalent line arrangement?

Arrangements

• f Pseudolines
• Topological lines in RP
• Two cross exactly once

2

Excursion: How to draw pseudoline arrangements?

2nd Method:

• Draw two copies
• n a sphere
• Circle pack
• connect the dots

Realizability / Strechability

Is there an equivalent line arrangement?

Arrangements

• f Pseudolines
• Topological lines in RP
• Two cross exactly once

2

Excursion: How to draw pseudoline arrangements?

2nd Method:

• Draw two copies
• n a sphere
• Circle pack
• connect the dots

Fact

Unique (!) representation

• f the arrangement

Question

What are the special properties?

Realizability / Strechability

Is there an equivalent line arrangement?

Arrangements

• f Pseudolines
• Topological lines in RP
• Two cross exactly once

2

Realizability / Strechability

Is there an equivalent line arrangement?

Arrangements

• f Pseudolines
• Topological lines in RP
• Two cross exactly once

2

Realizability / Strechability

Is there an equivalent line arrangement?

No !!

Pappus’ s Theorem

1 O O a b c d e f O 1 O g h i j k l O O 1 m n o p q r 1 2 3 4 5 6 7 8 9

Another proof by algebraic cancellation

1 3 2 6 5 4 9 8 7

Pappus’ s Theorem

1 O O a b c d e f O 1 O g h i j k l O O 1 m n o p q r 1 2 3 4 5 6 7 8 9 (123) (159) (168) (249) (267) (348) (357) (456) (789)

Another proof by algebraic cancellation

1 3 2 6 5 4 9 8 7

==> ce=bf ==> iq=hr ==> ko=ln ==> ar=cp ==> bj=ak ==> fm=do ==> dh=eg ==> gl=ij ==> mq=np

Pappus’ s Theorem

1 O O a b c d e f O 1 O g h i j k l O O 1 m n o p q r 1 2 3 4 5 6 7 8 9 (123) (159) (168) (249) (267) (348) (357) (456) (789)

Another proof by algebraic cancellation

1 3 2 6 5 4 9 8 7

==> ce=bf ==> iq=hr ==> ko=ln ==> ar=cp ==> bj=ak ==> fm=do ==> dh=eg ==> gl=ij ==> mq=np

Pappus’ s Theorem

1 O O a b c d e f O 1 O g h i j k l O O 1 m n o p q r 1 2 3 4 5 6 7 8 9 (123) (159) (168) (249) (267) (348) (357) (456) (789)

Another proof by algebraic cancellation

1 3 2 6 5 4 9 8 7

==> ce=bf ==> iq=hr ==> ko=ln ==> ar=cp ==> bj=ak ==> fm=do ==> dh=eg ==> gl=ij ==> mq=np

Pappus’ s Theorem

1 O O a b c d e f O 1 O g h i j k l O O 1 m n o p q r 1 2 3 4 5 6 7 8 9 (123) (159) (168) (249) (267) (348) (357) (456) (789)

Another proof by algebraic cancellation

1 3 2 6 5 4 9 8 7

==> ce=bf ==> iq=hr ==> ko=ln ==> ar=cp ==> bj=ak ==> fm=do ==> dh=eg ==> gl=ij <== mq=np

Pappus’ s Theorem

1 O O a b c d e f O 1 O g h i j k l O O 1 m n o p q r 1 2 3 4 5 6 7 8 9 (123) (159) (168) (249) (267) (348) (357) (456) (789)

1 3 2 6 5 4 9 8 7

Another proof by algebraic cancellation

Pappus’ s Theorem

(123) (159) (186) (429) (726) (483) (753) (456) (789)

Structure of the proof

1 3 2 6 5 4 9 8 7

Pappus’ s Theorem

(123) (159) (186) (429) (726) (483) (753) (456) (789)

Structure of the proof

1 3 2 6 5 4 9 8 7

Pappus’ s Theorem

(123) (159) (186) (429) (726) (483) (753) (456) (789)

c a b e d f r q p j l k h i g m n

• e

r j b g

• Structure of the proof

1 3 2 6 5 4 9 8 7

Pappus’ s Theorem

(123) (159) (186) (429) (726) (483) (753) (456) (789)

c a b e d f r q p j l k h i g m n

• e

r j b g

• Structure of the proof

1 3 2 6 5 4 9 8 7

Pappus’ s Theorem

(123) (159) (186) (429) (726) (483) (753) (456) (789)

c a b e d f r q p j l k h i g m n

• e

r j b g

• Structure of the proof

==> a torus

1 3 2 6 5 4 9 8 7

From: Self-Dual Configurations and Regular Graphs Coxeter, 1950

For Desargues’ s Theorem there is no such choice of a basis

Generalizing this proof

special choice of basis 2x2 determinants cancellation pattern

==> ce=bf ==> iq=hr ==> ko=ln ==> ar=cp ==> bj=ak ==> fm=do ==> dh=eg ==> gl=ij <== mq=np (123) (159) (168) (249) (267) (348) (357) (456) (789)

Problems:

Grassmann-Plücker Relations

1 2 3 x y

[123][1xy]-[12x][13y]+[12y][13x] = O

In every configuration of five points 1,2,3,x,y is satisfied ( with [abc]=det(a,b,c) ).

(123) collinear ==> [12x][13y]=[12y][13x] [12x][13y]=[12y][13x] ==> (123) collinear or (1xy) collinear

Pappos’ s Theorem

(123) ==> [124][137]=[127][134] (159) ==> [154][197]=[157][194] (168) ==> [184][167]=[187][164] (249) ==> [427][491]=[421][497] (456) ==> [457][461]=[451][467] (348) ==> [487][431]=[481][437] (267) ==> [721][764]=[724][761] (357) ==> [751][734]=[754][731] (789) <== [781][794]=[784][791]

1 3 2 6 5 4 9 8 7

Same cancellation pattern as before

Pappos’ s Theorem

(123) ==> [124][137]=[127][134] (159) ==> [154][197]=[157][194] (168) ==> [184][167]=[187][164] (249) ==> [427][491]=[421][497] (456) ==> [457][461]=[451][467] (348) ==> [487][431]=[481][437] (267) ==> [721][764]=[724][761] (357) ==> [751][734]=[754][731] (789) <== [781][794]=[784][791]

1 3 2 6 5 4 9 8 7

Same cancellation pattern as before

Pappos’ s Theorem

(123) ==> [124][137]=[127][134] (159) ==> [154][197]=[157][194] (168) ==> [184][167]=[187][164] (249) ==> [427][491]=[421][497] (456) ==> [457][461]=[451][467] (348) ==> [487][431]=[481][437] (267) ==> [721][764]=[724][761] (357) ==> [751][734]=[754][731] (789) <== [781][794]=[784][791]

1 3 2 6 5 4 9 8 7

Same cancellation pattern as before

An automatic method

Generate all “binomial equations” that come from the hypotheses Generate all “binomial equations” that come from the conclusion Find a suitable combination of the conclusion by the hypotheses

A linear problem: Is the conclusion in the span of the hypotheses ?

(binomial-proofs)

Desargues’ s by cancellation

==> (768) or (745) (479) ==> [471][496] = [476][491] (916) ==> [914][962] = [912][964] (259) ==> [256][291] = [251][296] (240) ==> [248][203] = [243][208] (083) ==> [082][035] = [085][032] (570) ==> [573][508] = [578][503] (213) ==> [215][234] = [214][235] (418) ==> [412][487] = [417][482] (536) ==> [532][567] = [537][562]

5 4 8 2 1 9 8 7 6

[764][785] = [765][784]

A non-realizable configuration

j h d g e f c b a i

(abi) ==> [abh][agi]=+[abg][ahi] (acf) ==> [adf][acj]=-[acd][afj] (adh) ==> [abd][afh]=-[abh][adf] (bce) ==> [bcd][bej]=-[bde][bcj] (bdg) ==> [abg][bde]=-[abd][beg] (cdj) ==> [acd][bcj]=+[acj][bcd] (efj) ==> [afj][egj]=+[aej][fgj] (egi) ==> [aeg][ghi]=-[agi][egh] (fhi) ==> [ahi][fgh]=-[afh][ghi] (ghj) ==> [egh][fgj]=+[egj][fgh] [aeg][bej]=+[aej][beg] ==> (abe) or (egj)

A non-realizable configuration

j h d g e f c b a i

(abi) ==> [abh][agi]=+[abg][ahi] (acf) ==> [adf][acj]=-[acd][afj] (adh) ==> [abd][afh]=-[abh][adf] (bce) ==> [bcd][bej]=-[bde][bcj] (bdg) ==> [abg][bde]=-[abd][beg] (cdj) ==> [acd][bcj]=+[acj][bcd] (efj) ==> [afj][egj]=+[aej][fgj] (egi) ==> [aeg][ghi]=-[agi][egh] (fhi) ==> [ahi][fgh]=-[afh][ghi] (ghj) ==> [egh][fgj]=+[egj][fgh] [aeg][bej]=+[aej][beg] ==> (abe) or (egj)

A non-realizable configuration

j h d g e f c b a i

(abi) ==> [abh][agi]=+[abg][ahi] (acf) ==> [adf][acj]=-[acd][afj] (adh) ==> [abd][afh]=-[abh][adf] (bce) ==> [bcd][bej]=-[bde][bcj] (bdg) ==> [abg][bde]=-[abd][beg] (cdj) ==> [acd][bcj]=+[acj][bcd] (efj) ==> [afj][egj]=+[aej][fgj] (egi) ==> [aeg][ghi]=-[agi][egh] (fhi) ==> [ahi][fgh]=-[afh][ghi] (ghj) ==> [egh][fgj]=+[egj][fgh] [aeg][bej]=+[aej][beg] ==> (abe) or (egj)

Six points on a conic

1 2 3 4 5 6

[123][156][426][453] = [456][126][254][423]

Six points 1,2,3,4,5,6 are on a conic <==>

Pascal’ s Theorem

[159][257] = -[125][579] [126][368] = +[136][268] [245][279] = -[249][257] [249][268] = -[246][289] [346][358] = +[345][368] [135][589] = -[159][358] [125][136][246][345] = +[126][135][245][346]

1 2 3 4 5 6 9 8 7

[289][579] = +[279][589]

Problems of this method

Usually large search space What is the “structure” of the proof How to cut down the search space in advance

(binomial-proofs)

Problems of this method

Usually large search space What is the “structure” of the proof How to cut down the search space in advance

(binomial-proofs)

A ambitious dream: Look at a theorem.... “see” its structure.... .... and produce a proof immediately!!

Coxeter’ s proof of Pappos’ s Theorem

The Theorems

• f Ceva and

Menelaos

Ceva and Menelaos

A B Y C Z X A B Y C Z X |AZ|·|BX|·|CY| |ZB|·|XC|·|YA| = 1 |AZ|·|BX|·|CY| |ZB|·|XC|·|YA| = -1

Ceva and Menelaos

A B Y C Z X A B Y C Z X |AZ|·|BX|·|CY| |ZB|·|XC|·|YA| = 1 |AZ|·|BX|·|CY| |ZB|·|XC|·|YA| = -1 · ·

[CDA] [ADB] [BDC] [CDB] [ADC] [BDA]= -1 [XYA] [XYB] [XYC] [XYB] [XYC] [XYA]

· ·

= 1

D

Glueing

|AX|·|BY|·|CZ| |XB|·|YC|·|ZA| = 1 A B C D X Y Z S R |AZ|·|CR|·|DS| |ZC|·|RD|·|SA| = 1

Glueing

|AX|·|BY|·|CZ| |XB|·|YC|·|ZA| = 1 A B C D X Y Z S R |AZ|·|CR|·|DS| |ZC|·|RD|·|SA| = 1 |AX|·|BY|·|CR|·|DS| |XB|·|YC|·|RD|·|SA| = 1

Glueing

a1 b2 b3 b4 b1 a2 a3 a4 = 1 a1 b2 b3 b4 b1 a2 a3 a4

Glueing

a1 b2 b3 b4 b5 b6 b1 a2 a3 a4 a5 a6 = 1 a1 b2 b3 b4 b5 b6 b1 a2 a3 a4 a5 a6

Glueing

+ = A “factory” for geometric theorems

A theorem on a tetrahedron

Front: two triangles with Ceva

A theorem on a tetrahedron

Back: two triangles with Ceva

A theorem on a tetrahedron

After glueing: an incidence theorem

4 Times Ceva

spatial interpretations interesting degenerate situations A special case of Pappus’ s Theorem as special case

Ceva on four sides

• f a tetrahedron

A census of incidence theorems

M M M M

+

C C C C

+

C C M M

+

C M M C

+

M M C C

+

M M C C

+

==> harmonic points ==> many interesing degenerate cases ==> Desargues’ s Thm.

Harmonic Quadruples

C C M M

+ Front: two triangles with Ceva

Harmonic Quadruples

C C M M

+ Back: two triangles with Menelaos

Harmonic Quadruples

C C M M

+ Glued: Uniques of harmonic point construction

Desargues by Glueing

M M M M

+ Front: two triangles with Menelaos

Desargues by Glueing

M M M M

+ Back: two triangles with Menelaos

Desargues by Glueing

M M M M

+ Glued: Desargues’ s Theorem

Six triangles

1 1 1 2 1 3 4 5 1 1 1 2 2 2

Double pyramid

• ver triangle

Degenerate torus

...and other degenerate spheres

Six triangles

1 1 1 2 1 3 4 5 1 1 1 2 2 2

Double pyramid

• ver triangle

Degenerate torus

...and other degenerate spheres

Six triangles

Six times Ceva

Six triangles

Six times Ceva Folding the triangles

Six triangles

Six times Ceva --> after identification

Six triangles

Pappus’ s Theorem

Moving points to infinity

Pappus affine Pappus

Grid theorems

+1

• 1
• 1
• 1

+1 +1

“row sums” = “colums sums” = “diagonal sums” = 0

Another Theorem

+1

• 1

+1 +1 +1

• 1
• 1
• 1

“row sums” = “colums sums” = “diagonal sums” = 0

Larger Grid Theorems

+ =

Composition of grid theorems “space of such theorems” is a vector space “little” Pappus configurations are a basis

Larger Grid Theorems

Composition can be interpreted topologically

„Geometrie der Waben“

compare Blaschke 1937

Arrangements of pseudolines

Rombic Tilings with three Directions

Rombic Tilings with three Directions

Conversion of proofs

1 2 3 4 4 4 5 6 7 7 8 9 A A B B C C 7

[124][137]=[127][134] [154][197]=[157][194] [184][167]=[187][164] [427][491]=[421][497] [457][461]=[451][467] [487][431]=[481][437] [721][764]=[724][761] [751][734]=[754][731] [781][794]=[784][791]

Conversion of proofs

1 2 3 4 4 4 5 6 7 7 8 9 A A B B C C 7

[124][137]=[127][134] [154][197]=[157][194] [184][167]=[187][164] [427][491]=[421][497] [457][461]=[451][467] [487][431]=[481][437] [721][764]=[724][761] [751][734]=[754][731] [781][794]=[784][791] 427 479 478 347 457 467 146 137 154 157 178 148 134 167 127 124 149 197

• Works in general
• Use Tutte-Groups
• and homotopy

A different Story

From: Self-Dual Configurations and Regular Graphs Coxeter, 1950

Graph: vertices -> brackets, edges -> bracktes differing by one letter

Glue versus matter

[12x][13y]-[12y][13x] Grassmann Menelaus Ceva

Glue versus matter

Glue versus matter

Glue versus matter

Glue versus matter

BFP <--> CM

And Conics ?

a1 b3 b1 a3 b5 a5 b2 b4 a2 a4 b6 a6 = 1 a1 b2 b3 b4 b5 b6 b1 a2 a3 a4 a5 a6

Carnot’ s Theorem

A Conic Theorem

Works for any orientable triangulated manifold

A Conic Theorem

Works for any orientable triangulated manifold

A Conic Theorem

Works for any orientable triangulated manifold

Pascal’ s Theorem

B A B C A C

M M M M Car

B A C

+

Loose Ends on Circles

K L M

Loose Ends on Circles

Loose Ends on Circles

Loose Ends on Circles

Loose Ends on Circles

Loose Ends on Circles

Loose Ends on Circles