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

A non-realizable configuration (abi) ==> [abh][agi]=+[abg][ahi] (acf) ==> [adf][acj]=-[acd][afj] (adh) ==> [abd][afh]=-[abh][adf] b a i (bce) ==> [bcd][bej]=-[bde][bcj] (bdg) ==> [abg][bde]=-[abd][beg] c (cdj) ==> [acd][bcj]=+[acj][bcd] (efj) ==> [afj][egj]=+[aej][fgj] (egi) ==> [aeg][ghi]=-[agi][egh] d (fhi) ==> [ahi][fgh]=-[afh][ghi] e (ghj) ==> [egh][fgj]=+[egj][fgh] f [aeg][bej]=+[aej][beg] ==> (abe) or (egj) j h g

Six points on a conic 2 1 3 6 4 5 Six points 1 , 2 , 3 , 4 , 5 , 6 are on a conic <==> [123][156][426][453] = [456][126][254][423]

Pascal’ s Theorem 2 1 3 8 7 9 6 4 [159][257] = -[125][579] [126][368] = +[136][268] [245][279] = -[249][257] 5 [249][268] = -[246][289] [346][358] = +[345][368] [135][589] = -[159][358] [125][136][246][345] = +[126][135][245][346] [289][579] = +[279][589]

Problems of this method (binomial-proofs) Usually large search space What is the “structure” of the proof How to cut down the search space in advance

Problems of this method (binomial-proofs) Usually large search space What is the “structure” of the proof How to cut down the search space in advance A ambitious dream: Look at a theorem.... “see” its structure.... .... and produce a proof immediately!!

Coxeter’ s proof of Pappos’ s Theorem

The Theorems of Ceva and Menelaos

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

Ceva and Menelaos C C Y Y X D X A B A Z B Z |AZ|·|BX|·|CY| |AZ|·|BX|·|CY| |ZB|·|XC|·|YA| = 1 |ZB|·|XC|·|YA| = -1 [CDA] [ADB] [BDC] [XYA] [XYB] [XYC] · · · · [CDB] [ADC] [BDA]= -1 = 1 [XYB] [XYC] [XYA]

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

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

Glueing a 1 b 4 a 4 b 1 a 1 a 2 a 3 a 4 = 1 b 1 b 2 b 3 b 4 b 3 a 2 a 3 b 2

Glueing b 6 a 1 a 1 a 2 a 3 a 4 a 5 a 6 a 6 = 1 b 1 b 1 b 2 b 3 b 4 b 5 b 6 b 5 a 2 a 5 b 2 b 4 a 3 a 4 b 3

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 of a tetrahedron

A census of incidence theorems M C M + + C C ==> harmonic points M C M M C ==> many interesing + + C C C C degenerate cases M C M C + + M M ==> Desargues’ s Thm. M C M M

Harmonic Quadruples C M + C M Front: two triangles with Ceva

Harmonic Quadruples C M + C M Back: two triangles with Menelaos

Harmonic Quadruples C M + C 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 2 2 5 1 2 1 1 3 4 1 2 1 Double pyramid Degenerate torus over triangle ...and other degenerate spheres

Six triangles Six times Ceva

Six triangles Folding the triangles Six times Ceva

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

Conversion of proofs C 4 7 [124][137]=[127][134] [154][197]=[157][194] 9 [184][167]=[187][164] [427][491]=[421][497] B A [457][461]=[451][467] 8 2 [487][431]=[481][437] 1 [721][764]=[724][761] [751][734]=[754][731] 7 4 [781][794]=[784][791] 6 3 A B 5 4 7 C

Conversion of proofs C 4 7 [124][137]=[127][134] [154][197]=[157][194] 149 197 9 [184][167]=[187][164] 124 178 [427][491]=[421][497] B A [457][461]=[451][467] 8 479 2 [487][431]=[481][437] 1 [721][764]=[724][761] 127 427 478 148 [751][734]=[754][731] 7 4 [781][794]=[784][791] 6 167 467 347 134 3 - Works in general 457 A B - Use Tutte-Groups 146 137 5 - and homotopy 154 157 A different Story 4 7 C

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

Glue versus matter Graph: vertices -> brackets, edges -> bracktes differing by one letter Grassmann Menelaus Ceva [12x][13y]-[12y][13x]

Glue versus matter

Glue versus matter BFP <--> CM

And Conics ? a 1 a 1 a 2 a 3 a 4 a 5 a 6 = 1 b 6 b 5 b 1 b 2 b 3 b 4 b 5 b 6 a 2 Carnot’ s Theorem b 1 a 6 b 2 b 4 a 4 a 5 a 3 b 3

A Conic Theorem Works for any orientable triangulated manifold

Pascal’ s Theorem A M B C M M M B C A B C Car + A

Loose Ends on Circles K L M

Loose Ends on Circles

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

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