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
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
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 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
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
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
A Conic Theorem Works for any orientable triangulated manifold
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
Loose Ends on Circles
Loose Ends on Circles
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