![Page 1: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/1.jpg)
Cyclic Structures andIncidence Theorems
Jürgen Richter-GebertTechnical University Munich
![Page 2: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/2.jpg)
Jürgen Richter-GebertTechnical University Munich
Stupid Proofs forInteresting (?) Theorems
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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
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Running Example1
32
654
9 8 7
The collinearity of (123), (456), (159), (168), (249), (267), (348), (357)implies the collineartity of (7,8,9).
Pappus‘s Theorem
![Page 5: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/5.jpg)
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
![Page 6: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/6.jpg)
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
![Page 7: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/7.jpg)
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
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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
![Page 9: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/9.jpg)
A Warmup
Area MethodDecomposing the cube
==> Discrete Königs NetsBobenko, Suris
![Page 10: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/10.jpg)
A Warmup
Area MethodReciprocal Diagrams
![Page 11: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/11.jpg)
A Warmup
Area MethodReciprocal Diagrams
![Page 12: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/12.jpg)
A Warmup
Area MethodReciprocal Diagrams
You never have a problem with the last edge
![Page 13: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/13.jpg)
Realizability / StrechabilityArrangements of Pseudolines
- Topological lines in RP- Two cross exactly once
Is there anequivalent
line arrangement?
2
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Realizability / Strechability
Is there anequivalent
line arrangement?
Arrangements of 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
ConjectureEvery such pseudolineis „a function graph“
==> no curls allowed!
![Page 15: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/15.jpg)
Realizability / Strechability
Is there anequivalent
line arrangement?
Arrangements of 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
ConjectureEvery such pseudolineis „a function graph“
==> no curls allowed!
![Page 16: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/16.jpg)
Realizability / Strechability
Is there anequivalent
line arrangement?
Arrangements of Pseudolines
- Topological lines in RP- Two cross exactly once
2
Excursion: How to draw pseudoline arrangements?
2nd Method:- Draw two copies on a sphere- Circle pack- connect the dots
![Page 17: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/17.jpg)
Realizability / Strechability
Is there anequivalent
line arrangement?
Arrangements of Pseudolines
- Topological lines in RP- Two cross exactly once
2
Excursion: How to draw pseudoline arrangements?
2nd Method:- Draw two copies on a sphere- Circle pack- connect the dots
FactUnique (!) representationof the arrangement
QuestionWhat are the special properties?
![Page 18: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/18.jpg)
Realizability / Strechability
Is there anequivalent
line arrangement?
Arrangements of Pseudolines
- Topological lines in RP- Two cross exactly once
2
![Page 19: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/19.jpg)
Realizability / Strechability
Is there anequivalent
line arrangement?
Arrangements of Pseudolines
- Topological lines in RP- Two cross exactly once
2
![Page 20: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/20.jpg)
Realizability / Strechability
Is there anequivalent
line arrangement?
No !!
![Page 21: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/21.jpg)
Pappus’s Theorem1 O Oa b cd e fO 1 Og h ij k lO O 1m n op q r
123456789
Another proof by algebraic cancellation
132
654
9 8 7
![Page 22: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/22.jpg)
Pappus’s Theorem1 O Oa b cd e fO 1 Og h ij k lO O 1m n op q r
123456789
(123)(159)(168)(249)(267)(348)(357)(456)(789)
Another proof by algebraic cancellation
132
654
9 8 7
![Page 23: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/23.jpg)
==> ce=bf==> iq=hr==> ko=ln==> ar=cp==> bj=ak==> fm=do==> dh=eg==> gl=ij==> mq=np
Pappus’s Theorem1 O Oa b cd e fO 1 Og h ij k lO O 1m n op q r
123456789
(123)(159)(168)(249)(267)(348)(357)(456)(789)
Another proof by algebraic cancellation
132
654
9 8 7
![Page 24: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/24.jpg)
==> ce=bf==> iq=hr==> ko=ln==> ar=cp==> bj=ak==> fm=do==> dh=eg==> gl=ij==> mq=np
Pappus’s Theorem1 O Oa b cd e fO 1 Og h ij k lO O 1m n op q r
123456789
(123)(159)(168)(249)(267)(348)(357)(456)(789)
Another proof by algebraic cancellation
132
654
9 8 7
![Page 25: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/25.jpg)
==> ce=bf==> iq=hr==> ko=ln==> ar=cp==> bj=ak==> fm=do==> dh=eg==> gl=ij==> mq=np
Pappus’s Theorem1 O Oa b cd e fO 1 Og h ij k lO O 1m n op q r
123456789
(123)(159)(168)(249)(267)(348)(357)(456)(789)
Another proof by algebraic cancellation
132
654
9 8 7
![Page 26: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/26.jpg)
==> ce=bf==> iq=hr==> ko=ln==> ar=cp==> bj=ak==> fm=do==> dh=eg==> gl=ij<== mq=np
Pappus’s Theorem1 O Oa b cd e fO 1 Og h ij k lO O 1m n op q r
123456789
(123)(159)(168)(249)(267)(348)(357)(456)(789)
132
654
9 8 7
Another proof by algebraic cancellation
![Page 27: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/27.jpg)
Pappus’s Theorem
(123)
(159)
(186)
(429) (726)
(483)
(753) (456)
(789)
Structure of the proof
13
2
654
9 8 7
![Page 28: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/28.jpg)
Pappus’s Theorem
(123)
(159)
(186)
(429) (726)
(483)
(753) (456)
(789)
Structure of the proof
13
2
654
9 8 7
![Page 29: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/29.jpg)
Pappus’s Theorem
(123)
(159)
(186)
(429) (726)
(483)
(753) (456)
(789)
c ab
e
d
f
r
q
p
j
l
k
h ig
m no
e r jb
g
o
Structure of the proof
13
2
654
9 8 7
![Page 30: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/30.jpg)
Pappus’s Theorem
(123)
(159)
(186)
(429) (726)
(483)
(753) (456)
(789)
c ab
e
d
f
r
q
p
j
l
k
h ig
m no
e r jb
g
o
Structure of the proof
13
2
654
9 8 7
![Page 31: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/31.jpg)
Pappus’s Theorem
(123)
(159)
(186)
(429) (726)
(483)
(753) (456)
(789)
c ab
e
d
f
r
q
p
j
l
k
h ig
m no
e r jb
g
o
Structure of the proof ==> a torus
13
2
654
9 8 7
![Page 32: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/32.jpg)
From: Self-Dual Configurations and Regular Graphs Coxeter, 1950
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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:
![Page 34: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/34.jpg)
Grassmann-Plücker Relations
1
2
3
x
y
[123][1xy]-[12x][13y]+[12y][13x] = OIn 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
![Page 35: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/35.jpg)
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]
132
654
9 8 7
Same cancellation pattern as before
![Page 36: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/36.jpg)
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]
132
654
9 8 7
Same cancellation pattern as before
![Page 37: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/37.jpg)
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]
132
654
9 8 7
Same cancellation pattern as before
![Page 38: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/38.jpg)
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)
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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]
54
821
09
876
[764][785] = [765][784]
![Page 40: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/40.jpg)
A non-realizable configuration
jh
d
g
e f
c
bai
(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)
![Page 41: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/41.jpg)
A non-realizable configuration
jh
d
g
e f
c
bai
(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)
![Page 42: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/42.jpg)
A non-realizable configuration
jh
d
g
e f
c
bai
(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)
![Page 43: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/43.jpg)
Six points on a conic1
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 <==>
![Page 44: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/44.jpg)
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]
12
3
4
5
6
98 7
[289][579] = +[279][589]
![Page 45: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/45.jpg)
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)
![Page 46: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/46.jpg)
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!!
![Page 47: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/47.jpg)
Coxeter’s proof of Pappos’s Theorem
![Page 48: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/48.jpg)
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The Theoremsof Ceva and Menelaos
![Page 51: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/51.jpg)
Ceva and Menelaosa1a2b1b2c1c2ABCDEFG
a1a2b1b2c1c2ABCDEH
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
![Page 52: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/52.jpg)
Ceva and Menelaosa1a2b1b2c1c2ABCDEFG
a1a2b1b2c1c2ABCDEH
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
![Page 53: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/53.jpg)
Glueing
|AX|·|BY|·|CZ||XB|·|YC|·|ZA| = 1
ABCDEFGHKLM
A
B
C
DX
Y
Z
S
R
|AZ|·|CR|·|DS||ZC|·|RD|·|SA| = 1
![Page 54: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/54.jpg)
Glueing
|AX|·|BY|·|CZ||XB|·|YC|·|ZA| = 1
ABCDEFGHKLM
A
B
C
DX
Y
Z
S
R
|AZ|·|CR|·|DS||ZC|·|RD|·|SA| = 1
|AX|·|BY|·|CR|·|DS||XB|·|YC|·|RD|·|SA| = 1
![Page 55: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/55.jpg)
Glueing
ABCDEFGHKLM
a1
b2
b3
b4
b1
a2a3
a4
= 1a1
b2 b3 b4b1
a2 a3 a4
![Page 56: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/56.jpg)
Glueinga1a2b1b2c1c2d1d2f2e1e2f1
a1
b2
b3
b4
b5
b6
b1
a2
a3 a4
a5
a6 = 1a1
b2 b3 b4 b5b6b1
a2 a3 a4 a5a6
![Page 57: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/57.jpg)
Glueing
ABCDEFHKMP
ABCDEFHKM
ABCDEFKMP+ =
A “factory” for geometric theorems
![Page 58: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/58.jpg)
A theorem on a tetrahedron
ABCDEFKMP
Front: two triangles with Ceva
![Page 59: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/59.jpg)
A theorem on a tetrahedron
ABCDEFHKM
Back: two triangles with Ceva
![Page 60: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/60.jpg)
A theorem on a tetrahedron
ABCDEFHKM
ABCDEFKMP
After glueing: an incidence theorem
![Page 61: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/61.jpg)
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
![Page 62: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/62.jpg)
A census of incidence theorems
MM M M+
CC C C+
CC
MM
+
CM M C+
MM C C+
MM C C+==> harmonic points
==> many interesing degenerate cases
==> Desargues’s Thm.
![Page 63: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/63.jpg)
Harmonic QuadruplesCC
MM
+
Front: two triangles with Ceva
![Page 64: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/64.jpg)
Harmonic QuadruplesCC
MM
+
Back: two triangles with Menelaos
![Page 65: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/65.jpg)
Harmonic QuadruplesCC
MM
+
Glued: Uniques of harmonic point construction
![Page 66: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/66.jpg)
Desargues by GlueingMM M M+
Front: two triangles with Menelaos
![Page 67: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/67.jpg)
Desargues by GlueingMM M M+
Back: two triangles with Menelaos
![Page 68: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/68.jpg)
Desargues by GlueingMM M M+
Glued: Desargues’s Theorem
![Page 69: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/69.jpg)
Six triangles
1
1
12
1
34
5
1
1
1 2
2
2
Double pyramid over triangle
Degenerate torus
...and other degenerate spheres
![Page 70: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/70.jpg)
Six triangles
1
1
12
1
34
5
1
1
1 2
2
2
Double pyramid over triangle
Degenerate torus
...and other degenerate spheres
![Page 71: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/71.jpg)
Six triangles
Six times Ceva
![Page 72: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/72.jpg)
Six triangles
Six times Ceva
Folding the triangles
![Page 73: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/73.jpg)
Six triangles
Six times Ceva -->after identification
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Six triangles
Pappus’s Theorem
![Page 75: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/75.jpg)
Moving points to infinity
Pappus affine Pappus
![Page 76: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/76.jpg)
Grid theorems
+1
-1
-1
-1
+1
+1
“row sums” = “colums sums” = “diagonal sums” = 0
![Page 77: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/77.jpg)
Another Theorem+1
-1
+1
+1
+1
-1
-1
-1
“row sums” = “colums sums” = “diagonal sums” = 0
![Page 78: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/78.jpg)
Larger Grid Theorems
+ =
Composition of grid theorems
“space of such theorems” is a vector space
“little” Pappus configurations are a basis
![Page 79: Cyclic Structures and Incidence Theorems - math.tu … · Pappus’s Theorem (123) (159) (186) (429) (726) (483) (753) (456) (789) b c a e d f r q p j l k g h i o m n e r j b g o](https://reader031.vdocuments.site/reader031/viewer/2022012914/5b5d03af7f8b9a65028ceb92/html5/thumbnails/79.jpg)
Larger Grid Theorems
Composition can be interpreted topologically
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„Geometrie der Waben“compare Blaschke 1937
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Arrangements of pseudolines
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Rombic Tilings with three Directions
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Rombic Tilings with three Directions
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Conversion of proofs
12
3
4
4
4
5
6
7
7
8
9
A
AB
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]
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Conversion of proofs
12
3
4
4
4
5
6
7
7
8
9
A
AB
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]
427479
478
347
457
467
146 137154 157
178
148
134167
127
124149 197
- Works in general- Use Tutte-Groups- and homotopy
A different Story
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From: Self-Dual Configurations and Regular Graphs Coxeter, 1950
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Graph: vertices -> brackets, edges -> bracktes differing by one letter
Glue versus matter
[12x][13y]-[12y][13x]
Grassmann Menelaus Ceva
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Glue versus matter
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Glue versus matter
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Glue versus matter
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Glue versus matter
BFP <--> CM
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And Conics ?a1
b3
b1
a3
b5
a5b2 b4
a2
a4
b6
a6
= 1a1
b2 b3 b4 b5b6b1
a2 a3 a4 a5a6
Carnot’s Theorem
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A Conic Theorem
Works for any orientable triangulated manifold
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A Conic Theorem
Works for any orientable triangulated manifold
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A Conic Theorem
Works for any orientable triangulated manifold
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Pascal’s Theorem
B
A
B
CA
C
MM
M
M
CarB
A
C
+
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Loose Ends on Circles
K
L
M
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Loose Ends on Circles
P7
P8
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Loose Ends on Circles
P7
P8
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Loose Ends on Circles
P7
P8
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Loose Ends on Circles
P7
P8
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Loose Ends on Circles
P7
P8