.1/109 p – Introduction An uence: r Cong Scissors

Introduction

An College Mathematics Leibon y of tmouth tment Gregor Dar Depar

Congruence:

Scissors

INDIRECT, IS Exp .2/109 p – Introduction An uence: r Cong Scissors } n H , n S , n E { let ∈ n

Congruence G talk, this Scissors or F .3/109 p – Introduction to to An uence: r chop Cong uent pieces r Scissors of can pieces cong e sided, w umber these n scissors (finite geodesic vided le n is G be P pro finite Q in ) a

Congruence a Q that . y and ∼ into Q reassemb sa P Scissors P m ( up e or f and compact). W Let polyhedr Q P .4/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .5/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .6/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .7/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .8/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .9/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .10/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .11/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .12/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .13/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .14/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .15/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .16/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .17/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .18/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .19/109 p – Introduction An uence: r Cong Scissors ) P ( or r ir M ∼ P

Example: .20/109 p – Introduction An uence: r Cong Scissors 0 1 2 3 . A

Notation polygon a

Some see e 0 1 2 3 w Here .21/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1 2 . 3 C polygon 0 Notation 4 5 6 1 2 3 7 8 9 second a

Some see e 0 1 2 3 4 5 6 7 8 9 w Here .22/109 p – Introduction An 6 uence: r Cong . Scissors 5 C F 9 A see e w 4 7 Here . 8 ∅ = ) Q 0 T 1 P 2 ( int 3 vided pro Q

Notation S P be Q

Some F 1 0 2 3 4 5 6 7 8 9 P let e W .23/109 p – Introduction An 6 uence: r Cong Scissors 5 , B 4 polygon this with t 0 star e W . C F A of w vie 0 4 5 Notation 6 w ne a e

Some tak e w w No .24/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1 2 3 0 Notation 4 5 6 1 2 3 7 8 9 . D

Some is 0 1 2 3 4 5 6 7 8 9 here and .25/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1 2 3 . 0 4 5 6 1 2 3 7 8 9 Notation D F B e v

Some ha e w Here .26/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1 equal. 2 are 3 y the since , D 0 Notation 4 5 6 1 2 3 7 8 9 F B ∼ C Some F 0 1 2 3 4 5 6 7 8 9 A Note .27/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 anslation. tr 0 a 1 y b 2 er 3 diff y the since 0 Notation 4 5 6 1 2 3 7 8 9 D ∼ C

Some more 0 1 2 3 4 5 6 7 8 9 ther Fur .28/109 p – Introduction An 6 . B uence: r ∼ Cong A Scissors 5 then , D 9 ∼ C and 4 D 7 F 8 B ∼ C 0 F 1 A 2 if , n 3 G in that ts Theorem s 0 4 5 6 1 2 3 7 8 9 asser v'

Zyle Theorem 0 1 2 3 4 5 6 7 8 9 s v' Zyle .29/109 p – . Introduction D An 6 and uence: r C Cong up Scissors 5 pair e w 9 Here . 4 7 8 correspondences 0 1

Theorem our 2 s

v' 3 using

Zyle region the 0 4 5 6 1 2 3 7 8 9 divide e w

Realizing it e 0 1 2 3 4 5 6 7 8 9 v pro o T .30/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1

Theorem 2 s . D v' 3 F B

Zyle and C 0 4 5 6 1 2 3 7 8 9 F A up

Realizing pair e 0 1 2 3 4 5 6 7 8 9 w Here .31/109 p – . D Introduction F An 6 B uence: ∼ r C Cong F A Scissors 5 via 9 paired are 4 7 8 regions 0 the 0 1

Theorem 2 Note s

v' 3 act".

Zyle "subtr us let 0 4 5 6 1 2 3 7 8 9 , ings pair Realizing 0 1 2 3 4 5 6 7 8 9 our Using .32/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 . D F 0 B 1

Theorem ∼ 2

s C F v' 3 A via

Zyle D in 0 4 5 6 1 2 3 7 8 9 ed w vie be

Realizing can 0 1 2 3 4 5 6 7 8 9 1 Note .33/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1 .

Theorem 4 2 s with

v' 3

Zyle identi®ed be 0 4 5 6 1 2 3 7 8 9 can 1 , D

Realizing ∼ 0 1 2 3 4 5 6 7 8 9 C rom F .34/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 . D F 0 B 1

Theorem ∼ 2 s C F v' 3 A via

Zyle B in 0 4 5 6 1 2 3 7 8 9 ed w vie be

Realizing 0 1 2 3 4 5 6 7 8 9 can 4 w No .35/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1

Theorem 2 s . 3 v' 4 and 1 Zyle 0 4 5 6 1 2 3 7 8 9 identi®ed e v ha

Realizing e w 0 1 2 3 4 5 6 7 8 9 Hence .36/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1

Theorem 2 s . 2 v' 3 or f

Zyle same the 0 4 5 6 1 2 3 7 8 9 do can e

Realizing w 0 1 2 3 4 5 6 7 8 9 ly Similar .37/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1

Theorem 2 s

v' 3 action"...

Zyle "subtr 0 4 5 6 1 2 3 7 8 9 called is

Realizing 0 1 2 3 4 5 6 7 8 9 process This .38/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 old. 0 1 from Theorem 2 s

v' 3 bijection w

Zyle ne uct 0 4 5 6 1 2 3 7 8 9 constr to y a Realizing w 0 1 2 3 4 5 6 7 8 9 a is and .39/109 p – Introduction An 6 uence: r Cong Scissors 5 9 .... D F 4 B 7 ∼ 8 C F A 0 1

Theorem using 2 s ight 3

v' r the to Zyle ving 0 4 5 6 1 2 3 7 8 9 mo y b that

Realizing is 0 1 2 3 4 5 6 7 8 9 y e k The .40/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1

Theorem 2 s

v' 3 ....

Zyle D ∼ C 0 4 5 6 1 2 3 7 8 9 using left

Realizing 0 1 2 3 4 5 6 7 8 9 the to and .41/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1

Theorem 2 . s

v' 3 loop in

Zyle caught 0 4 5 6 1 2 3 7 8 9 get er v ne

Realizing can 0 1 2 3 4 5 6 7 8 9 e w that .42/109 p – .... Introduction D An 6 F B uence: r ∼ Cong C F Scissors 5 A 9 from path ing 4 7 enter 8 one 0 has 1

Theorem 2 s node

v' 3 each

Zyle namely 0 4 5 6 1 2 3 7 8 9 simple is

Realizing 0 1 2 3 4 5 6 7 8 9 reason The .43/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1

Theorem 2 s . v' 3 D ∼ C Zyle from 0 4 5 6 1 2 3 7 8 9 path xiting Realizing e 0 1 2 3 4 5 6 7 8 9 one and .44/109 p – Introduction An 6 uence: r Cong Scissors 5 9 .... 4 A 7 in 8 ing star 0 1

Theorem loop 2 a s

v' 3 such

Zyle enter to y a 0 4 5 6 1 2 3 7 8 9 w no is

Realizing there 0 1 2 3 4 5 6 7 8 9 Hence .45/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1

Theorem 2 s

v' 3

Zyle 0 4 5 6 1 2 3 7 8 9

Realizing 0 1 2 3 4 5 6 7 8 9 and... .46/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1

Theorem 2 s

v' 3

Zyle 0 4 5 6 1 2 3 7 8 9

Realizing 0 1 2 3 4 5 6 7 8 9 and... .47/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1

Theorem 2 s

v' 3

Zyle 0 4 5 6 1 2 3 7 8 9 ly

Realizing 0 1 2 3 4 5 6 7 8 9 Similar and... .48/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1

Theorem 2 s

v' 3

Zyle 0 4 5 6 1 2 3 7 8 9 . B in

Realizing land 0 1 2 3 4 5 6 7 8 9 ust m e w .49/109 p – Introduction An 6 uence: r Cong Scissors 5 9 4 7 8 0 1 bijection.

Theorem 2 s xplicit v' 3 e an e

Zyle v ha e w 0 4 5 6 1 2 3 7 8 9 and , B

Realizing ∼ A 0 1 2 3 4 5 6 7 8 9 Hence .50/109 p – Introduction An ated uence: r Cong Scissors gener roup g a. > P < Abelian polyhedr

Group free

Free the

Big geodesic

A all y b Let denote .51/109 p – Introduction An . ated uence: r Cong roup Scissors g gener roup a. g > LARGE P y < er Abelian polyhedr v a

Group free is

Free the this

Big geodesic

A all y Let denote Notice b .52/109 p – Introduction An uence: r Cong Scissors S R − m or − f ) S the R Q ( in + som P elements I all y b ated . R Relations = gener Q F

The roup P subg the and where Let .53/109 p – Introduction An uence: r Cong Scissors , > R P < ≡ ) n G (

-group is S K

First

The Let .54/109 p – Introduction An uence: r Cong Scissors , > R P . ] < Q Q [ ≡ ∼ = ) ] n P P [ G (

-group is S K if

First e v only

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Shift . 2

Hyperbolic H in

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of xample e our at

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pieces of 8

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of orced f are e w Limit in®nity At .62/109 p – Introduction An uence: r Cong

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Scissors 8

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Hyperbolic

the anslation. tr of ®nite a y b Limit produced .64/109 p – Introduction An uence: r in®nite Cong Scissors with Letting uch. classes ) n m H ( ∞ is loose S scissors .... le e don't v e stab w ha

Points e w the

Ideal tunately tices or er F denote v .65/109 p – Introduction An uence: r Cong Scissors . ) n H ( is S = ∼ ) Sah): n H ( ∞ is S

Points (Dupont,

Ideal Theorem .66/109 p – e Introduction v An remo uence: r y Cong the Scissors isingly) pr sur less

en

v e 8 (perhaps and nothing, add surprising a

too polyhedr

Not ideal Notice nothing. .67/109 p – Introduction An uence: r Cong Scissors , ) if Q , ( 2 . ea G ] Q In [ Ar = = ] ) P P [

Theorem ( ea (Euclid, Ar ,Classical):

Fundamental Gauss Theorem then .68/109 p – Introduction An and uence: , r Cong B , Scissors A ) angles C , B

Theorem with A, ( T iangle tr

Fundamental the

Proof: . Let denote C .69/109 p – Introduction An and uence: r , Cong 0 , Scissors A angles ) A ( with C iangle tr the

Notation . Let denote 0 .70/109 p – , 0 Introduction , An 0 uence: r Cong Scissors ideal angles with unique . a y I is iangle tr there isometr ideal to up the Recall .

Notation 0 iangle Let denote and tr .71/109 p – ] Introduction I An 2[ uence: r − Cong )] Scissors C ( C [ + )] B ( C [ + )] A ( C [ =

1 )] C , B 1:

Lemma A, ( T [ Lemma .72/109 p – Introduction An uence: r Cong Scissors T . 1 iangle B tr

Lemma + A yperbolic

of h C ®nite a

Proof with Begin .73/109 p – Introduction An uence: r Cong Scissors . A T−C(A)

1 angle B with x te + Lemma er v A A − of the C to )] A (

Proof C [ − Adjoin .74/109 p – Introduction An uence: r Cong Scissors − . B T−C(A)−C(B) B

1 angle B with x te

Lemma er v + A A

of the − C to )] B (

Proof C [ − Adjoin .75/109 p – Introduction An uence: r Cong Scissors . − C T−C(A)−C(B)−C(C) B

1 angle B with x te

Lemma er v + A A

of the C − to )] C C (

Proof C [ − − Adjoin .76/109 p – Introduction An uence: r Cong Scissors − +I + − T−C(A)−C(B)−C(C) B

1 B +

Lemma + A + . A of ] I [ C − red C

Proof this − add w No .77/109 p – Introduction An uence: r Cong Scissors _ + +I+I + − + T−C(A)−C(B)−C(C) B

1 B + +

Lemma + A A

of + C − . ] I [ C

Proof red − this and .78/109 p – Introduction An uence: r Cong Scissors out.... _ + +I+I+I cancel + − + T−C(A)−C(B)−C(C) B

1 signs of lot B + a +

Lemma that + − + A + A

of + C Notice − . ] I [ C + −

Proof + red + + − this and .79/109 p – Introduction An uence: r Cong Scissors needed. T=−2I+ +I+I+I−I=0 as , ] C(A)+C(B)+C(C) I [ T−C(A)−C(B)−C(C)

1 of class +

Lemma scissors of the with

Proof us ving ...lea .80/109 p – )] Introduction An D uence: r ( Cong C [ Scissors + . )] D C ( + C [ C = = )] B B ( + if C [ A

2 + 2: only )]

Lemma A ( and C [ if Lemma .81/109 p – Introduction An uence: r Cong Scissors )] B ( C [ +

2 )] A ( C [ B with t

Lemma A star of lemma

Proof this e v pro o T .82/109 p – Introduction An uence: r Cong Scissors

2 B

Lemma A

of

Proof and... .83/109 p – Introduction An uence: r Cong Scissors

2 B

Lemma A

of

Proof and... .84/109 p – Introduction An uence: r Cong Scissors

2 B

Lemma A

of

Proof and... .85/109 p – Introduction An uence: r Cong Scissors

2 B

Lemma A

of

Proof and.. .86/109 p – Introduction An uence: r Cong Scissors

2 B

Lemma A

of

Proof and.. .87/109 p – Introduction An uence: r Cong Scissors

2 B

Lemma A

of . together

Proof them k stic .88/109 p – Introduction An uence: r Cong Scissors

2 ... top B the at Lemma A

of iangle tr ideal

Proof the Notice .89/109 p – Introduction An uence: r Cong Scissors . )] B + A ( C [ + ] 2 I [ a as A+B Lemma region

of this w vie to

Proof us wing allo .90/109 p – Introduction An uence: r Cong Scissors ic.... isometr all

2 are that since

Lemma ] A+B I [ of the shift

Proof can e w w No .91/109 p – Introduction An uence: r Cong Scissors

2

Lemma A+B

of

Proof until... .92/109 p – Introduction An uence: r Cong Scissors

2

Lemma A+B

of

Proof until... .93/109 p – Introduction An uence: r Cong Scissors

2 ... to

Lemma A+B=C+D enient of v con it

Proof ®nd e w until .94/109 p – Introduction An uence: r Cong Scissors

2

Lemma C+D

of as: region

Proof our xpress e .95/109 p – Introduction An uence: r Cong Scissors

2 D needed. as ,

Lemma C )] D ( of C [ + )] C

Proof ( C [ Hence .96/109 p – 1 easily Introduction An ws if uence: r lemma ollo f Cong only By . this Scissors ely and if (and hold sum respectiv } i . B )] i { angles equality B . ( i the and C this B [ } ] i to Q N N =1 =1 A [ i i X X and { = = = ] i )] P related i A [ A around angles ( N =1 i X C [ with N =1 af®nely , i X angles is our 2) e iangles area v

etch tr the mo N

Sk lemma can that into of e Q w wn if 2

Proof and proof kno P only ell the w lemma m is and or It f By Break if .97/109 p – Introduction Q An uence: r Cong and Scissors P a ) Q ( . ] ol Q V [ polyhedr = 6 = ) ] P P [ problem ( ol V

Third geodesic s ®nd that ,

Hilbert' 3 E ut In such b .98/109 p – Introduction An uence: r Cong Scissors . ) Q ( . ] ehn Q [ D if = = , ] 3 ) P P G [ ( ehn

Answer D s (Dehn)

Dehn' then Theorem .99/109 p – Introduction An uence: r Cong ) Scissors e ( θ ⊗ . ) e Z ( l π R 2 P ∈ Z e X ⊗ = R )

ariant P ∈

v (

In of ehn D Dehn element an

The as ed w vie Where .100/109 p – with Introduction e An uence: edge r Cong the Scissors label e W polyhedron. a chop us let iant . ariant ar ϕ v v in

In is ) ϕ angle e l ( al ϕ ⊗ ) Dehn e dihedr ( l P and ∈ e The l P see length o T its .101/109 p – Introduction An uence: r Cong Scissors .

ariant v plane

In reen g the

Dehn along

The sliced e e'v W .102/109 p – Introduction An uence: r Cong Scissors

ariant v

In

Dehn

The and... .103/109 p – Introduction An uence: r Cong Scissors

ariant v

In

Dehn

The and... .104/109 p – Introduction An uence: r Cong Scissors

ariant v

In

Dehn

The and... .105/109 p – Introduction An uence: r Cong Scissors ϕ 2 , l Z π R 2 Z ϕ ⊗ R 1 in l ϕ 2 ⊗ ariant l v = 1 In ϕ ⊗ 2 l=l +l l +

Dehn ϕ ⊗ 1 l and .106/109 p – Introduction An uence: r Cong Scissors . Z h π R 2 Z π−φ ⊗ h R in φ 0 = π ⊗ h = )

ariant ψ v − π

In ( ⊗ h + 1 Dehn ψ ⊗ h and .107/109 p – Introduction An Note uence: r . Cong olume v Scissors 0 0 same 6 = = π the 3)) / e ⊗ v (1 h 6 ha = ) arccos 2 π ⊗ ⊗ ahedron h 6(1 ational. tetr 1 = irr 12( ) = be et. ) to regular .T ube arcos(1/3) eg wn C and ( R ( sho ehn ehn cube D π/2 D Example easily the h s is that 3) / such

Dehn' h arccos(1 ut Choose since b .108/109 p – Introduction An uence: r Cong Scissors ]) Q ]) ([ Q ([ . if ] ehn ol , Q 3 V D [ E = = = ] In ]) ]) P P P [ ([ ([ ol V Theorem ehn s D (Sydler):

Sydler' Theorem and then .109/109 p – Introduction An uence: r Cong if , Scissors 3 ]) G Q ]) ([ Q In ([ : . ] ehn ol Q V D [ = = = ] ]) ]) P P P [ Conjecture ([ ([ Conjecture y ol V ehn D Sufficienc

Fundamental Dehn and then