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Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

iling with unique vertex oron
joint work with hirk prettl¤ from fielefeld niversity oh elexey qrer
Moscow State University and Delone Lab oratory of Yaroslavl State University, Russia

pifth hisrete qeometry nd elgeri gomintoris gonferene epril IVD PHIQ

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

ilings
he(nition e olletion T of @onvexA polytopes in Rd is lled tiling if union of ll polytopes from T is Rd Y they do not interset in internl pointsY every ll intersets only (nite numer of polytopes from T F
(lo cally nite)

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

ilings
he(nition e olletion T of @onvexA polytopes in Rd is lled tiling if union of ll polytopes from T is Rd Y they do not interset in internl pointsY every ll intersets only (nite numer of polytopes from T F he(nition e tiling is lled face-to-face or tiles is fe of othF
normal (lo cally nite)

if intersetion of ny two

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

eriodi nd periodi tilings
he(nition e tiling of Rd is lled groupF
p erio dic

if it hs d Edimensionl trnsltion

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

eriodi nd periodi tilings
he(nition e tiling of Rd is lled groupF
p erio dic

if it hs d Edimensionl trnsltion

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

eriodi nd periodi tilings
he(nition e tiling of Rd is lled
ap erio dic

if it hs no trnsltion groupF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

eriodi nd periodi tilings
he(nition e tiling of Rd is lled
ap erio dic

if it hs no trnsltion groupF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

grystllogrphi tilings

he(nition e tiling T is lled crystallographic if fundmentl domin of its symmetry group hs ompt fundmentl dominF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

grystllogrphi tilings

he(nition e tiling T is lled crystallographic if fundmentl domin of its symmetry group hs ompt fundmentl dominF sn iuliden spe rystllogrphi tiling is the sme s periodi tilingF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

grystllogrphi tilings

he(nition e tiling T is lled crystallographic if fundmentl domin of its symmetry group hs ompt fundmentl dominF sn iuliden spe rystllogrphi tiling is the sme s periodi tilingF fut this de(nition n e used in ryperoli spe Hd tooF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

gorons of tile

gonsider n ritrry tile

P

of tiling T

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

gorons of tile

gonsider n ritrry tile he(nition

P

of tiling T

he k -th corona of P is the olletion of ll tiles of T tht n e rehed from P e t most k steps ross fets of the tilingF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

vol heorem

heorem @qenerlized vol heorem y xF holilin nd wF htogrinA
A tiling of

Rd

(or

Hd

) is crystallographic i for some k the

following conditions hold. For the numb er N (k

)

of k -coronas we have: N (k

)=

N (k

+ I)

and this numb er is nite.

(k + I)

For every i the symmetry groups Si (k

)

and Si (k

+ I)

of k - and

-coronas of the i -th typ e coincides.

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

ertex oron
gonsider n ritrry vertex he(nition he set of ll polytopes ontins
A A

of the tiling T F is lled
the vertex corona

of

A

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

ertex oron
gonsider n ritrry vertex he(nition he set of ll polytopes ontins he(nition e tiling T is sid to e unique vertex corona tiling if ll its vertex orons re ongruentF his mens not only olletions of polytopes re the sme ut lso tht they rrnged t orrespondent verties in the sme wyF
A A

of the tiling T F is lled
the vertex corona

of

A

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

nique vertex oron nd periodiity

uestion
Is it true that unique vertex corona of a tiling p erio dic (crystallographic)?

T

implies that

T

is

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

nique vertex oron nd periodiity

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

nique vertex oron nd periodiity

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

sde of feEtoEfe lssi(tionX topologil struture

vemm
If every vertex corona contains n p olygons and ki of them are i -gons then ki i n

=

-P . P

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

sde of feEtoEfe lssi(tionX topologil struture

vemm
If every vertex corona contains n p olygons and ki of them are i -gons then ki i n

=

-P . P

prom this equlity we n derive only (nitely mny lol strutures of oronsF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

sde of feEtoEfe lssi(tionX topologil struture

vemm
If every vertex corona contains n p olygons and ki of them are i -gons then ki i n

=

-P . P

prom this equlity we n derive only (nitely mny lol strutures of oronsF end only (nitely mny glol topologil strutures of the whole tilingF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

sde of feEtoEfe lssi(tionX topologil struture
por exmpleD there is solution
k
3

= I,

k4

= P,

k6

= IF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

sde of feEtoEfe lssi(tionX topologil struture
por exmpleD there is solution
k
3

= I,

k4

= P,

k6

= IF

wo theoretil lol strutures re

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

sde of feEtoEfe lssi(tionX topologil struture
por exmpleD there is solution
k
3

= I,

k4

= P,

k6

= IF

end the only possile topologil struture is the followingX

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

sde of feEtoEfe lssi(tionX segments nd ngles
e mrk segments tht re equl with the sme olorF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

sde of feEtoEfe lssi(tionX segments nd ngles
e mrk segments tht re equl with the sme olorF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

sde of feEtoEfe lssi(tionX segments nd ngles
e mrk segments tht re equl with the sme olorF

sn this prtiulr se there re two possiilitiesF

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

ome exmples of single oron tilings

Figure : Tiling with regular triangle, hexagon and trapezoid.
A.Garb er Tiling with unique vertex corona MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

ome exmples of single oron tilings

Figure : Tiling with two dierent rectangles and trapezoid.
A.Garb er Tiling with unique vertex corona MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

ome exmples of single oron tilings

Figure : Tiling with three hexagons.
A.Garb er Tiling with unique vertex corona MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

ixmple of nonEperiodi tiling

Figure : Tiling with one-dimensional translation group.
A.Garb er Tiling with unique vertex corona MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

sde of non feEtoEfe lssi(tion

vemm
Assume every vertex corona of A contains n p olygons one of which contains vertex on its side. And ki of them that has A as a vertex are i -gons. Then ki i n

=

-P . P

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

roperties of twoEdimensionl tilings
glim
If a two-dimensional tiling could b e non-p erio dic.

T

has a unique vertex corona then it

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

roperties of twoEdimensionl tilings
glim
If a two-dimensional tiling could b e non-p erio dic.

T

has a unique vertex corona then it

glim
But

T

always has at least one-dimensional translation group GT .

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

roperties of twoEdimensionl tilings
glim
If a two-dimensional tiling could b e non-p erio dic.

T

has a unique vertex corona then it

glim
But

T

always has at least one-dimensional translation group GT .

glim
If GT is one-dimensional then we will need to use corona CT and its reected image (rotations are not enough).

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

purther questions out tilings in ritrry dimensions

uestion
What is the minimal dimension of translation group GT a tiling with unique vertex corona can have?

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

purther questions out tilings in ritrry dimensions

uestion
What is the minimal dimension of translation group GT a tiling with unique vertex corona can have?

uestion
Can a tiling with unique non-reected vertex corona b e non-p erio dic?

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

istimtes for feEtoEfe tilings

heorem @hF prettl¤ eFqFA ohD
There are d -dimensional face-to-face tilings with unique vertex corona with translation group of dimension at most d

P

.

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

istimtes for feEtoEfe tilings

heorem @hF prettl¤ eFqFA ohD
There are d -dimensional face-to-face tilings with unique vertex corona with translation group of dimension at most d

But for now we need to use b oth reections of corona.

P

.

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

eperiodi non feEtoEfe tiling

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

eperiodi non feEtoEfe tiling

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

eperiodi non feEtoEfe tiling

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

eperiodi non feEtoEfe tiling

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

f¤ ozky tiling or¤

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

hul tiling is nonErystllogrphi

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU


Perio dic and ap erio dic tilings

Unique vertex corona

Planar tilings

Higher dimensions

Hyp erb olic space

THANK YOU!

A.Garb er Tiling with unique vertex corona

MSU and Delone Lab of YSU