The Math Behind Escher's Picture Gallery

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The Math behind Eschers Print Gallery

Enrique Acosta

Department of Mathematics

University of Arizona

September 2009


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What Escher wanted to do


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A self referential picture


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A self referential picture


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A self referential picture where the self reference is thepicture itself.


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As you go around the picture the scale increases so that whenyou return to where you started, you are now inside the selfreference.


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A town with a gallery and a guy looking at a picture of the townwith the gallery and him looking at the picture ...

but he is both the guy looking at the picture and the guy in the

picture!


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As you go along each side the scale increases by a factor of 4.


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How he did it


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How he did it

Draw a straight picture and copy it to a special grid thattwists it.


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How he did it

Draw a straight picture and copy it to a special grid thattwists it.

Use curved grid to preserve angles.


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Eschers Grid


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The Straight picture

The self reference of Eschers picture is 256 times smaller

than the outside picture, so painting the whole straightpicture was virtually impossible.


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Instead, he drew portions of the picture at varying scales.

Th S i h i


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Instead, he drew portions of the picture at varying scales.

Using his grid we can unwind his final picture to see thestraight one.


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The Math


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The Math

The Math


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The Math

Every image that contains a copy of itself has a center.

The Math


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e at


The Math


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Now place the picture in the complex plane with the center at theorigin.

The Math


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The fact that the picture contains a copy of itself just meansthat it is invariant under multiplication by a scalar r, which we

will call the period.

The Math


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The fact that the picture contains a copy of itself just meansthat it is invariant under multiplication by a scalar r, which we

will call the period. Eschers twisted picture SHOULD have a complex period.

That is, it should be a picture that is invariant under both arotation and a scaling.


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What we will try to do


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Start with a straight self referential picture (i.e., with a realperiod r).



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Transform it with a map C C so that:



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Transform it with a map C C so that: The map is conformal.



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Transform it with a map C C so that: The map is conformal. The transformed picture has complex period.



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Transform it with a map C C so that: The map is conformal. The transformed picture has complex period. A loop in the transformed picture around the center

corresponds to a path around the center of the original picture

going from a point z to rz (r is the real period).


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Images with complex period


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A black and white image with a complex period can be

though of as a map

f : C {black, white}

such that

f(z) = f(z)

for all z C.



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A black and white image with a complex period can be

though of as a map

f : C {black, white}

such that

f(z) = f(z)

for all z C.

If we drop the origin we can actually think of this f as afunction

f : C

/ {black, white}

where = {n | n Z} is the subgroup ofC generated by.

What is C/ for || > 1?


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Let = rei. We can picture C/ like below:

What is C/ for || > 1?


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Let = rei. We can picture C/ like below:

It is a Torus!

Coverings, Fundamental Groups


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exp : C C/

z [ez

] = ez



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exp : C C/

z [ez

] = ez

exp is a covering space map and also a group homomorphism.



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exp : C C/

z [ez

] = ez


The kernel is the lattice L = 2iZ+ log()Z.



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exp : C C/

z [ez

] = ez



The isomorphism C/L = C/ is actually an isomorphism

of Riemann surfaces (i.e. it is biholomorphic) and gives usanother way to picture the torus C/.



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exp : C C/

z [ez

] = ez



The isomorphism C/L = C/ is actually an isomorphism

of Riemann surfaces (i.e. it is biholomorphic) and gives usanother way to picture the torus C/.

If || = 1, we can think ofL as the fundamental group of the

torus.

The Fundamental Group ofC/


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Every element ofL corresponds to a unique loop ofC/.

The Fundamental Group ofC/


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Every element ofL corresponds to a unique loop ofC/.

This loop is the image under the exponential map of any pathstarting at 0 and ending at the element ofL.

Alternatively, L are the endpoints of the lifts of loops in1(C

/, 1).

Back to Eschers picture


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Start with a picture with real period r,

f : C/r {black, white}

Transform it with a map C C so that: The map is conformal. The map sends 1 to 1. The transformed picture has complex period . The process can be reversed.



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Start with a picture with real period r,

f : C/r {black, white}

Transform it with a map C C so that: The map is conformal. The map sends 1 to 1. The transformed picture has complex period . The process can be reversed.

If this happens then we get an isomorphism (biholomorphic)

C/r C/ between Riemann surfaces sending 1 to 1.


By the theory of covering spaces this map lifts to a map


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By the theory of covering spaces this map lifts to a map

C //

exp

C

exp

C/r // C/

The top map is holomorphic and we may assume that it sends0 to 0.

This map induces isomorphism C/Lr C/L betweenRiemann surfaces.

The theory of complex tori as Riemann surfaces (or complexelliptic curves) implies that the map C C is multiplicationby a scalar which satisfies

Lr = L.

Loop Correspondence

Th l d


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The loop correspondence

Loop Correspondence

Th l d


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Loop Correspondence

Th l d


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implies that multiplication by sends the element2i + log(r) to the element 2i.

Loop Correspondence



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implies that multiplication by sends the element2i + log(r) to the element 2i.

So

=2i

2i + log(r)

Loop Correspondence


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=2i

2i + log(r)

Finally, since we know , and Lr = L allows us to find . So, starting with r and the loop correspondence that Escher

wanted, we can find and . So we can produce Escherspicture mathematically.

The mathematically precise grid


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The mathematically precise picture


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The mathematically precise picture


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VIDEO

Credits


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Hendrik Lenstra (Elliptic curve factorization) started towonder about this on a plane when he found the picture on amagazine.

Him and Bart de Smit wrote the article The MathematicalStructure of Eschers Print Gallery which was published in theNotices of the AMS on April 2003.

They also created a webpage with lots of images and videos.http://escherdroste.math.leidenuniv.nl/

Google Escher Droste to see what people have done with

this.


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Variations


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Variations


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Variations


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Form Flickrs Escher Droste Print Gallery group (1540 images)trufflepig droste copy by manyone1


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VIDEO

The Math Behind Escher's Picture Gallery

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