escher exhibition-escher and topology _4º eso_

8/3/2019 Escher Exhibition-Escher and Topology _4 ESO_

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Activity: Escher Exhibition (4 ESO) Mathematics and History Departments

IES Albayzn (Granada) Page 1

Before visiting Escher Exhibition:Escher and Topology1

Escher was very interested in visual aspects of Topology, a branch of

mathematics just coming into full flower during his lifetime. The Mbius strip is

perhaps the prime example, and Escher made many representations of it. It has

the curious property that it has only one side, and one edge. Thus, if you trace

the path of the ants in Mbius Strip II, you will discover that they are not walking

on opposite sides of the strip at all they are all walking on the same side.

Eschers Topological Images

Balcony Print Gallery

Mbius Strip IIPrint Gallery, (grid-paper sketch)

1 This activity has been developed using different materials selected from the followingbibliographic sources:http://www.mathacademy.com/pr/minitext/escher/index.asp

AAVV, A Survey of Mathematics with Applications, Eighth Edition (Pearson Education, 2009)


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Another very remarkable lithograph, called Print Gallery, explores both the

logic and the topology of space. Here a young man in an art gallery is looking

at a print of a seaside town with a shop along the docks, and in the shop is an

art gallery, with a young man looking at a print of a seaside town . . . but wait!

What's happened?

All of Escher's works reward a prolonged stare, but this one does especially.

Somehow, Escher has turned space back into itself, so that the young man is

both inside the picture and outside of it simultaneously. The secret of its making

can be rendered somewhat less obscure by examining the grid-paper sketch

the artist made in preparation for this lithograph. Note how the scale of the grid

grows continuously in a clockwise direction. And note especially what this trick

entails: A hole in the middle. A mathematician would call this a singularity, a

place where the fabric of the space no longer holds together. There is just no

way to knit this bizarre space into a seamless whole, and Escher, rather than try

to obscure it in some way, has put his trademark initials smack in the center of

it.

As we have seen Escher was very interested in Topology. Now we will try to

clarify the meaning of this branch of mathematics.

Topological Equivalence

Someone once said that topologist is a person who does not

know the difference between a doughnut and a coffee cup. Two

geometric figures are said to be topologically equivalent if one

figure can be elastically twisted (torcida), stretched (estirada),

bent (doblada), or shrunk (encogida) into the other figure without

puncturing (perforar) or ripping (rasgar) the original figure. If a

doughnut is made of elastic material, it can be stretched, twisted,

bent, shrunk, and distorted until it resembles a coffee cup with a

handle, as shown in the picture below.

In topology, figures are classified according to their genus. Thegenusof an object is determined by the number of holes that go through the


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object. A cup and a doughnut each have one hole and are of genus 1 (and are

therefore topologically equivalent). Notice that the cup handle is considered a

hole, whereas the opening at the rim of the cup (borde de la taza) is not

considered a hole.

The following chart illustrates the genus of several objects.

Marble. Genus 0 Doughnut.

Genus 1

Strainer. Genus 3 or more.

Bowling ball.

Genus 0

Coffee cup.

Genus 1

Kettle. Genus 2 Scissors. Genus 2


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Activities 1:

Give the name and the genus of the following objects. If the object has a genus

larger than 5, write larger than 5.

ame:

Genus:ame:

Genus:ame:

Genus:ame:

Genus:ame:

Genus:

ame:Genus:

ame:Genus:

ame:Genus:

ame:Genus:

Activity 2: Jordan Curves

A Jordan Curveis a topological object that can be thought of as a circle twisted

out of shape. Like a circle, it has an inside and an outside. To get from one side

to the other, at least one line must be crossed. Consider the following Jordan

curve; are points A and B inside or outside the curve? Could you establish ageneral rule to know whether a point is inside or outside the Jordan curve?


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Activity 3: Mbius Strip

If you place a pencil on one surface of a sheet of paper and do not remove it

from the sheet, you must across the edge to get to the other surface. Thus, a

sheet of paper has one edge and two surfaces. The sheet retains these

properties even when crumpled into a ball. The Mbius strip, also called a

Mbius band, is a one-sided, one-edged surface. You can construct one by:

a) Taking a strip of paper

b) Giving one end a half twist

c) Taping the ends together

The Mbius strip has some very interesting properties. To better understand

these properties, perform the following experiments.


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Experiment 1: Take a sheet of paper, a strip of paper and construct a paper

ring as shown in the picture.

Could you tell how many edges and how many sides these different surfaces

have?

Surface Number of edges Number of sides

Sheet of paper

Strip of paper

Ring of paper

Hints:

How to count the edges: Start coloring an edge at one point with your felt-tip pen, if you color

the entire edge and never have to lift the pen from the paper then the paper has one edge. A

pointy vertex does not divide an edge into two parts.

How to count the sides: Start coloring one side, fill it with color but don't cross over any sharp

edges. When you are done, one side will be colored the other will not. So, the strip has 2 sides.

A simpler way to test for the number of sides is to draw a line along one side. If any point can

be reached from the line without crossing an edge then that point is on the same side as the

line. Draw a line on one side of the paper, points on the other side cannot be reached without

crossing an edge, this means the paper has two sides.


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Experiment 2: Make a Mbius strip using a strip of paper and tape as

illustrated above. Check a Mbius band is a one-sided, one-edged surface.

Experiment 3: Make a Mbius strip. Use scissors to make a small slit in the

middle of the strip. Starting at the slit, cut along the strip, keeping the scissors in

the middle of the strip. Continue cutting and observe what happens.

Experiment 4: Make a Mbius strip. Make a small slit at a point about one-third

of the width of the strip. Cut along the strip, keeping the scissors the same

distance from the edge. Continue cutting and observe what happens.

escher exhibition-escher and topology _4º eso_

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