Optical illusionsand undecidable

figuresA small project to

start with GeoGebra

by students of grade 8 of Pestalozzi Gymnasium Unna

Optical illusions and undecidable figures

Comment from the teacher: The following constructions were made from grade 8 who learn for some weeks how to work with Geogebra. I looked for smaller projects of 1-3 lessons (45 min. each lessons).

I choosed this project as one of the first steps to teach how to use Geogebra for dynamic constructions and make tools to facilitate work.

The project was to to choose an optical illusion and after that a undecidable figure (I had some cards and books) and construct this figures. In addition to the construction they should try to explain the illusion shortly (the translation was made by older students).

I made for the MMM project annotation on each slide of what I intended and what I pointed out when students presented her work to the others. Important for me was to to clarify what it means to construct dynamically and to show easy ways using reflection, rotation …

Optical illusion 1

What is the illusion?Straight lines seem to be crooked and not parallel (but they are constructed as parallels).

Easy constructions , using coordinate system as if

does not need to be dynamic, reflections

Optical illusion 2

What is the illusion?The lenght of the upper horizontal line seems to be longer than the llenght of the lower horizonal line. (This construction was made dynamically)

Learn, how to construct dynamically (horizontale

line was divided in 4 parts)

Optical illusion 3

What is the illusion?The sides of the square seem to be crooked (but they are straight lines). (This construction was made dynamically by adding the half of the first radius to the previous radius.)

How to get construct dynamically by using the the first radius, using the

radius to fix the eges of the square

Optical illusions 5

What is the illusion?The left line seems to be longer, but they have the same length.

Geometric construction using parallel lines and

circles for the same length (dynamic construction)

Optical illusions 6

What is the illusion?he shapes distract from seeing the right sizes. Each shape seems to be a little bit smaller than the left neighbour. If you look from the right rather parallel to the picture the illusion disappears.

Easy constructing to use dilations

Optical illusion 8

What is the illusion?he shapes distract from seeing the right sizes. Each shape seems to be a little bit smaller than the left neighbour. If you look from the right rather parallel to the picture the illusion disappears.

Easy to construct by using the coordinate system

Optical illusion 8

What is the illusion?The two parts of a ring of circles sseem to be different with different radius. But the two shapes are equal (congruent). (This construction was made dynamically.)

Learning how to construct arcs of a circle, and using

dilations

Optical illusion 9

What is the illusion?he shapes distract from seeing the right sizes. Each shape seems to be a little bit smaller than the left neighbour. If you look from the right rather parallel to the picture the illusion disappears.

Optical illusion 10

What is the illusion?The the right inner circle seems to have a smaller radius than the inner circle left. But both inner circles really have the same radius.

Optical illusion 10

What is the illusion?There are arrows which are oriented to the top. But the outlines of these arrows go to bottom.

Dynamic construction of one arrow then reflections

and dilations

Optical illusion 10

What is the illusion?The the right inner circle seems to have a smaller radius than the inner circle left. But both inner circles really have the same radius.

Optical illusion 11

What is the illusion?It seems that there are different circles. But they have the same radius. (Watch the circle line only between the two white lines).

Learn to analyse the geometry of this picture and work with arcs of a

circle

Optical illusion 10

What is the illusion?Sometimes you think to look into an open book, sometimes it is as you look at the back of an standing book. Both views switch.

A dynamic construction using 3 basic points and

a reflection.

Optical illusion 10

What is the illusion?If you imagine a horizontale line between the black forms you think it is not straight. The forms don‘t seem to be trapeziods but they are squares!

The student made a tool for the different

coloumns.

Undecidable figure 1

Whyis it impssible?

Easily to „construct“ by using the coordinate

system. More complex if it has be to constructed

dynamically.

Undecidable figure 2

What is the illusion?The shapes distract from seeing the right sizes. Each shape seems to be a little bit smaller than the left neighbour. If you look from the right rather parallel to the picture the illusion disappears.

Easily to „construct“ by using the coordinate

system. More complex if it has be to constructed

dynamically.

Undeciderable figure 3

What is the illusion?It seems to be 3 dimensional but at the edge left there is confusion because it does not figure for a 3D figure.

Easily to „construct“ by using the coordinate

system.

Undecidable figure 4

Why is it indecidable? This 3d figure cannot exist. If you built it with wood the parts are not in the same plane. From special points of view it looks this this picture.

Undecidarable figure 5

What is the illusion?It seems to be a frame but when you watch the last edge there is confusion.

The aim was to to construct it dyna-mially using parallels and circles of the the same radius.

Construction needs a lot of concentration!

Undecidable figure 6

What is the illusion?These stairs don‘t exist in reality. The picture causes confusion in our mind.

Students constructed it using points of the coordinate

system. One tries it dynamically but made a

mistake art the beginning,-)

We hope you enjoy this presentation!