Grade 8 students in Germany constructed optical illusions and undecidable figures using the GeoGebra software. They explained 13 optical illusions involving lines, shapes, circles and perspective that appear distorted but are constructed accurately. They also explained 6 undecidable figures that involve impossible geometric configurations or objects that could not exist in reality despite appearing coherent. The teacher's comments indicate this was one of the students' first projects using GeoGebra to create dynamic geometric constructions.

1. Optical illusions
and undecidable figures
A small project to start
with GeoGebra
by students of grade 8
of Pestalozzi Gymnasium Unna

2. 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 …

3. 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

4. Optical illusion 2
Learn, how to construct
dynamically (horizontale line
was divided in 4 parts)
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)

5. Optical illusion 3
How to get construct
dynamically by using the the
first radius, using the radius to
fix the eges of the square
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.)

6. Optical illusions 4
Geometric construction using
parallel lines and circles for the
What is the illusion?
The left line seems to be longer, but they have the same length.
same length (dynamic
construction)

7. Optical illusions 5
Easy constructing to use
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.
dilations

8. Optical illusion 6
Easy to construct by using the
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.
coordinate system

9. Optical illusion 7
Learning how to construct arcs
of a circle, and using dilations
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.)

10. 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.

11. Optical illusion 9
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.

12. Optical illusion 10
Dynamic construction of one
arrow then reflections and
What is the illusion?
There are arrows which are oriented to the top. But the outlines of
these arrows go to bottom.
dilations

13. Optical illusion 11
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.

14. Optical illusion 11
Learn to analyse the geometry
of this picture and work with
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).
arcs of a circle

15. Optical illusion 12
A dynamic construction
using 3 basic points and a
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.
reflection.

16. Optical illusion 13
The student made a tool for
the different coloumns.
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!

17. Undecidable figure 1
Whyis it impssible?
Easily to „construct“ by using
the coordinate system. More
complex if it has be to
constructed dynamically.

18. Undecidable figure 2
Easily to „construct“ by using
the coordinate system. More
complex if it has be to
constructed dynamically.
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.

19. Undeciderable figure 3
Easily to „construct“ by using
the coordinate system.
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.

20. 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.

21. Undecidarable figure 5
The aim was to to construct it dyna-mially
using parallels and circles of
the the same radius. Construction
needs a lot of concentration!
What is the illusion?
It seems to be a frame but when you watch the last edge there is
confusion.

22. Undecidable figure 6
Students constructed it using
points of the coordinate system.
One tries it dynamically but made
a mistake art the beginning,-)
What is the illusion?
These stairs don‘t exist in reality. The picture causes confusion in
our mind.

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