This document summarizes research on impossible constructions in mathematics and architecture. It discusses M.C. Escher's work with impossible constructions and defines different types of impossible shapes like impossible cubes, Penrose stairs, the Penrose triangle, and the impossible trident. The document then describes an experiment where the researchers attempted to draw Penrose triangles from memory without references. Most of the triangles drawn were successful, showing that these impossible shapes can be reproduced without copying. The conclusion discusses the collaboration between the researchers and challenges of meeting deadlines during the holiday season.

1. Mathematics in
architecture
J.J.A.C. EMANS, Y.F.W. LAMBERS,
A.E.S. VAN DE POLL, G.K.W. ZUIJDGEEST

2. Table of contents
Introduction
Escher
Impossible constructions
• Impossible cubes
• Penrose stairs
• Penrose triangle
• Impossible trident
Experiment
Discussion

3. Introduction
We decided to do a research on impossible constructions.
These constructions can be called impossible for several reasons; the gravity can be adjusted
wrong in the drawings or the walls/stairs can be constructed in a way that looks real, but is
nothing less than impossible.

4. Escher (1898-1972)
Escher was a Dutch woodworker who was famous for his mathematically inspired woodcuts and
impossible constructions.
“Only those who try to reach for the absurd,
will achieve the impossible.”

5. Impossible constructions
Impossible constructions are constructions that are against the law of nature and logics.
It is an optical illusion and always 2-dimentional, it is possible to form a sort of 3-D illusion, but
in real, impossible shapes can’t be recreated into 3-dimentional shapes.
The most famous shapes are:
Impossible cubes
Penrose stairs
Penrose triangle
Impossible trident

6. Impossible cube

7. Penrose stairs

8. Penrose triangle

9. Impossible trident

10. Experiment
Can we draw an impossible shape without the help of any sort?
We expect ourselves to be able to draw a Penrose triangle, since these figures are easier to
understand and draw than the other shapes.

11. Results

12. Conclussion
As you can see, we aren’t really good in drawing triangles, but we tried and except for 1, all the
triangles are succeeded, which means that people can draw penrose triangles without
something to copy it from.

13. Discussion
We think that the cooperation between the four of us went pretty good; we didn’t have a lot of
disagreements and we had a lot of fun doing this project. We chose a little bit of a hard subject,
but we figured it out anyway.
Making the deadline was a little harder; in the middle of the Christmas holiday trying to
communicate with each other isn’t or idea of a successful team.