The work of M.C. Escher needs no introduction. We have all learned to
appreciate the impossibilities that this master of illusion's artwork presents to the
layman's eye. Nevertheless, it may come as a surprise for some, but
many of the so-called 'impossible' drawings of M. C. Escher can be
realized as actual physical objects. These objects will resemble the
Escher's drawing, of the same name, from a certain viewing direction. This work
below presents
some of these three-dimensional models that were designed and built using geometric
modeling and computer graphics tools.
In the following sequences, figures are frequently
presented in pairs. The left figure in each pair is the front view-Escher's
drawing's direction, whereas the right figure gives a general
view. Whenever a real, tangible model has been created, it will be presented as
a second pair to the right of the pair of computer rendered
images. The objects were physically realized with the aid of layered
manufacturing systems: a Z402 3D Printer from Zcorporation and a Stratasys FDM3000 printing
machine. Click on any image below to get the full size version
of these images. In addition, some models are also accompanied by AVI movies
that present them from a multitude of directions.
We
start with the L. S. Penrose triangle object. There are several ways to
build a real geometry that will look like the Penrose triangle from a
certain viewing direction. This specific shape is reconstructed as a
C^0 continuous sweep surface with a square cross section that rotates
as we move along the edges. As will be shown below, the Penrose
triangle plays a majosr role in M.C. Escher's drawings
The impossible shape conveyed by the Penrose triangle is the most well-known
one. However, one can, with similar ease and with the aid of a geometric
modeling system, construct more complex natural extensions to the Penrose
triangle. Herein a Penrose rectangle is presented.
Similarly and following the construction of a Penrose triangle and
Penrose rectangle, one can easily create an arbitrary Penrose n-gon.
Here, we will stop at a Penrose pentagon.
Here we present another way to simulate and realize geometry
that looks like the Penrose triangle from a certain view. Here is an
avi movie that shows this model rotated.
Here is our realizable variant of Escher's Cube. If you will look
carefully enough, you will find this cube in Escher's original Belvedere
drawing. Here is an avi movie that shows this model
rotated.
Here is our physical realization of Escher's Waterfall. You
can also watch this object rotating in space in this avi movie. This realization stems
from using three joint Penrose triangles along the water stream.
Note also the way the house is warped so as to look natural from this
viewing direction (only).
To better understand this, examine the pictures on the left. The
rightmost image shows the Penrose triangle only, from above; the leftmost
image shows the original Waterfall scene; and the middle image is a blend of
the two. In fact, the original Waterfall model presents three different and connected such
triangles. Also interesting in this image is the house. When we examine the
original Waterfall drawing, we see that the house ends up behind the middle corner of
the water path. In order to accomplish this in our constructed image, the house's geometry is warped
so as to look straight only from the proper viewing direction. The house, as
well as the S shaped rods that look vertical from the original viewing direction,
were modeled as generalized sweeps by the geometric modeler.
Here is our physical realization of Escher's Belvedere drawing. Again, this
model looks like the original Escher drawing from one direction only,
whereas the (not so) vertical poles stretch from the far top to the
near bottom sides and vice-versa. This trick is somewhat similar to
the trick we used in the Penrose triangle but is somewhat simpler.
This specific model was realized using a
Stratasys FDM3000 machine.
You can also watch this object rotating in space in this avi movie.