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. Many of
these movies are using the DivX CODEC which you can get free
of charge for private use from http://www.divx.com/divx/download.
We start with the Penrose triangle object (also independently invented by
Oscar Reutersvard). 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. An STL geometry file, for those of you
with layered manufacturing devices, of this model, is available
here .
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.
An STL geometry file, for those of you
with layered manufacturing devices, of this model, is available
here .
And here is a wooden variation. See also my wood
working page.
Here is our realizable variant of Escher's/Necker's Cube. If you will look
carefully enough, you will find this cube in Escher's original
Belvedere drawing. And here is an avi movie
that shows this model rotated.
This is a result of my woodworking hobby, making a version of this
cube from wood. The plans I used to make this wood cube could be
found here. The images on the right
are from my office and our CS lobby - you are welcome to drop by and see this!
Here is a variant of the Moebius Strip formed out of Escher's duck tiling.
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.
You can also watch this object rotating in space in this avi movie.
Here is our physical realization of Escher's Relativity drawing that
was modeled with the aid of Oded Fuhrmann, Technion. This model is
different compared to many of the above in the sense that it is a
regular model (with stair cases all over the place :-). We need no
special view direction, for this model to look like the original
Escher drawing. The STL file of this model is
here .