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.
A small clipping at the edge of the triangle that makes the
echnion's version.
The impossible torus is, in fact, a very possible object. One can easily
construct it in any solid modeler by sweeping a square shape along a circle
trajectory, while the square is rotated. The four instances of the impossible
torus differ from each other only by the amount the square cross section is
rotated while swept along the 360 degrees of the circle, starting from 90
degrees rotation on the left, all the way to 360 degree in the right
example.
The hex nut is another quite known example of impossible shapes. Here again,
one can construct a real 3D object that from a certain view will look like
the impossible hexnut.
Here is another, continuous way, of making a 3D object that looks
like the infamous impossible hexnut for a certain view location.
The original "impossible" art work of Sandro Del-Prete called the
"Garden Fence" is realized here as a 3D model that is similar to the
original drawing from one view point.
ZCorp (Can you recognize the
logo?) has the ability to build colored models nowadays, so here we
take advantage of this feature.
Another original "impossible" art work of Sandro Del-Prete called the
"Folded Chess Set" is realized here as a 3D model that is similar to the
original drawing from one view point. This is a simple one as it is merely
a folded chess board...
An "impossible" cylinder art work by Istvan Orosz (A poster design) is
realized here. This cylinder is, in fact, a double-twist Moebius ring.
A simple variation on an "impossible" art work by Jos de Mey ("Curious
Wagtail in a Strange Small Window") is realized here. This is one
window out of quite a few renditions out there.
Try this animation, to see this window
from different angles.
Combining the above two drawings, we have these impossible windows
inside the impossible cylinder.
Try this animation, to see this cylinder
with windows from different angles.
The fact that in three-space we have three orthogonal views for a model,
allows one to design three-dimensional objects that look completely
different from different direction and are completely independent.
Herein, we mold together the david-star and the Menorah emblem of Israel.
(C) Copyright, Gershon Elber 2003-12.
Here is the real, tangible, piece.
Here is another variant with the Menora on one view and the Technion's
logo on the other. (C) Copyright, Gershon Elber 2004-12.
Here is the real, tangible, (colored!) piece.
Victor vasarely has a (not so known) drawing of a horse in which
parallel lines are shifted in the planar drawing to give the illusion of
a three-dimensional object, or depth.
Here we exploit a similar idea to construct bars that are twisted
independently in X and Y so that two independent pictures could be
seen from the Y and X directions. The logo of the Technion (right)
and of its CS department (left) are etched into these two directions.
(C) Copyright, Gershon Elber 2003-12.
Here is the real, tangible, piece.
Yaacov agam introduced the idea of presenting two different images via
a zigzag structure (also known as lenticular printing in the micro
level). Herein we extend this idea to three different images from
three different views. Shown: a general view (top left), a view of
Ben Gurion (top middle), a view of Herzl (bottom left), and a view of
Rabin (bottom right).
The top right animation shows the model as it is continuously rotated.
Made using ZCorp technology.
(C) Copyright, Gershon Elber 2003-12.