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Escher for Real

December 11, 2009

via on 12/10/09

Escher for Real

(C) Copyright 2002-9 Gershon Elber, Computer Science Department, Technion
The work of M.C. Escher needs no introduction. We have all learned toappreciate the impossibilities that this master of illusion’s artworkpresents to the layman’s eye. Nevertheless, it may come as a surprisefor some, but many of the so-called ‘impossible’ drawings ofM. C. Escher can be realized as actual physical objects. Theseobjects will resemble the Escher’s drawing, of the same name, from acertain viewing direction. This work below presents some of thesethree-dimensional models that were designed and built using geometricmodeling and computer graphics tools.


In the following sequences, figures are frequentlypresented in pairs. The left figure in each pair is the frontview-Escher's drawing's direction, whereasthe right figure gives a general view. Whenever a real,tangible model has been created, it will be presented as a secondpair to the right of the pair of computer rendered images. The objectswere physically realized with the aid of layered manufacturingsystems: a Z402 3D Printer from Zcorporation and a Stratasys FDM3000 printingmachine. Click on any image below to get the full size version ofthese images. In addition, some models are also accompanied by AVImovies that present them from a multitude of directions. Many ofthese movies are using the DivX CODEC which you can get freeof charge for private use from

The Penrose Triangle:

We start with the Penrose triangle object (also independently invented byOscar Reutersvard). There are several ways tobuild a real geometry that will look like the Penrose triangle from acertain viewing direction. This specific shape is reconstructed as aC^0 continuous sweep surface with a square cross section that rotatesas we move along the edges. As will be shown below, the Penrosetriangle plays a majosr role in M.C. Escher’s drawings. An STL geometry file, for those of youwith layered manufacturing devices, of this model, is available here .

The Penrose Rectangle:

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.

The Penrose Pentagon:

Similarly and following the construction of a Penrose triangle andPenrose rectangle, one can easily create an arbitrary Penrose n-gon.Here, we will stop at a Penrose pentagon.

The Penrose Triangle II:

  Here we present another way to simulate and realize geometrythat looks like the Penrose triangle from a certain view. Here is anavi movie that shows this model rotated.An STL geometry file, for those of youwith layered manufacturing devices, of this model, is available here .

The Penrose Triangle III:

And here is a wooden variation. See also my woodworking page.

Escher’s (Louis Albert Necker’s) Cube:

Here is our realizable variant of Escher’s/Necker’s Cube. If you will lookcarefully enough, you will find this cube in Escher’s originalBelvedere drawing. And here is an avi moviethat shows this model rotated.

This is a result of my woodworking hobby, making a version of thiscube from wood. The plans I used to make this wood cube could befound here. The image on the rightis from my office – you are welcome to drop by and see this!

Escher’s Moebius Strip – Ducks:

Here is a variant of the Moebius Strip formed out of Escher’s duck tiling.

Escher’s Moebius Strip – Ants:

Here is a variant of the Moebius Strip formed out of Escher’s antswalking forever and covering both sides of the ring. Click on thissmall movie on the right to see the real large version.
Here are also two raytraced versions (these are periodic movies, so repeatedplay will create a continuous animation):

Escher’s Waterfall:

Here is our physical realization of Escher’s Waterfall. Youcan also watch this object rotating in space in this avi movie. This realization stemsfrom using three joint Penrose triangles along the water stream.Note also the way the house is warped so as to look natural from thisviewing direction (only).

To better understand this, examine the pictures on the left. Therightmost image shows the Penrose triangle only, from above; the leftmostimage shows the original Waterfall scene; and the middle image is a blend ofthe 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.

Escher’s Belvedere:

Here is our physical realization of Escher’s Belvedere drawing. Again, thismodel looks like the original Escher drawing from one direction only,whereas the (not so) vertical poles stretch from the far top to thenear bottom sides and vice-versa. This trick is somewhat similar tothe trick we used in the Penrose triangle but is somewhat simpler.You can also watch this object rotating in space in this avi movie.

Escher’s Relativity:

Here is our physical realization of Escher’s Relativity drawing thatwas modeled with the aid of Oded Fuhrmann, Technion. This model isdifferent compared to many of the above in the sense that it is aregular model (with stair cases all over the place :-). We need nospecial view direction, for this model to look like the originalEscher drawing.

See also Beyond Escher for Real

I would like to acknowledge Ronit Schneor, ME, Technion and Sam Drake,CS/ME, University of Utah for their help in the making of theseobjects. I am also grateful to Zvika Grinberg, Caliber, ZCorp Israeli rep.,for his aid in making the colored models.

This page is (C) Copyrighted to Gershon Elber, 2002-9. Permission tocopy in parts or as a whole the content of this page must be given by Gershon Elber prior toany such use.

All M.C. Escher works (C) the M.C. Escher Company B.V. – Baarn – the Netherlands.Used by permission. All rights reserved.

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