Escher’s “Relativity” etching dates back to 1953, and has fascinated me for many years. Unlike many of his pictures, the image represents a scene that is completely valid structurally - it is not an ‘impossible shape’, although the perspective is quite contorted.
Looking on the internet finds several different attempts to model this in three dimensions. Since the image can be seen from three different points of view, the steps have equal height and depth, so that the whole thing can be seen as a three-dimensional grid of cubes. It is convenient to make the model with a unit-cell equal to one step.
Here is Escher’s original print for comparison:
There are some tricky aspects of the model that have to be worked out from the graphic. The bottom two-thirds of the picture can be worked out by counting steps up and down, and so can the top third, but the two sections are not connected. There is a band a third of the way down that has to be scaled in some way, and some models on the internet don’t appear to have allowed enough space for this.
The wall with the open door appears to be co-planar with the wall below (with the stairs to the basement), but actually it is not. In other terms, the stair with the man with a bottle on a tray is narrower than the stairs below it, by two cell-widths.
The ‘bridge of sighs’ has its railings coming out from the sides. This is needed, because the bridge is only three cells wide, but also because some of the banisters continue through to form the railings on the balcony beyond. Escher must have chuckled to himself as he added this detail.
Anyway, here is my model for inspection, and I would appreciate any comments. I have not attempted to create the people, nor the plants and tables, but I did do all the railings, which are key to the design. If anyone has a good parameterised posable human model that would allow for creating people going up and down stairs and hanging onto railings, I would like to see it.
Escher5.2.3dm (16.6 MB)
Escher5.2int.gh (68.3 KB)
I actually did all the object creation in a series of spreadsheets, and imported them using a Lunchbox component, but to avoid the need for this plugin the above definition has internalized all the data. I used the Human plugin for display positioning purposes, but I think the definition should work just fine even if you don’t have it.
The .STL file appears to be too big to upload. For 3D printing purposes, the only change I needed to make was to fatten up the railings, since the fragility of these is beyond my printer. There are sliders for this, and also to scale it up to 4mm per step, the limits of my printer.
Finally, for completeness, here is a picture from 1960 taken by my father in Cambridge at a conference on crystallography. Escher was an amazing man! He certainly liked railings!
Bob
