In this computationally inclined space we inhabit it’s quite easy to avoid any tangible encounter with an exotic mathematical surface – the nearest we get is probably in architecture. We are treated to fine models of ‘hard to imagine topologies’ on the computer screen but really never get the chance to ‘feel’ them in real space. The 19th centuary, however, saw a great interest in geometry in maths and from the mid 1880’s the great age of mathematical model making began (but more on this a little later down the line).
The wonderful Institute for Figuring brings to us the Business Card Menger Sponge exhibit, where we see a three level Menger Sponge brought into existence by Dr Jeannine Mosely. According to the Institute this incredible object took 9 years of effort involving hundreds of people spread across America and….66.048 buisness cards!
Most will be familiar with the form and algorithm – it is a 3d extension of Sierpinksi’s Carpet which itself is a generalisation of the Cantor Set. It has, of course, a self-similar geometry, it is a recursive fractal object.
From wikipedia, the Menger Sponge (3level) algorithm is a follows:
1.Begin with a cube. Divide every face of the cube into 9 squares. This will sub-divide the cube into 27 smaller cubes, like a Rubik’s Cube
2.Remove the cube at the middle of every face, and remove the cube in the centre. This is a Level 1 Menger sponge.
3.Repeat steps 1-3 for each of the remaining smaller cubes.
Conceptually we could continue to pass level 3 and keep removing cubes from successively more minute cubes – the resulting enigma is eloquently explained by the Institute:
‘Taken to its infinite conclusion the Sierpinski Carpet (And Menger Sponge) dissolves into a foam whose final structure has no area whatever yet possesses a perimeter that is infinitely long. Like the skeleton of a beast whose flesh has vanished, the concluding form is without substance – it occupies a planar surface, but no longer fills it.
And radically that:
‘Any geometry of loop quantum gravity can be embedded in a Menger Sponge’ and that ‘the structure of space-time may be allied with this foam-like form.’ Great!
As well as giving precise instructions for making your own Menger Sponge Dr Mosely explains the procedure for making the 3 Level Model using a divide and conquer strategy of sub assembly using component units called tripods!
Jeannine Mosely is a driving force behind a new movement of computational origami – you can view her personal documentation of this fascinating project here!
Margaret Wertheim’s Flickr set of the exhibit.
Mark Allen’s Sponge set of the IFF exhibit.
Murky’s drawings of the Menger Sponge, some Fractals and other Maths surfaces.
Friends of Folding’s business card origami, including an inverse sponge, no less!