(Dali, Searching for the 4th dimension, 1979)
This project was done in collaboration with artist-educator, Jane Elliott.
Photographs of the workshop held at LKL on 16 January 2008:
Information about Zometool: www.zometool.com
There is a piece of software called 'virtual Zome' which allows you to make and manipulate Zome objects on the computer: vZome.
If you click on the following link the vZome program will install and run automatically: www.vorthmann.org/zome/jnlp/v21latest/vZome-v21latest.jnlp.
There is a good book of mathematical learning activities with Zometool: Zome Geometry by Hart & Picciotto [website].
Information about LKL's Zome model: Visitor from the 4th Dimension.
Photos of the model in construction (August 2006): http://homepages.wmich.edu/%7Edrichter/bridgeszome2006.htm.
Photographs in high resolution of the model (taken by Gary Woodley, UCL Slade School of Art):
(These photos are Copyright; to be used only with explicit permission.)
We talked about how it is possible to build mathematically a 4th dimension of space. This is different from the four-dimensional spacetime of Einstein's theory of relativity!
We talked about the 3 ways that it is possible to "see" shapes in the 4th dimension inside our own 3-dimensional space:
Slicing
Projection (shadows)
Unfolding
The best description of these are in the book by Banchoff (see below).
Geometrical shapes in 4D space are called polytopes. In the session at Tate Modern, we looked at models of the five regular polyhedra in 3D space - tetrahedron, cube, octahedron, dodecahedron, and icosahedron. 'Regular' means that all the sides are the same regular polygon shape, and the same number of polygons meets at every vertex. There are many more 'semi-regular' polyhedra, and infinitely-many 'unregular' ones. In 4D space, there are six regular polytopes (simplex, hypercube, 16-cell, 24-cell, 120-cell, 600-cell). LKL's model is based on the 600-cell, but it is more complicated because it has been 'truncated' (the corners are cut off).
Wikipedia has pages on polyhedra and polytopes, but the language is very technical. It is probably more interesting to explore visually - for example with Google Images - search polytope, search polyhedron. Also recommended: George Hart's website which has an Encylopeadia of Polyhedra; Hart is co-author of a book on Zome Geometry, and is also an active sculptor of beautiful geometrically-inspired works (see gallery).
I demonstrated two Java applets. This one does slicing: http://dogfeathers.com/java/hyperstar.html
And this one does projection (shadows): http://darkwing.uoregon.edu/~koch/java/FourD.html
Both work with the 3D 'anaglyph' glasses (blue lens goes on the right eye)
Here are some pictures of the unfolded 'regular polytopes': www.weimholt.com/andrew/polytope.shtml
Software (free demo) for unfolding 3D polyhedra and 4D polytopes: www.software3d.com/Stella.php
Software (free) for unfolding polyhedra: JavaGami
As we discussed, the 4th dimension can become an obsession. Here is a letter written by an man in his 60s, remembering 40 years before using special visualisation cubes developed by the mystical mathematician, Charles Hinton (letter published in Martin Gardner's book Mathematical Carnival):
Dear Mr. Gardner: A shudder ran down my spine when I read your reference to Hinton's cubes. I nearly got hooked on them myself in the nineteen-twenties. Please believe me when I say that they are completely mind-destroying. The only person I ever met who had worked with them seriously was Francis Sedlak, a Czech neo-Hegelian Philosopher (he wrote a book called The Creation of Heaven and Earth) who lived in an Oneida-like community near Stroud, in Gloucestershire. As you must know, the technique consists essentially in the sequential visualizing of the adjoint internal faces of the poly-colored unit cubes making up the larger cube. It is not difficult to acquire considerable facility in this, but the process is one of autohypnosis and, after a while, the sequences begin to parade themselves through one's mind of their own accord. This is pleasurable, in a way, and it was not until I went to see Sedlak in 1929 that I realized the dangers of setting up an autonomous process in one's own brain. For the record, the way out is to establish consciously a countersystem differing from the first in that the core cube shows different colored faces, but withdrawal is slow and I wouldn't recommend anyone to play around with the cubes at all.
Salvador Dali famously used the hypercube unfolding in the 1955 painting, Crucifixion: Corpus Hypercubus. In his final years, he took a great interest in 4D geometry - and one of the paintings is shown at the top of this page.
The American artist Tony Robbin seems to be one of few artists who has worked in a mathematical way with 4D geometry to create artworks; he has also written several books about it.
It is claimed (for example by Tony Robbin) that much of the inspiration for cubism as developed by Picasso and Braque, was based on 4D geometry. This is arguable, but if you look at the drawings of polytopes (see around pages 152-154) by the mathematician, Jouffret, which were circulating among artists in the early 1900s, the visual resemblance is intriguing.
The mathematician Thomas Banchoff was a pioneer of visualising the 4th dimension using computer graphics, and has produced some interesting mathematical artwork.
There are many "science fiction" stories about the fourth dimension, often inspired by Abbott's Flatland (see the Literature section in Wikipedia on Flatland). The short story "And He Built a Crooked House" by Robert Heinlein is entertaining, very short and available online.
We watched the movie version of Flatland, released in 2007. More information, and the DVD can be bought, on the movie website: www.flatlandthemovie.com
The original Flatland story was written by an English schoolmaster, Edwin Abbott, in 1884. It is not long, and well worth reading. There are versions to read online (e-text), and it can be bought as a book published by Dover Publications, Oxford University Press, and others. There is an annotated edition with explanatory notes about mathematics, culture and history.
Edwin A. Abbott, Flatland: A romance of many dimensions, 1884. Available online as e-text.
Edwin A. Abbott and Ian Stewart, The Annotated Flatland: A romance of many dimensions, Perseus Books, 2001.
Thomas F. Banchoff, Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, W. H. Freeman, 1990. [Chapters 1 and 2 online]
H. M. Cundy & R. M. Rollett, Mathematical Models, 1961, reprint by Tarquin Publications.
Michele Emmer (editor), The Visual Mind II, MIT Press, 2005.
George Hart and Henri Picciotto, Zome Geometry: Hands-on Learning with Zome Models, Key Curriculum Press, 2001. [Author's website]
Robert Heinlein, And He Built a Crooked House, 1940. Available as e-text.
Michio Kaku, Hyperspace:A Scientific Odyssey Through Parallel Universes, Time Warps and the Tenth Dimension, Oxford University Press, 1994.
Tony Robbin, Fourfield: Computers, Art, and the Fourth Dimension, 1992.
Tony Robbin, Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought, Yale University Press, 2006.
Rudy Rucker, The Fourth Dimension: Toward a Geometry of Higher Reality, Houghton-Mifflin, 1984.